Boveri-Sutton Chromosome Theory

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Kind: captions
Language: en

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- [Voiceover] Let's give
ourselves a reminder

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of how important Gregor Mendel's work was

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in 1866, that he published in 1866.

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And it's important to
realize it wasn't like

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immediately in 1866 or 1867
the whole world changed,

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and everyone said, "Oh, Gregor
Mendel figured it all out."

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Like a lot of times in
science the big discoveries,

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the ones that really
change people's thinkings,

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aren't really taken
that seriously at first.

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And actually, Mendel's work,

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a lot of people either
didn't take it seriously

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or kind of ignored it in 1866.

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And it wasn't until the
early 1900's that people

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rediscovered his work and realized,

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"Wait. Wait. There is something
very, very powerful here,

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"and we might be able
to connect it to things

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"that we are actually
observing inside of cells."

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But let's just remind
ourselves about Mendel's work.

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So, for most of human history,

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we've recognized probably that, okay.

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It looks like animals,
or not just animals,

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any type of living creature

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seems to pass on traits
to their offspring.

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I could look at you and I'd say,

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"Oh, you know, your hair
is kind of like your dad's,

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"and your eyes are kind
of like your mom's."

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Maybe your nose looks
like something in between.

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You walk a little bit like your uncle.

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So, we've always recognized
that we pass on traits

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to our offspring,

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but we didn't really
have a rigorous way of

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thinking about it.

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And we definitely didn't have
a way to make predictions

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that were testable based on those traits,

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and that's what Mendel gave us.

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He said, "Well, look. I'm observing,"

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and he did this with pea plants,

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"I observed these heritable factors,

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"and there might be heritable factors on,

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"let's say, height."

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If we're talking about plant,

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it would be the height of a plant.

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There might be heritable
factors on, let's say,

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flower color.

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So, flower color.

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And he recognized that there
were different versions

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of those factors,

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and, so, a given plant might have

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one of the tall versions.

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So, they might have a tall
version for the height factor,

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and they might have a short version.

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Or they might have two talls,

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or they might have two shorts.

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Or they might have a red factor,

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and they have a pink factor.

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Or they could have two reds, two reds.

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It would look like that.

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Or two pinks would look like that,

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but the important realization was that

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there were these versions of the factors.

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Today, we call these factors, we say,

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"Hey, there's a gene for height."

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If there is one.

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Or that there is a gene for flower colors,

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and those variations of the genes, today,

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we call these alleles.

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So, we'd say, "Hey,
you have the variation.

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"You have one copy of the tall allele

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"and one copy of the short."

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Let me just write it this way.

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Let me just say these are
all alleles right here.

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So, you have one tall
allele, one short allele.

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And what Mendel did is he realized,

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"Well, look. These things are,"

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he didn't know how,

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"but these things are the
things that get passed on

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"from a parent to their offspring."

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And he started to describe
about how they got passed on.

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He observed that, even
if you have two of these,

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that they tend to segregate when you go

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to the next generation,

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and what do we mean by segregation,

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or I guess we can say
the law of segregation?

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Law of segregation.

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Well, that means if this
was, if I'm a pea plant

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and these are the version that I have,

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to my offspring I might pass on

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an A, a capital A, the tall one,

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or I could pass on the lowercase A.

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I might pass on the tall,

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or I might pass on the red version

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of the flower color factor,

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or I might pass on

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the pink one.

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And he also realized that whether or not

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I pass on the capital
A or the lowercase A,

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it's independent of whether I pass on

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the capital B or the lowercase B.

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So, they independently assort.

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How this one assorts is independent of

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how this one assorts.

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So, independent assortment.

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Independent.

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Independent Assortment.

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Assortment.

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Law.

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Law of independent assortment.

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And he also observed that

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some of these versions
dominate the other one.

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So, if an offspring has a tall version

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and a short one, if the
tall one is dominant,

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the observed trait will still look tall,

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and the only way they'd look short is

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if they have two versions
of the short one.

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And, so, that one he described as his

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law of dominance.

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Law of dominance.

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And if all of this is
completely new to you,

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I encourage you to watch the videos on

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Mendelian genetics on Khan Academy,

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but this is just gonna appreciate

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a little bit of a historical appreciation.

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But as big of a deal as Mendel's work was,

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it's also important to
realize what he didn't know.

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He had no idea of how this
was actually happening

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at a molecular level
or at a cellular level,

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and it wasn't until the early 1900's

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that people started to have

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fairly robust theories
of how this happens.

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And, so, in 1902 and 1903.

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So, 1902-1903 these two gentlemen

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independently start coming up with the

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chromosome theory of inheritance,

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and it's called the Boveri-Sutton

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Chromosome Theory of Inheritance,

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because right around the same time,

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they both started to realize

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that maybe chromosomes were the actual

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molecular mechanism,
the cellular mechanism,

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by which these factors segregate
and independently assort.

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And, so, let me write this down.

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This is the Boveri-Sutton,
it's sometimes called the

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Sutton-Boveri.

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Boveri-Sutton Chromosome Theory.

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Chromosome Theory.

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And even though they're starting to say,

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"Maybe chromosomes have
something to do with it."

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They still don't know
exactly what is inside

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the chromosome that are allowing somehow

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this information to be encoded,

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and we'll get to that.

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And we will get to that in a little bit,

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but let me just underline this.

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Boveri-Sutton Chromosome Theory.

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And, so, what was their key insight?

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Well, they started to
look inside of cells,

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meiosis was observed actually after

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Mendel published his laws of inheritance,

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and then chromosomes,

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or how chromosomes behave in meiosis,

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was discovered after that.

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And then these guys, they
independently studied

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different organisms.

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Walter Sutton, he studied grasshoppers.

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Theodor Boveri, he studied sea urchins,

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but they looked at meiosis,

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and they looked at the reproduction

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and the fertilization
during these processes.

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And they saw that the chromosomes

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seemed to do things that were very similar

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to these laws of segregation,

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laws of independent assortment,

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laws of dominance.

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And actually, the law of dominance

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we'll talk more about in future videos.

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But he saw that, let's say
that you had a organism here,

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and in this particular organism,

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I just did it for simplification,

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it has two pairs of
homologous chromosomes.

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So, what does homologous
chromosomes means?

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Well, these two are different chromosomes,

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but they seem to be very similar.

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It seems like they're
kind of the same length,

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same size, same shape.

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So, that's one pair of
homologous chromosomes.

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That's another pair of
homologous chromosomes.

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So, notice. Homologous chromosomes,

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two things that are kind
of looking the same,

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but maybe they're a little bit different.

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We're not sure.

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Well, maybe, this is
what fits what's going on

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right over here with these factors.

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Maybe, just maybe, one
of these chromosomes

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somehow has on it some place

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what encodes for the capital A.

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Capital A.

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And maybe, the other
chromosome in a similar part

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of the chromosome.

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In a similar part of the chromosome,

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has what encodes a similar
part of a chromosome,

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has what encodes for a lower case A.

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Now, this is starting to make sense

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because they would be
homologous chromosomes

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similar the chromosomes
look like they code

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for the same thing,

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for the same factors, for the same genes,

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but there might be some variation

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between these chromosomes.

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And these guys weren't
able to somehow sequence

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the chromosomes.

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So, they didn't know.

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They didn't even know that
the DNA was what was important

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in the chromosomes,

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but they said, "Well, looks
like these two things,

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"if we look through
the process of meiosis,

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"it seems like they
segregate from each other."

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For example, this capital
A one, it will replicate.

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So, you have capital A, and
then this is the lowercase A one

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right over here.

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You might have some cross over,

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and we'll talk about that when

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you can review meiosis,

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if it looks unfamiliar.

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But then they segregate.

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You could have you capital
A ones right over here,

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and then these sister
chromatids split apart.

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So, capital A capital A,

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and then you have lowercase A.

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These lower case A ones, they segregate.

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And they independently assort

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from the other chromosomes.

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So, this one right over
here might be the capital B,

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this might be the lowercase B.

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And whether or not this gets a,

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whether or not this gets a
capital B or a lowercase B,

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is independent of whether
it got a capital A

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or a lowercase A.

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So, it seems like these chromosomes

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independently assort.

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And, so, they came up with
this chromosomal theory

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that it looks like maybe
chromosomes are what

00:09:49.797 --> 00:09:52.457
contain these heritable factors that

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Mendel was talking about,

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because it seems like chromosomes behave

00:09:55.545 --> 00:09:57.820
very similar to those heritable factors.

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Maybe chromosomes code for multiple

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of these heritable factors that segregate

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and independently assort.

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And as we know now, they were right.

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So, this is a very, very big deal,

00:10:08.502 --> 00:10:10.638
but it's important to realize
that they weren't sure.

00:10:10.638 --> 00:10:11.962
They established a theory,

00:10:11.962 --> 00:10:13.564
they were able to make some observations

00:10:13.564 --> 00:10:15.480
with the grasshoppers and the sea urchins,

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and they saw the patterns between

00:10:17.301 --> 00:10:18.684
what Mendel was describing,

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and the way chromosomes
behave during meiosis.

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And then they know that each
of these products of meiosis,

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each of these gametes will then go

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and form with other gametes to form

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the next organisms who say,

00:10:31.048 --> 00:10:32.767
"Oh, look. Parents will
contribute either capital A

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"or lowercase A. Either a
capital B or a lowercase B."

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So, this is very similar to
what Mendel was describing.

00:10:38.792 --> 00:10:41.090
So, they laid the
groundwork for the theory,

00:10:41.090 --> 00:10:42.879
but they still weren't sure.

00:10:42.879 --> 00:10:44.818
They didn't definitively prove it,

00:10:44.818 --> 00:10:47.198
and it will take another decade or so

00:10:47.198 --> 00:10:49.137
until it's definitively proven.

00:10:49.137 --> 00:10:51.830
And even then, no one was
really sure exactly how

00:10:51.830 --> 00:10:54.059
these different traits were encoded,

00:10:54.059 --> 00:10:57.079
and for that we would have
to wait a little bit longer.


Circuit terminology

WEBVTT
Kind: captions
Language: en

00:00:00.267 --> 00:00:01.684
- In this video, we're going to talk about

00:00:01.684 --> 00:00:04.292
some terminology that we use to talk about

00:00:04.292 --> 00:00:06.647
how circuits are put together.

00:00:06.647 --> 00:00:08.620
In previous videos, we've talked about

00:00:08.620 --> 00:00:10.694
the components or elements that are used

00:00:10.694 --> 00:00:13.419
to make up circuits, so for example,

00:00:13.419 --> 00:00:18.419
resistor, capacitor, and inductor

00:00:18.705 --> 00:00:22.152
are circuit components, we
also call those elements.

00:00:22.152 --> 00:00:24.893
In addition, we have some sources

00:00:26.682 --> 00:00:31.085
like a voltage source or for
example, a current source.

00:00:31.085 --> 00:00:34.028
Those would be the components

00:00:38.833 --> 00:00:43.833
or elements of a circuit, el-e-ments.

00:00:48.194 --> 00:00:49.862
Now we're going to start
assembling these things

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into circuits, and we need a
few more words to talk about.

00:00:53.493 --> 00:00:56.260
Here's some circuit
components that are laying

00:00:56.260 --> 00:00:57.858
out on the tabletop,

00:00:57.858 --> 00:01:01.657
and we're going to connect
those up with a wire.

00:01:01.657 --> 00:01:06.044
For example, I could
connect this one to this one

00:01:06.044 --> 00:01:08.823
with this yellow ideal wire, and an ideal

00:01:08.823 --> 00:01:12.682
wire has zero resistance
and is perfect everywhere,

00:01:12.682 --> 00:01:17.647
and this forms a junction
between these two components

00:01:17.647 --> 00:01:21.524
and that is called a node.

00:01:21.524 --> 00:01:25.279
Node is the word for junction,
meaning the same thing.

00:01:28.577 --> 00:01:31.042
And that's what a node is.

00:01:31.042 --> 00:01:35.590
If I connect up these
two other components,

00:01:35.590 --> 00:01:39.066
I still have one node
because I have one junction

00:01:39.066 --> 00:01:41.379
that's the same voltage everywhere,

00:01:41.379 --> 00:01:43.794
so that's what a node is.

00:01:44.651 --> 00:01:46.126
We're going to go over
to this circuit here

00:01:46.126 --> 00:01:47.451
and we'll identify the nodes.

00:01:47.451 --> 00:01:50.632
This is a little more
normal-looking, tidy circuit.

00:01:50.632 --> 00:01:54.598
Here's a junction right here between

00:01:54.598 --> 00:01:57.579
a resistor and this voltage source,

00:01:57.579 --> 00:01:59.519
so that's one node.

00:01:59.519 --> 00:02:02.390
If I move over here, I see resistors

00:02:02.390 --> 00:02:04.310
connected together by ideal wires,

00:02:04.310 --> 00:02:07.966
so that forms one single node like that.

00:02:07.966 --> 00:02:11.328
So that's our second node, and down below,

00:02:11.328 --> 00:02:14.091
same sort of thing, I see an ideal wire

00:02:14.091 --> 00:02:17.359
connecting the resistors and the source.

00:02:17.359 --> 00:02:20.664
So I can color that in and that's

00:02:20.664 --> 00:02:23.173
node number three.

00:02:23.173 --> 00:02:25.739
So this circuit has three nodes.

00:02:29.833 --> 00:02:31.076
Now what's connecting the nodes?

00:02:31.076 --> 00:02:34.877
The thing that's connecting
the nodes is called a branch.

00:02:36.484 --> 00:02:39.482
And a branch is the same
thing as an element.

00:02:39.482 --> 00:02:43.255
We'll cap the... the branches
are elements in this thing.

00:02:43.255 --> 00:02:45.931
This voltage source
connects the third node

00:02:45.931 --> 00:02:50.193
to the first node, so that's one element.

00:02:50.193 --> 00:02:52.976
This resistor connects
node one and node two,

00:02:52.976 --> 00:02:56.141
so that's the second branch.

00:02:56.141 --> 00:02:58.830
This resistor connects
node two and node three,

00:02:58.830 --> 00:03:02.136
so there's the third
branch, and this resistor

00:03:02.136 --> 00:03:05.161
with a separate current also
connects those two nodes

00:03:05.161 --> 00:03:07.133
so that's the fourth branch.

00:03:07.133 --> 00:03:12.076
So this circuit has one, two, three, four

00:03:13.486 --> 00:03:17.541
elements in it, and it also
means it has four branches.

00:03:17.541 --> 00:03:21.472
Four branches, so that's
what a branch and a node are.

00:03:22.554 --> 00:03:24.010
I can move the picture over a little bit

00:03:24.010 --> 00:03:26.810
so that we can do this again on

00:03:28.350 --> 00:03:29.870
a little more fancy circuit.

00:03:29.870 --> 00:03:31.452
So the first thing we're going to do again

00:03:31.452 --> 00:03:34.258
just to repeat the process,
we're going to count the nodes.

00:03:34.258 --> 00:03:37.839
Here's a junction between a resistor

00:03:37.839 --> 00:03:41.187
and a source, here's three resistors

00:03:41.187 --> 00:03:44.095
connected by a perfect wire.

00:03:44.095 --> 00:03:45.909
So that's the second node.

00:03:45.909 --> 00:03:50.679
Here's three more resistors, connected

00:03:50.679 --> 00:03:53.263
by a perfect wire, there's a third.

00:03:53.263 --> 00:03:55.553
Down here we have, there's a junction

00:03:55.553 --> 00:03:59.197
between two resistors, so
that's our fourth node.

00:03:59.197 --> 00:04:02.375
And finally we have this node here

00:04:02.375 --> 00:04:05.924
connecting these four elements

00:04:05.924 --> 00:04:09.397
with one node.

00:04:09.397 --> 00:04:12.520
This is sometimes called a distributed,

00:04:12.520 --> 00:04:14.611
a distributed node when
it's all spread out

00:04:14.611 --> 00:04:17.859
on the page like that, but
it's still just one node.

00:04:17.859 --> 00:04:20.069
So this circuit has five nodes.

00:04:25.589 --> 00:04:26.707
And if we count up the elements,

00:04:26.707 --> 00:04:29.005
that will tell us how
many branches there are.

00:04:29.005 --> 00:04:33.421
One element, two, three, four,

00:04:33.421 --> 00:04:38.065
five, six, seven.

00:04:38.065 --> 00:04:40.495
Seven elements.

00:04:46.798 --> 00:04:48.972
All right, there's our two, two key words.

00:04:48.972 --> 00:04:52.307
Elements and nodes.

00:04:55.760 --> 00:04:57.802
Now I'm going to quickly move again

00:04:57.802 --> 00:05:01.541
down, bring in another circuit here.

00:05:01.541 --> 00:05:05.698
Now we're going to talk
about the idea of a mesh.

00:05:08.525 --> 00:05:10.034
And the other thing
we're going to talk about

00:05:10.034 --> 00:05:13.120
is the word loop.

00:05:14.957 --> 00:05:17.601
The word mesh comes from screen doors

00:05:17.601 --> 00:05:20.186
or screens that you put on your windows

00:05:20.186 --> 00:05:21.139
to keep the bugs out.

00:05:21.139 --> 00:05:23.860
If I draw a screen like this, this is what

00:05:23.860 --> 00:05:27.175
it looks like, a bunch of crossing wires

00:05:27.175 --> 00:05:29.682
and this little space
right here, that little gap

00:05:29.682 --> 00:05:34.020
is called a mesh, that's
what that word comes from.

00:05:34.020 --> 00:05:36.012
So we're going to find
the meshes of our circuit,

00:05:36.012 --> 00:05:38.678
and what we look for, here's the branches,

00:05:38.678 --> 00:05:42.777
and the mesh, it is a kind of a loop

00:05:42.777 --> 00:05:46.289
that fills up this open space.

00:05:46.289 --> 00:05:48.328
This circuit has one mesh,

00:05:49.954 --> 00:05:54.863
two mesh, three meshes.

00:05:54.863 --> 00:05:56.226
That's how that looks.

00:05:56.226 --> 00:05:58.615
To draw a mesh, you start on a node,

00:05:58.615 --> 00:06:01.203
you go through elements
until you come back

00:06:01.203 --> 00:06:02.820
to where you started.

00:06:02.820 --> 00:06:04.234
That's how we did those three,

00:06:04.234 --> 00:06:07.090
and they fill the open
windows of the circuit.

00:06:07.090 --> 00:06:09.693
So this circuit has three meshes.

00:06:11.449 --> 00:06:14.129
Now a mesh is a loop, and we can have

00:06:14.129 --> 00:06:16.413
other kinds of loops too.

00:06:16.413 --> 00:06:18.019
They don't have to be just the ones

00:06:18.019 --> 00:06:20.834
that fill the windows, so in general,

00:06:20.834 --> 00:06:23.051
this circuit has other
loops, and we'll identify

00:06:23.051 --> 00:06:26.807
some of those, let's just start
at one of these nodes here,

00:06:26.807 --> 00:06:30.758
and go around like that
and this is a loop.

00:06:32.413 --> 00:06:33.927
And I could draw other loops in here.

00:06:33.927 --> 00:06:35.720
We'll make them all different colors.

00:06:35.720 --> 00:06:39.854
There's a loop if I start right here,

00:06:39.854 --> 00:06:43.176
I can draw a loop through these elements.

00:06:46.593 --> 00:06:48.608
And finally, if I have a sharp eye,

00:06:48.608 --> 00:06:51.005
there's one more loop in this circuit.

00:06:51.005 --> 00:06:53.265
Let's just start right
here, and it actually

00:06:53.265 --> 00:06:56.138
goes all the way around the outside.

00:06:59.014 --> 00:07:01.859
So this circuit has three, actually,

00:07:01.859 --> 00:07:04.818
if I add them together,
this has three loops

00:07:04.818 --> 00:07:07.192
that I drew here plus the three loops

00:07:07.192 --> 00:07:11.033
that were the meshes, so
this circuit has six loops.

00:07:12.878 --> 00:07:14.423
Circuits always have a lot of loops,

00:07:14.423 --> 00:07:17.120
and so usually we don't talk about these.

00:07:17.120 --> 00:07:19.871
More often, it's more
organized and straightforward

00:07:19.871 --> 00:07:23.243
to talk about how many
meshes are in a circuit.

00:07:23.243 --> 00:07:26.039
All right, so that does it for this video.

00:07:26.039 --> 00:07:29.486
We got mesh and loop, we talked

00:07:29.486 --> 00:07:33.163
about components and elements.

00:07:41.555 --> 00:07:44.393
And we finished up with the idea

00:07:46.146 --> 00:07:48.996
also of nodes

00:07:51.568 --> 00:07:53.451
and branches.

00:07:59.773 --> 00:08:01.818
So that'll do it, there's
our new vocabulary

00:08:01.818 --> 00:08:03.354
for talking about circuits.


The scientific method

WEBVTT
Kind: captions
Language: en

00:00:00.194 --> 00:00:04.124
- [Voiceover] Let's explore the scientific

00:00:04.124 --> 00:00:07.821
scientific method

00:00:07.821 --> 00:00:09.730
which, at first might seem a little

00:00:09.730 --> 00:00:11.518
bit intimidating, but when we walk

00:00:11.518 --> 00:00:13.363
through it, you'll see that it's actually

00:00:13.363 --> 00:00:15.561
almost a common-sense way of looking

00:00:15.561 --> 00:00:17.448
at the world and making progress in our

00:00:17.448 --> 00:00:20.397
understanding of the
world and feeling good

00:00:20.397 --> 00:00:22.365
about that progress of our understanding

00:00:22.365 --> 00:00:23.743
of the world.

00:00:23.743 --> 00:00:25.508
So, let's just use a tangible example

00:00:25.508 --> 00:00:27.454
here, and we'll walk through what we could

00:00:27.454 --> 00:00:29.933
consider the steps of the scientific

00:00:29.933 --> 00:00:31.396
method, and you'll see different steps

00:00:31.396 --> 00:00:33.326
articulated in different
ways, but they all

00:00:33.326 --> 00:00:35.541
boil down to the same thing.

00:00:35.541 --> 00:00:38.263
You observe something
about reality, and you say,

00:00:38.263 --> 00:00:40.722
well, let me try to come up with

00:00:40.722 --> 00:00:44.694
a reason for why that observation

00:00:44.694 --> 00:00:47.831
happens, and then you try
to test that explanation.

00:00:47.831 --> 00:00:48.937
It's very important that you come up with

00:00:48.937 --> 00:00:50.949
explanations that you
can test, and then you

00:00:50.949 --> 00:00:52.683
can see if they're true, and then based on

00:00:52.683 --> 00:00:55.723
whether they're true, you keep iterating.

00:00:55.723 --> 00:00:57.639
If it's not true, you come
up with another explanation.

00:00:57.639 --> 00:00:59.670
If it is true, but it
doesn't explain everything,

00:00:59.670 --> 00:01:02.374
well once again, you try
to explain more of it.

00:01:02.374 --> 00:01:06.449
So, as a tangible example,
let's say that you live in,

00:01:06.449 --> 00:01:08.931
in I don't know, northern
Canada or something, and let's

00:01:08.931 --> 00:01:10.211
say that you live near
the beach, but there's

00:01:10.211 --> 00:01:13.488
also a pond near your
house, and you notice that

00:01:13.488 --> 00:01:15.778
the pond, it tends to
freeze over sooner in

00:01:15.778 --> 00:01:18.876
the Winter than the ocean does.

00:01:18.876 --> 00:01:21.680
It does that faster and
even does it at higher

00:01:21.680 --> 00:01:24.057
temperatures than when the ocean seems

00:01:24.057 --> 00:01:25.601
to freeze over.

00:01:25.601 --> 00:01:27.691
So, you could view that
as your observation.

00:01:27.691 --> 00:01:30.803
So, the first step is you're
making an observation.

00:01:30.803 --> 00:01:33.210
Observation.

00:01:33.210 --> 00:01:35.892
In our particular case
is that the pond freezes

00:01:35.892 --> 00:01:38.208
over at higher temperatures than the ocean

00:01:38.208 --> 00:01:41.695
does, and it freezes over
sooner in the Winter.

00:01:41.695 --> 00:01:43.592
Well, the next question that you might

00:01:43.592 --> 00:01:45.931
wanna, or the next step you could view as

00:01:45.931 --> 00:01:47.068
a scientific method.

00:01:47.068 --> 00:01:48.775
It doesn't have to be
this regimented, but this

00:01:48.775 --> 00:01:50.540
is a structured way of thinking about it.

00:01:50.540 --> 00:01:53.062
Well, ask yourself a question.

00:01:53.062 --> 00:01:55.866
Ask a question.

00:01:55.866 --> 00:01:59.993
Why does, so in this
particular question, or in

00:01:59.993 --> 00:02:03.589
this particular scenario,
why does the pond tend

00:02:03.589 --> 00:02:06.457
to freeze over faster and
at higher temperatures

00:02:06.457 --> 00:02:08.739
than the ocean does?

00:02:08.739 --> 00:02:11.211
Well, you then try to answer
that question, and this

00:02:11.211 --> 00:02:13.982
is a key part of the scientific method

00:02:13.982 --> 00:02:16.511
is what you do in this third step

00:02:16.511 --> 00:02:19.651
is that you try to create
an explanation, but what's

00:02:19.651 --> 00:02:23.126
key is that it is a testable explanation.

00:02:23.126 --> 00:02:27.268
So, you try to, you create a testable

00:02:28.207 --> 00:02:30.147
explanation.

00:02:30.147 --> 00:02:32.524
Testable explanation,
and this is kind of the

00:02:32.524 --> 00:02:34.901
core, one of the core
pillars of the scientific

00:02:34.901 --> 00:02:37.761
method, and this testable
explanation is called

00:02:37.761 --> 00:02:39.991
your hypothesis.

00:02:39.991 --> 00:02:42.439
Your hypothesis.

00:02:42.439 --> 00:02:45.141
And so, in this particular
case, a testable

00:02:45.141 --> 00:02:49.631
explanation could be
that, well the ocean is

00:02:49.631 --> 00:02:52.302
made up of salt water, and this pond is

00:02:52.302 --> 00:02:54.620
fresh water, so your testable explanation

00:02:54.620 --> 00:02:56.671
could be salt water,

00:02:56.671 --> 00:03:01.671
salt water has lower freezing point.

00:03:01.780 --> 00:03:05.214
Has lower freezing,

00:03:05.214 --> 00:03:08.384
freezing point.

00:03:08.384 --> 00:03:09.785
Lower freezing point, so it takes colder

00:03:09.785 --> 00:03:14.785
temperatures to freeze
it than fresh water.

00:03:14.849 --> 00:03:17.173
Than fresh water.

00:03:17.173 --> 00:03:18.959
So, this, right over here, this would

00:03:18.959 --> 00:03:20.483
be a good hypothesis.

00:03:20.483 --> 00:03:22.196
It doesn't matter
whether the hypothesis is

00:03:22.196 --> 00:03:23.017
actually true or not.

00:03:23.017 --> 00:03:24.869
We haven't actually run
the experiment, but it's

00:03:24.869 --> 00:03:26.998
a good one, because we can construct an

00:03:26.998 --> 00:03:29.965
experiment that tests this very well.

00:03:29.965 --> 00:03:32.402
Now, what would be an example of a bad

00:03:32.402 --> 00:03:34.124
hypothesis or of something
that you couldn't

00:03:34.124 --> 00:03:35.844
even necessarily consider as part of the

00:03:35.844 --> 00:03:37.112
scientific method?

00:03:37.112 --> 00:03:42.081
Well, you could say that there is a

00:03:42.081 --> 00:03:46.832
fairy that blesses that, let's

00:03:46.832 --> 00:03:49.719
say that performs magic

00:03:49.719 --> 00:03:54.719
performs magic

00:03:55.463 --> 00:03:57.038
on the pond

00:03:59.286 --> 00:04:01.499
to freeze it faster.

00:04:01.499 --> 00:04:05.597
Freeze it faster.

00:04:05.597 --> 00:04:08.401
And, the reason why this
isn't so good is that

00:04:08.401 --> 00:04:11.367
this is not so testable,
because it's depending

00:04:11.367 --> 00:04:14.293
on this fairy, and you
don't know how to convince

00:04:14.293 --> 00:04:16.223
the fairy to try to do it again.

00:04:16.223 --> 00:04:17.310
You haven't seen the fairy.

00:04:17.310 --> 00:04:19.017
You haven't observed the fairy.

00:04:19.017 --> 00:04:21.516
It's not based on any observation, and so

00:04:21.516 --> 00:04:23.147
this one right over here, this would not

00:04:23.147 --> 00:04:26.311
be a good hypothesis for the scientific

00:04:26.311 --> 00:04:30.747
method, so we would
wanna rule that one out.

00:04:30.747 --> 00:04:34.093
So, let's go back to our
testable explanation,

00:04:34.093 --> 00:04:35.266
our hypothesis.

00:04:35.266 --> 00:04:38.227
Salt water has a lower freezing
point than fresh water.

00:04:38.227 --> 00:04:41.855
Well, the next step would be to

00:04:42.839 --> 00:04:45.279
make a prediction based
on that, and this is the

00:04:45.279 --> 00:04:47.946
part where we're really
designing an experiment.

00:04:47.946 --> 00:04:50.474
So, you could just view
all of this as designing.

00:04:50.474 --> 00:04:52.434
Let me do this in a different color.

00:04:52.434 --> 00:04:54.421
Where we wanna design an experiment.

00:04:55.541 --> 00:04:58.516
Design an experiment.

00:05:00.654 --> 00:05:03.194
And in that experiments
lets say, and let's see,

00:05:03.194 --> 00:05:06.019
the next two steps I will put as

00:05:06.019 --> 00:05:09.483
part of this experimental.

00:05:09.483 --> 00:05:10.641
Whoops.

00:05:10.641 --> 00:05:11.187
I messed up.

00:05:11.187 --> 00:05:13.957
Let me, I did my undo step.

00:05:13.957 --> 00:05:17.306
So, the next part that I
will do is the experiment.

00:05:18.458 --> 00:05:21.069
Experiment.

00:05:21.069 --> 00:05:23.882
And there you go.

00:05:23.882 --> 00:05:27.214
So, the first thing is,
we'll say I take, you know,

00:05:27.214 --> 00:05:29.470
there's all sorts of things
that are going on outside.

00:05:29.470 --> 00:05:31.816
The ocean has waves.

00:05:31.816 --> 00:05:34.336
You know, maybe there are boats going by

00:05:34.336 --> 00:05:36.083
that might potentially break up the ice.

00:05:36.083 --> 00:05:39.189
So, I just wanna isolate
that one variable that

00:05:39.189 --> 00:05:40.817
I care about, whether something is salt

00:05:40.817 --> 00:05:42.402
water or not, and I want a control for

00:05:42.402 --> 00:05:43.621
everything else.

00:05:43.621 --> 00:05:46.282
So, I want a control for whether there's

00:05:46.282 --> 00:05:48.659
waves or not or whether
there's wind or any

00:05:48.659 --> 00:05:51.402
other possible explanation for why the

00:05:51.402 --> 00:05:53.400
pond freezes over faster.

00:05:53.400 --> 00:05:56.021
So, what I do, in a very
controlled environment

00:05:56.021 --> 00:05:58.107
I take two cups.

00:05:58.107 --> 00:05:59.195
I take two cups.

00:05:59.195 --> 00:06:02.405
That's one cup and two cups, and I put

00:06:02.405 --> 00:06:04.665
water in those cups.

00:06:04.665 --> 00:06:07.036
I put water in those cups.

00:06:07.036 --> 00:06:10.952
Now, let's say I start with
distilled water, but then

00:06:10.952 --> 00:06:13.985
this one stays, the
first one right over here

00:06:13.985 --> 00:06:15.936
stays distilled, and
distilled means that through

00:06:15.936 --> 00:06:17.914
evaporation I've taken
out all of the impurities

00:06:17.914 --> 00:06:20.084
of that water, and in the second one

00:06:20.084 --> 00:06:23.392
I take that distilled
water, and I throw a bunch

00:06:23.392 --> 00:06:24.848
of salt in it.

00:06:24.848 --> 00:06:28.888
So, this one is fresh,
very fresh, and in fact,

00:06:28.888 --> 00:06:30.345
far fresher than you would find in a pond.

00:06:30.345 --> 00:06:31.726
It's distilled water.

00:06:31.726 --> 00:06:35.922
And then this is over
here, this is salt water.

00:06:35.922 --> 00:06:38.481
So, you wouldn't see the
salt, but just for our

00:06:38.481 --> 00:06:39.599
visuals, you depict it.

00:06:39.599 --> 00:06:41.885
Then we would make a prediction, and we

00:06:41.885 --> 00:06:44.039
could even view this as Step Four,

00:06:44.039 --> 00:06:45.756
our prediction.

00:06:45.756 --> 00:06:49.392
We predict that the fresh
water will freeze at

00:06:49.392 --> 00:06:51.343
a higher temperature than the salt water.

00:06:51.343 --> 00:06:56.079
So, we'll prediction,
let's say the fresh freezes

00:06:56.079 --> 00:06:57.894
fresh freezes

00:06:59.470 --> 00:07:04.470
at zero degrees Celsius,

00:07:04.492 --> 00:07:07.824
but salt doesn't.

00:07:07.824 --> 00:07:09.293
Salt water doesn't.

00:07:10.380 --> 00:07:11.796
Whoop, not sat.

00:07:11.796 --> 00:07:15.703
Salt water doesn't.

00:07:17.146 --> 00:07:19.608
So, what you then do is that
you test your prediction.

00:07:19.608 --> 00:07:21.254
So, then you test it.

00:07:21.254 --> 00:07:22.290
And how would you test it?

00:07:22.290 --> 00:07:24.220
Well, you could have a
very accurate freezer that

00:07:24.220 --> 00:07:27.776
is exactly at zero degrees
Celsius, and you put

00:07:27.776 --> 00:07:31.616
both of these cups into
it, and you wanna make

00:07:31.616 --> 00:07:32.804
sure that they're identical
and everything where

00:07:32.804 --> 00:07:33.911
you control for everything else.

00:07:33.911 --> 00:07:35.603
You control for the surface area.

00:07:35.603 --> 00:07:37.589
You control for the material of the glass.

00:07:37.589 --> 00:07:39.559
You control for how much water there is.

00:07:39.559 --> 00:07:41.006
But, then you test it.

00:07:41.006 --> 00:07:44.279
Then you see what happened from your test.

00:07:44.279 --> 00:07:46.543
Leave it in overnight,
and if you see that the

00:07:46.543 --> 00:07:49.573
fresh water has frozen
over, so it's frozen

00:07:49.573 --> 00:07:51.828
over, but the salt water
hasn't, well then that

00:07:51.828 --> 00:07:56.003
seems to validate your
testable explanation

00:07:56.003 --> 00:07:58.644
that salt water has a
lower freezing point than

00:07:58.644 --> 00:08:01.469
fresh water, and if it didn't
freeze, well it's like,

00:08:01.469 --> 00:08:04.354
okay, well may that, or if
there isn't a difference,

00:08:04.354 --> 00:08:05.959
maybe either both of them
didn't freeze or both

00:08:05.959 --> 00:08:08.498
of them did freeze, then
you might say, well, okay,

00:08:08.498 --> 00:08:10.246
that wasn't a good explanation.

00:08:10.246 --> 00:08:11.505
I have to find another explanation for why

00:08:11.505 --> 00:08:15.280
the ocean seems to freeze
at a lower temperature.

00:08:15.280 --> 00:08:16.798
Or, you might say, well that's part of the

00:08:16.798 --> 00:08:18.260
explanation, but that by itself doesn't

00:08:18.260 --> 00:08:20.117
explain it, or you
might now wanna ask even

00:08:20.117 --> 00:08:21.775
further questions about, well, when does

00:08:21.775 --> 00:08:24.559
salt water freeze, and what
else is it dependent on?

00:08:24.559 --> 00:08:25.798
Do the waves have an impact?

00:08:25.798 --> 00:08:27.440
Does the wind have an impact?

00:08:27.440 --> 00:08:30.082
So, then you can go into the process

00:08:30.082 --> 00:08:32.666
of iterating and refining.

00:08:32.666 --> 00:08:37.049
So, you then refine

00:08:37.049 --> 00:08:38.194
refine

00:08:38.194 --> 00:08:41.650
refine and iterate on the process.

00:08:41.650 --> 00:08:42.656
When I'm talking about iterate, you're

00:08:42.656 --> 00:08:44.733
doing it over again, but then, based on

00:08:44.733 --> 00:08:46.049
the things that you've learned.

00:08:46.049 --> 00:08:48.122
So, you might come up with a more refined

00:08:48.122 --> 00:08:50.336
testable explanation, or you might come up

00:08:50.336 --> 00:08:51.962
with more experiments that could get you

00:08:51.962 --> 00:08:53.648
a better understanding of the difference

00:08:53.648 --> 00:08:55.111
between fresh and salt water, or you might

00:08:55.111 --> 00:08:57.205
try to come up with experiments for why

00:08:57.205 --> 00:09:00.167
exactly, what is it
about the salt that makes

00:09:00.167 --> 00:09:02.598
this water harder to freeze?

00:09:02.598 --> 00:09:04.223
So, that's essentially the essence of

00:09:04.223 --> 00:09:06.366
the scientific method,
and I wanna emphasize

00:09:06.366 --> 00:09:08.957
this isn't some, you know, bizarre thing.

00:09:08.957 --> 00:09:12.194
This is logical reasoning.

00:09:12.194 --> 00:09:14.666
Make a testable explanation for something

00:09:14.666 --> 00:09:16.799
that you're observing
in the world, and then

00:09:16.799 --> 00:09:19.102
you test it, and you
see if your explanation

00:09:19.102 --> 00:09:22.239
seems to hold up based
on the data from your

00:09:22.239 --> 00:09:24.365
test, and then whether or not it holds up,

00:09:24.365 --> 00:09:27.121
you then keep going, and
you keep refining, and you

00:09:27.121 --> 00:09:29.234
keep learning more
about the world, and the

00:09:29.234 --> 00:09:31.428
reason why this is better
than just saying, oh well,

00:09:31.428 --> 00:09:33.907
look, okay, I see the pond
has frozen over and the

00:09:33.907 --> 00:09:36.548
ocean hasn't, it must be the
salt water, and you know,

00:09:36.548 --> 00:09:38.438
I just feel good about
that, is that you can't

00:09:38.438 --> 00:09:39.677
feel good about that.

00:09:39.677 --> 00:09:41.119
There's a million
different reasons, and you

00:09:41.119 --> 00:09:43.700
shouldn't just go on
your gut, 'cause at some

00:09:43.700 --> 00:09:45.488
point, your gut might be right 90% of

00:09:45.488 --> 00:09:47.459
the time, but that 10%
that it's wrong, you're

00:09:47.459 --> 00:09:51.481
going to be passing on
knowledge or assumptions

00:09:51.481 --> 00:09:53.736
about the world that
aren't true, and then other

00:09:53.736 --> 00:09:55.441
people are going to build
on that, and then all

00:09:55.441 --> 00:09:57.699
of our knowledge is going
to be built on kind of

00:09:57.699 --> 00:10:01.136
a shaky foundation, and
so the scientific method

00:10:01.136 --> 00:10:04.830
ensures that our foundation is strong.

00:10:04.830 --> 00:10:07.472
And I'll leave you with
the gentleman who's often

00:10:07.472 --> 00:10:10.183
considered to be the father, or one of the

00:10:10.183 --> 00:10:12.126
fathers, of the scientific method.

00:10:12.126 --> 00:10:16.373
He lived in Cairo, and in what is now Iraq,

00:10:16.373 --> 00:10:20.422
oh nearly 1,000 or roughly
1,000 years ago, and he

00:10:20.422 --> 00:10:24.211
was a famous astronomer and
physist and mathematician.

00:10:24.211 --> 00:10:26.584
And his quote is a pretty
powerful one, 'cause

00:10:26.584 --> 00:10:28.479
I think it even stands today:

00:10:28.479 --> 00:10:31.641
"The duty of the man who
investigates the writings

00:10:31.641 --> 00:10:36.641
of scientists, if learning
the truth is his goal, ..."

00:10:36.874 --> 00:10:38.433
Let me start over, just
so I can get the dramatic

00:10:38.433 --> 00:10:39.469
effect right.

00:10:39.469 --> 00:10:42.029
"The duty of the man who
investigates the writings

00:10:42.029 --> 00:10:45.260
of scientists, if learning
the truth is his goal,

00:10:45.260 --> 00:10:48.755
is to make himself an enemy of all that

00:10:48.755 --> 00:10:52.714
he reads, and attack it from every side.

00:10:52.714 --> 00:10:55.413
He should also suspect
himself as he performs

00:10:55.413 --> 00:10:58.535
his critical examination
of it, so that he may

00:10:58.535 --> 00:11:02.490
avoid falling into either
prejudice or leniency."

00:11:02.490 --> 00:11:05.964
Hasan Ibn al-Haytham,
and his Latinized name

00:11:05.964 --> 00:11:07.834
is Alhazen.

00:11:07.834 --> 00:11:09.968
So, he's saying be skeptical,
and not just skeptical

00:11:09.968 --> 00:11:12.253
of what other people
write and read, but even

00:11:12.253 --> 00:11:15.093
of yourself, and another
aspect of the scientific

00:11:15.093 --> 00:11:17.190
method which is super
important is, if someone

00:11:17.190 --> 00:11:19.648
says they made a hypothesis
and they tested and they

00:11:19.648 --> 00:11:23.069
got a result, in order for
that to be a good test and in

00:11:23.069 --> 00:11:25.613
order for that to be a good
hypothesis, that experiment

00:11:25.613 --> 00:11:27.419
has to be reproducible.

00:11:27.419 --> 00:11:28.851
Someone can't say, oh it's
only, you know, a certain

00:11:28.851 --> 00:11:31.895
time that only happens once
every 100 years and not,

00:11:31.895 --> 00:11:33.626
that that's why it happened that day.

00:11:33.626 --> 00:11:35.981
It has to be reproducible,
and reproducible

00:11:35.981 --> 00:11:37.962
is key, because then
another skeptical scientist

00:11:37.962 --> 00:11:40.148
like yourself can say, let me see if I can

00:11:40.148 --> 00:11:40.942
reproduce it.

00:11:40.942 --> 00:11:43.299
Let me not just believe
it, because that person

00:11:43.299 --> 00:11:45.468
looks like they're
smart, and they said that

00:11:45.468 --> 00:11:47.348
it is true.


pH and pKa relationship for buffers

WEBVTT
Kind: captions
Language: en

00:00:00.445 --> 00:00:02.558
- [Voiceover] We're gonna
talk about the relationship

00:00:02.558 --> 00:00:06.249
between pH and pK_a

00:00:06.249 --> 00:00:08.362
and buffers.

00:00:08.362 --> 00:00:12.081
And specifically, we're
going to be talking mostly

00:00:12.081 --> 00:00:16.730
about this in terms of the
Henderson-Hasselbalch equation.

00:00:16.730 --> 00:00:19.850
But before we go to the
Henderson-Hasselbalch equation,

00:00:19.850 --> 00:00:21.866
which I am gonna assume
you've seen before,

00:00:21.866 --> 00:00:24.239
and if not we have some
other videos introducing it

00:00:24.239 --> 00:00:27.424
and also deriving it,
let's do a quick review

00:00:27.424 --> 00:00:29.277
of what exactly is a buffer.

00:00:29.277 --> 00:00:32.687
So a buffer is something that contains

00:00:32.687 --> 00:00:36.725
an aqueous solution.

00:00:36.725 --> 00:00:41.086
It's something that
contains both a weak acid,

00:00:41.086 --> 00:00:44.044
which generically we write as HA,

00:00:44.044 --> 00:00:48.920
and it also contains in equilibrium

00:00:48.920 --> 00:00:53.601
the conjugate base of
our acid, so A minus.

00:00:55.273 --> 00:00:57.219
This is, as written,

00:00:57.219 --> 00:01:02.030
it's the acid dissociation
on reaction for HA

00:01:02.030 --> 00:01:04.774
and since it's an equilibrium

00:01:04.774 --> 00:01:08.967
we can write an expression called K_a,

00:01:08.967 --> 00:01:13.967
which is just the equilibrium
constant for this equation.

00:01:14.006 --> 00:01:16.509
And it just has a special
name, 'cause it happens to be

00:01:16.509 --> 00:01:19.500
for the dissociation of an acid.

00:01:19.500 --> 00:01:21.190
K_a is just equal to

00:01:21.190 --> 00:01:24.733
H_three O plus times A minus

00:01:24.733 --> 00:01:27.185
all over the concentration of HA,

00:01:27.185 --> 00:01:31.151
and we don't include water
because it's a pure liquid,

00:01:31.151 --> 00:01:34.207
so we assume that the
concentration is always one.

00:01:34.207 --> 00:01:38.725
Based on this expression
for K_a, we can, and do,

00:01:38.725 --> 00:01:40.461
in the separate video, derive the

00:01:40.461 --> 00:01:42.555
Henderson-Hasselbalch equation.

00:01:42.555 --> 00:01:44.700
So the Henderson-Hasselbalch equation

00:01:44.700 --> 00:01:47.886
just says that the pH is equal to

00:01:47.886 --> 00:01:51.917
the pK_a plus the log

00:01:51.917 --> 00:01:55.892
of A minus over HA,

00:01:57.313 --> 00:01:59.453
where HA is our weak acid

00:01:59.453 --> 00:02:02.606
and A minus is its conjugate base.

00:02:02.606 --> 00:02:04.849
And as you can see up here,

00:02:04.849 --> 00:02:06.312
an acid and its conjugate base

00:02:06.312 --> 00:02:10.245
are just related by the fact
that the acid has an extra H.

00:02:10.245 --> 00:02:13.431
We can rearrange the
Henderson-Hasselbalch equation

00:02:13.431 --> 00:02:16.584
to get a lot of different
kinds of information.

00:02:16.584 --> 00:02:17.930
One kind of problem you see a lot

00:02:17.930 --> 00:02:20.790
is for some buffer,

00:02:20.790 --> 00:02:21.831
you know, they might ask you,

00:02:21.831 --> 00:02:23.521
oh, what's the pH?

00:02:23.521 --> 00:02:25.536
And then that means you
probably know the pK_a

00:02:25.536 --> 00:02:29.172
and you know the concentrations
of A minus and HA.

00:02:29.172 --> 00:02:30.830
The other thing that you can use the

00:02:30.830 --> 00:02:33.105
Henderson-Hasselbalch equation to tell you

00:02:33.105 --> 00:02:34.538
is the relationship

00:02:34.538 --> 00:02:38.594
between A minus and HA

00:02:38.594 --> 00:02:40.616
which is something you might wanna know.

00:02:40.616 --> 00:02:42.339
A lot of times, you just wanna know,

00:02:42.339 --> 00:02:43.835
you know, what's in your solution,

00:02:43.835 --> 00:02:46.143
depending on what you
wanna do to your solution,

00:02:46.143 --> 00:02:47.573
if you wanna add things to it,

00:02:47.573 --> 00:02:49.816
maybe you wanna add some
acid, you wanna add some base.

00:02:49.816 --> 00:02:50.856
You wanna know what's going on.

00:02:50.856 --> 00:02:51.929
The Henderson-Hasselbalch equation

00:02:51.929 --> 00:02:55.407
gives you a really quick
and easy way of doing that.

00:02:55.407 --> 00:02:57.200
So what we're gonna do, is we're gonna

00:02:57.200 --> 00:03:00.329
rearrange this equation
to solve for this ratio

00:03:00.329 --> 00:03:02.247
that we might be interested in.

00:03:02.247 --> 00:03:05.571
So we're gonna subtract

00:03:05.571 --> 00:03:08.984
pK_a from both sides

00:03:08.984 --> 00:03:10.674
and that gives us a log

00:03:10.674 --> 00:03:14.170
of A minus over HA.

00:03:14.170 --> 00:03:17.696
And I don't know about
you, but I actually find,

00:03:17.696 --> 00:03:21.928
well, (laughs) I find logs
not super-intuitive sometimes.

00:03:21.928 --> 00:03:24.269
So I'm actually going
to get rid of the log

00:03:24.269 --> 00:03:26.740
by raising both sides to the 10th power.

00:03:26.740 --> 00:03:28.527
So that gives us

00:03:28.527 --> 00:03:32.526
10 to the pH minus pK_a

00:03:32.526 --> 00:03:33.956
is equal to

00:03:33.956 --> 00:03:38.410
A minus over HA.

00:03:38.410 --> 00:03:39.937
So what does this tell us?

00:03:39.937 --> 00:03:42.083
It may not look like it
tells us a whole lot more,

00:03:42.083 --> 00:03:43.253
but actually, it tells us a lot.

00:03:43.253 --> 00:03:46.276
It tells us about the
relative relationship and size

00:03:46.276 --> 00:03:50.307
between A minus and HA concentration.

00:03:50.307 --> 00:03:53.655
And it's saying that these
two things are related

00:03:53.655 --> 00:03:57.036
to the relative size of pH and pK_a.

00:03:57.036 --> 00:04:00.956
So if we look at this, we can
derive a couple relationships.

00:04:02.786 --> 00:04:05.842
We're saying that pH versus pK_a,

00:04:05.842 --> 00:04:08.995
this relationship, can tell us

00:04:08.995 --> 00:04:12.269
about A minus

00:04:12.269 --> 00:04:14.999
over HA, which is a ratio,

00:04:14.999 --> 00:04:18.608
but that in turn we can relate to

00:04:18.608 --> 00:04:23.592
just the relative size of HA versus A.

00:04:23.592 --> 00:04:27.996
So let's go ahead and look
at all the possible scenarios

00:04:27.996 --> 00:04:30.564
for these three things.

00:04:30.564 --> 00:04:33.535
We're going to start with the
simplest possible scenario,

00:04:33.535 --> 00:04:38.327
which is that pH is equal to pK_a.

00:04:38.327 --> 00:04:41.058
When pH is equal to pK_a, we're raising

00:04:41.058 --> 00:04:42.943
10 to the zeroth power.

00:04:42.943 --> 00:04:46.031
So anything to the zeroth
power is equal to one.

00:04:46.031 --> 00:04:49.087
Which tells us that this
ratio is equal to one.

00:04:49.087 --> 00:04:50.953
And if A minus concentration

00:04:50.953 --> 00:04:53.228
over HA concentration is equal to one,

00:04:53.228 --> 00:04:57.129
that means that they have
the same concentration.

00:04:57.129 --> 00:04:58.397
I forgot a minus sign there.

00:04:58.397 --> 00:05:00.413
This is a really helpful
thing to remember.

00:05:00.413 --> 00:05:02.558
Any time you have a buffer

00:05:02.558 --> 00:05:04.573
and the pH of your solution

00:05:04.573 --> 00:05:06.654
is equal to the pK_a of your buffer,

00:05:06.654 --> 00:05:08.864
you immediately know
that the concentration

00:05:08.864 --> 00:05:11.660
of your acid and its
conjugate base are the same.

00:05:11.660 --> 00:05:13.832
And this comes up a lot not
just when you're talking

00:05:13.832 --> 00:05:15.750
about buffers by themselves, but also when

00:05:15.750 --> 00:05:17.161
you're doing titrations.

00:05:17.161 --> 00:05:18.656
And the point in your titration

00:05:18.656 --> 00:05:21.257
where the HA is equal to A minus

00:05:21.257 --> 00:05:24.394
is called the half-equivalence point.

00:05:26.062 --> 00:05:29.833
Half-equivalence point.

00:05:29.833 --> 00:05:32.043
And if you haven't learnt
about buffers, that's okay.

00:05:32.043 --> 00:05:34.183
Oh, sorry, if you haven't
learn about titrations yet,

00:05:34.183 --> 00:05:35.386
that's fully fine.

00:05:35.386 --> 00:05:37.173
Just ignore what I just said (laughs),

00:05:37.173 --> 00:05:38.701
but if you have, the moral is just that,

00:05:38.701 --> 00:05:42.147
this is a really, really
important relationship

00:05:42.147 --> 00:05:44.125
that is really helpful to remember.

00:05:44.125 --> 00:05:46.172
And I said really a lot there.

00:05:46.172 --> 00:05:48.285
So there are two other possibilities

00:05:48.285 --> 00:05:50.138
for pH and pK_a.

00:05:50.138 --> 00:05:53.191
We can have a pH that's greater than pK_a

00:05:53.191 --> 00:05:55.500
for your buffer, and you can have a pH

00:05:55.500 --> 00:05:58.649
that is less than you
pK_a for your buffer.

00:05:58.649 --> 00:06:01.510
So if your pH is bigger than your pK_a,

00:06:01.510 --> 00:06:03.265
then this term up here,

00:06:03.265 --> 00:06:06.159
10 to the pH minus pK_a,
is going to be positive.

00:06:06.159 --> 00:06:10.429
And when you raise 10
to a positive number,

00:06:13.752 --> 00:06:16.366
when you raise 10 to a positive number,

00:06:16.366 --> 00:06:19.237
you get a ratio that is

00:06:19.237 --> 00:06:22.683
greater than one.

00:06:22.683 --> 00:06:27.572
So if our ratio A minus over HA

00:06:27.572 --> 00:06:31.896
is greater than one, that tells us that

00:06:31.896 --> 00:06:35.959
A minus, the numerator,
is actually greater than

00:06:35.959 --> 00:06:38.573
the denominator, HA.

00:06:38.573 --> 00:06:39.952
So if you know the pH

00:06:39.952 --> 00:06:43.365
and you know it's bigger
than the pK_a of your buffer,

00:06:43.365 --> 00:06:45.250
buffer's acid, to be more specific,

00:06:45.250 --> 00:06:47.201
then you immediately know that you have

00:06:47.201 --> 00:06:49.444
more conjugate base around than your acid.

00:06:49.444 --> 00:06:53.052
The last scenario, pH is less than pK_a,

00:06:53.052 --> 00:06:55.360
well in that case, we're raising 10

00:06:55.360 --> 00:06:58.481
to a negative number, because

00:06:58.481 --> 00:07:02.329
we're subtracting a bigger
number from a smaller number.

00:07:02.329 --> 00:07:04.749
And that means that our ratio,

00:07:04.749 --> 00:07:08.997
A minus over HA, is
actually less than one.

00:07:08.997 --> 00:07:12.125
So that tells us that our denominator, HA,

00:07:14.703 --> 00:07:17.501
is actually bigger than our numerator.

00:07:18.844 --> 00:07:20.665
So just to wrap up, we can look at

00:07:20.665 --> 00:07:22.433
the Henderson-Hasselbalch equation

00:07:22.433 --> 00:07:24.546
and we can just look at the relationship

00:07:24.546 --> 00:07:26.724
between pK_a and pH,

00:07:26.724 --> 00:07:28.869
and depending on whether
they're equal to each other

00:07:28.869 --> 00:07:30.592
or one is bigger than the other,

00:07:30.592 --> 00:07:33.160
we can immediately know
what the relationship is

00:07:33.160 --> 00:07:35.872
between our acid and its
conjugate base in our solution.

00:07:35.872 --> 00:07:37.887
It's very easy to derive.

00:07:37.887 --> 00:07:39.350
We did it in just a few minutes,

00:07:39.350 --> 00:07:41.463
so it's okay if you don't
remember this all the time.

00:07:41.463 --> 00:07:43.933
I usually just remember
that pH equals pK_a

00:07:43.933 --> 00:07:45.689
means you have the same concentration,

00:07:45.689 --> 00:07:49.182
and then, if I forget, I will
derive the other relation.


Buffer capacity

WEBVTT
Kind: captions
Language: en

00:00:00.561 --> 00:00:02.895
- [Voiceover] Let's talk
about buffer capacity.

00:00:04.083 --> 00:00:07.614
Buffer capacity is a property of a buffer

00:00:07.614 --> 00:00:11.073
and it tells you how much

00:00:11.073 --> 00:00:15.089
acid or base you can add
before the pH starts changing.

00:00:15.089 --> 00:00:17.737
Basically, as

00:00:17.737 --> 00:00:20.175
your buffer capacity goes up,

00:00:20.175 --> 00:00:21.730
which I'm going to abbreviate BC,

00:00:21.730 --> 00:00:24.493
as your buffer capacity goes up,

00:00:24.493 --> 00:00:26.189
you can

00:00:26.189 --> 00:00:27.605
add

00:00:27.605 --> 00:00:31.100
more of your acid or base

00:00:34.355 --> 00:00:37.108
before the pH starts changing a lot.

00:00:41.872 --> 00:00:43.405
That might seem like a pretty vague

00:00:43.405 --> 00:00:47.794
and qualitative definition,
so let's go through an example

00:00:47.794 --> 00:00:51.546
to see what that might look like exactly.

00:00:51.546 --> 00:00:53.330
The example we're gonna look at

00:00:53.330 --> 00:00:56.743
is going to be using
an acetic acid buffer.

00:00:56.743 --> 00:01:01.743
So acetic acid is CH three COOH.

00:01:04.197 --> 00:01:09.084
And we're gonna abbreviate
that in this talk using HA.

00:01:10.507 --> 00:01:12.758
And this is an aqueous solution

00:01:12.758 --> 00:01:15.894
and that is reversibly reacting

00:01:15.894 --> 00:01:20.885
to form H plus ion

00:01:20.885 --> 00:01:25.885
and CH three COO minus,

00:01:26.517 --> 00:01:28.283
or acetate.

00:01:28.283 --> 00:01:31.093
And so we're gonna abbreviate
acetate in this talk

00:01:31.093 --> 00:01:32.903
as A minus.

00:01:32.903 --> 00:01:35.841
So more information about
this particular buffer.

00:01:36.869 --> 00:01:41.412
Acetic acid, the Ka is equal to

00:01:41.412 --> 00:01:45.336
1.8 times 10 to the minus five.

00:01:45.336 --> 00:01:47.148
And if we take the negative log of that,

00:01:47.148 --> 00:01:49.261
that will give us the pKa,

00:01:49.261 --> 00:01:50.839
which is also useful.

00:01:50.839 --> 00:01:55.839
And the pKa of acetic acid is 4.74.

00:01:56.342 --> 00:02:00.522
So that'll tell us a lot about
the behavior of this buffer.

00:02:00.522 --> 00:02:03.215
And the last thing we need to know for

00:02:03.215 --> 00:02:05.862
predicting the behavior of our buffer

00:02:05.862 --> 00:02:10.862
is what the ratio is
between our HA and A minus.

00:02:11.993 --> 00:02:16.993
So we're going to be
looking at a buffer where

00:02:17.136 --> 00:02:20.928
A minus over HA, this ratio,

00:02:21.861 --> 00:02:26.060
is equal to 1.82.

00:02:26.060 --> 00:02:29.043
Based on this ration,
we can calculate the pH.

00:02:30.233 --> 00:02:33.831
So the initial pH of our buffer

00:02:33.831 --> 00:02:35.039
before we do anything to it,

00:02:35.039 --> 00:02:38.197
before we add any acid or we add any base,

00:02:38.197 --> 00:02:40.937
the initial pH of our
buffer we can calculate

00:02:40.937 --> 00:02:43.318
using the Henderson–Hasselbalch equation.

00:02:44.276 --> 00:02:48.524
And so that equation tells
that pH is equal to pKa

00:02:48.524 --> 00:02:53.456
plus log of A minus concentration,

00:02:53.456 --> 00:02:57.449
or acetate, over acetic
acid concentration.

00:02:57.449 --> 00:03:01.420
And if you're not 100% confident
with using this equation,

00:03:01.420 --> 00:03:02.929
or you want to know where it comes from,

00:03:02.929 --> 00:03:04.879
we actually have separate videos on it,

00:03:04.879 --> 00:03:06.644
so I would recommend checking those out.

00:03:06.644 --> 00:03:11.055
In this video, we're just going
to use this equation as is.

00:03:11.055 --> 00:03:12.820
So if we plug in our values here,

00:03:12.820 --> 00:03:17.023
we get that the initial pKa is equal to

00:03:17.023 --> 00:03:20.529
4.74, our pKa,

00:03:20.529 --> 00:03:24.174
plus log of 1.82,

00:03:24.174 --> 00:03:27.030
which is our ratio of A minus over HA.

00:03:27.030 --> 00:03:30.573
So then we know our initial pH is equal to

00:03:32.226 --> 00:03:34.658
5.00.

00:03:34.658 --> 00:03:38.072
What we're gonna do next
is we're gonna look at

00:03:38.072 --> 00:03:40.255
two different buffers,

00:03:40.255 --> 00:03:43.687
and both of them are going
to be made with acetic acid

00:03:43.687 --> 00:03:46.682
and with acetate.

00:03:46.682 --> 00:03:49.561
And we're gonna call them
buffer one and buffer two.

00:03:49.561 --> 00:03:53.270
So buffer one has a ratio

00:03:53.270 --> 00:03:56.581
of A minus NHA that is 1.82.

00:03:57.701 --> 00:04:00.441
So the pH is going to be five.

00:04:00.441 --> 00:04:03.599
And the A minus concentration,

00:04:03.599 --> 00:04:05.293
before we do anything to it,

00:04:05.293 --> 00:04:06.269
before we add anything,

00:04:06.269 --> 00:04:08.370
is 0.90 molar.

00:04:09.491 --> 00:04:11.453
And the HA concentration

00:04:12.875 --> 00:04:15.488
is going to be 0.49 molar.

00:04:16.516 --> 00:04:19.767
And our second buffer that
we're gonna compare it to

00:04:19.767 --> 00:04:23.147
we will call buffer two.

00:04:24.174 --> 00:04:29.096
And buffer two also has the
same ratio of acid to base,

00:04:29.096 --> 00:04:32.045
except this time both concentrations

00:04:32.045 --> 00:04:35.527
are gonna be 10 times
smaller than buffer one.

00:04:35.527 --> 00:04:40.527
So our A minus concentration
is 0.090 molar.

00:04:42.861 --> 00:04:47.861
And our acetic acid
concentration is 0.049 molar.

00:04:49.821 --> 00:04:51.122
And what we're gonna do
here is we're gonna see

00:04:51.122 --> 00:04:53.444
what happens to the pH of both of these,

00:04:53.444 --> 00:04:55.765
So they both start out with a pH of five,

00:04:55.765 --> 00:04:59.260
but how much do they change when we add...

00:05:00.125 --> 00:05:01.531
We're going to

00:05:02.689 --> 00:05:07.668
add 0.04 moles

00:05:07.668 --> 00:05:11.397
of a strong base, sodium hydroxide,

00:05:11.397 --> 00:05:13.782
to one liter buffer.

00:05:15.947 --> 00:05:18.222
So when you do that,

00:05:18.222 --> 00:05:19.361
well it's a buffer.

00:05:19.361 --> 00:05:22.170
We know that it's going
to resist the pH change,

00:05:22.170 --> 00:05:23.446
but what exactly is happening?

00:05:23.446 --> 00:05:26.768
What's the reaction that
goes on when you add this in?

00:05:26.768 --> 00:05:28.881
So when you add your strong base,

00:05:28.881 --> 00:05:32.526
which is going to
dissociate to form OH minus,

00:05:32.526 --> 00:05:36.613
it's going to react with
the acid in your buffer.

00:05:36.613 --> 00:05:39.351
So our acid is

00:05:39.351 --> 00:05:43.357
CH three COOH.

00:05:44.779 --> 00:05:48.168
And we always assume that a strong base

00:05:48.168 --> 00:05:52.324
is going to react irreversibly.

00:05:52.324 --> 00:05:55.194
So one-sided arrow with a weak acid.

00:05:56.128 --> 00:05:58.542
So what happens is this proton

00:05:58.542 --> 00:06:00.656
is going to react with OH
minus and we're gonna get

00:06:00.656 --> 00:06:02.862
H two O, or water.

00:06:02.862 --> 00:06:06.846
And we're gonna make CH three COO minus.

00:06:08.012 --> 00:06:09.555
So that's our base.

00:06:10.698 --> 00:06:12.766
And so we can see that the hydroxide

00:06:12.766 --> 00:06:16.295
is going to react one to
one with our weak acid

00:06:16.295 --> 00:06:20.196
and it's gonna produce one
equivalent of our base.

00:06:20.196 --> 00:06:21.727
So now we can use this information

00:06:21.727 --> 00:06:23.864
to calculate what happens to our buffer

00:06:23.864 --> 00:06:27.289
when we add .04 moles of sodium hydroxide.

00:06:28.571 --> 00:06:30.917
What's gonna happen is it's gonna react

00:06:30.917 --> 00:06:32.937
with our acid.

00:06:32.937 --> 00:06:36.043
Which means the concentration
of the acetic acid

00:06:36.043 --> 00:06:38.610
is gonna go down by .04 molar.

00:06:40.727 --> 00:06:43.004
Our new concentration after it's reacted

00:06:43.004 --> 00:06:48.004
is gonna be 0.45 molar.

00:06:49.946 --> 00:06:52.174
Sorry, I tried to do that in my head.

00:06:53.309 --> 00:06:58.301
And what happens to our
concentration of the base?

00:06:58.301 --> 00:07:02.410
Remember that when the acid
reacts with sodium hydroxide

00:07:02.410 --> 00:07:04.630
we actually make more base,

00:07:04.630 --> 00:07:09.630
So we have to adjust the
concentration of A minus upward.

00:07:10.253 --> 00:07:14.415
It actually makes .04 molar acetate

00:07:14.415 --> 00:07:16.267
when this reaction happens.

00:07:16.267 --> 00:07:19.750
So our new concentration
of our acetate ion

00:07:19.750 --> 00:07:23.408
is 0.94 molar.

00:07:24.319 --> 00:07:26.409
So these are our final concentrations.

00:07:26.409 --> 00:07:29.775
We can no plug them back in to the

00:07:29.775 --> 00:07:33.003
Henderson–Hasselbalch
equation to get the pH.

00:07:33.003 --> 00:07:35.488
So the pH of buffer one,

00:07:35.488 --> 00:07:37.920
I'll put a little subscript one,

00:07:37.920 --> 00:07:41.392
is going to be 4.74, the pKa,

00:07:42.230 --> 00:07:47.230
plus log of 0.94 molar

00:07:48.410 --> 00:07:50.970
divided by 0.45 molar.

00:07:52.532 --> 00:07:54.761
And if we plug that into our calculator,

00:07:54.761 --> 00:07:58.012
we get that the pH after
we add that hydroxide

00:07:58.012 --> 00:07:59.934
is 5.06.

00:07:59.934 --> 00:08:01.117
So it's a little bit higher.

00:08:01.117 --> 00:08:03.208
It went up by .06.

00:08:03.208 --> 00:08:04.717
That makes sense.

00:08:04.717 --> 00:08:06.993
We do imagine it would be higher.

00:08:06.993 --> 00:08:08.664
If you add base, it becomes more basic

00:08:08.664 --> 00:08:10.475
and the pH should go up.

00:08:10.475 --> 00:08:11.799
But it didn't change a whole lot,

00:08:11.799 --> 00:08:14.209
because it's a buffer.

00:08:14.209 --> 00:08:17.785
Now let's compare that with
what happens to buffer two.

00:08:17.785 --> 00:08:22.785
We add .04 mole sodium
hydroxide to buffer two.

00:08:22.846 --> 00:08:27.457
Our acetic acid concentration
is gonna go down by .04 molar.

00:08:28.716 --> 00:08:33.598
So our new concentration
of HA is .009 molar.

00:08:33.598 --> 00:08:37.568
By the same reaction,
our base concentration

00:08:37.568 --> 00:08:39.659
is gonna go up,

00:08:39.659 --> 00:08:43.526
since when the acid reacts,
it makes more conjugate base.

00:08:43.526 --> 00:08:48.077
So we're gonna have to add 0.04 molar

00:08:48.077 --> 00:08:50.260
to A minus concentration.

00:08:50.260 --> 00:08:54.012
And so the new concentration
is going to be 0.13 molar.

00:08:55.178 --> 00:08:58.290
And we can plug those concentrations into

00:08:58.290 --> 00:09:00.682
the Henderson–Hasselbalch equation

00:09:00.682 --> 00:09:04.606
and we get that pH of our buffer two

00:09:04.606 --> 00:09:08.855
is equal to 4.74, our pKa,

00:09:10.567 --> 00:09:14.674
plus log of 0.13 molar

00:09:14.674 --> 00:09:16.294
divided by

00:09:16.294 --> 00:09:20.813
0.009 molar.

00:09:20.813 --> 00:09:24.718
And so that would be, if we
plugged this into our calculator

00:09:24.718 --> 00:09:27.481
this is equal to 1.16.

00:09:27.481 --> 00:09:30.104
So if we add that together with our pKa,

00:09:30.104 --> 00:09:33.948
we get that the new pH is 5.90 molar.

00:09:34.977 --> 00:09:36.764
So the pH changed a whole lot more

00:09:36.764 --> 00:09:39.136
for buffer two than for buffer one.

00:09:39.136 --> 00:09:43.242
The pH went up by .9 instead of by .06.

00:09:43.242 --> 00:09:46.284
So if we were comparing
the buffer capacities

00:09:46.284 --> 00:09:48.005
of both of these buffers,

00:09:48.005 --> 00:09:50.467
we would say that

00:09:50.467 --> 00:09:52.046
buffer one,

00:09:52.046 --> 00:09:55.226
well, the buffer changes less when you add

00:09:55.226 --> 00:09:57.108
the same amount of acid or base.

00:09:57.108 --> 00:10:00.858
So buffer one has the
higher buffer capacity.

00:10:03.115 --> 00:10:06.621
Since you don't want these concentrations,

00:10:06.621 --> 00:10:08.850
A minus and HA, to be too low,

00:10:08.850 --> 00:10:12.182
the general rule of thumb is that you want

00:10:13.930 --> 00:10:16.274
your concentrations of HA

00:10:16.274 --> 00:10:20.286
and A minus between

00:10:20.286 --> 00:10:24.595
0.10 molar and

00:10:26.086 --> 00:10:28.352
1.0 molar.

00:10:31.050 --> 00:10:33.303
And by following that rule of thumb,

00:10:33.303 --> 00:10:34.835
you can make a buffer

00:10:34.835 --> 00:10:36.832
where you don't have to worry too much

00:10:36.832 --> 00:10:39.270
about adding too much acid or base

00:10:39.270 --> 00:10:41.011
before you pH changes,

00:10:41.011 --> 00:10:42.771
since that's usually why
you're making a buffer

00:10:42.771 --> 00:10:43.657
in the first place.


Biology overview

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.000
- [Voiceover] I would like to welcome you

00:00:02.000 --> 00:00:06.274
to Biology at Khan Academy.

00:00:06.274 --> 00:00:08.318
And biology, as you might now,

00:00:08.318 --> 00:00:10.789
is the study of life.

00:00:10.789 --> 00:00:13.526
And I can't really imagine
anything more interesting

00:00:13.526 --> 00:00:16.124
than the study of life.

00:00:16.124 --> 00:00:17.348
And when I say "life,"

00:00:17.348 --> 00:00:20.331
I'm not just talking
about us, human beings.

00:00:20.331 --> 00:00:21.810
I'm talking about all animals.

00:00:21.810 --> 00:00:22.921
I'm talking about plants.

00:00:22.921 --> 00:00:24.258
I'm talking about bacteria.

00:00:24.258 --> 00:00:26.464
And it really is fascinating.

00:00:26.464 --> 00:00:30.929
How do we start off with
inanimate molecules and atoms?

00:00:30.929 --> 00:00:33.162
You know, this right here
is a molecule of DNA.

00:00:33.162 --> 00:00:34.825
How do we start with things like that,

00:00:34.825 --> 00:00:38.217
and we get the complexity
of living things?

00:00:38.217 --> 00:00:39.181
And you might be saying,

00:00:39.181 --> 00:00:41.703
well, what makes something living?

00:00:41.703 --> 00:00:46.669
Well, living things convert
energy from one form to another.

00:00:46.669 --> 00:00:49.758
They use that energy to grow.

00:00:49.758 --> 00:00:52.010
They use that energy to change.

00:00:52.010 --> 00:00:54.390
And I guess growth is a form of change.

00:00:54.390 --> 00:00:58.356
They use that energy to reproduce.

00:00:58.356 --> 00:01:01.357
And these are all, in and of themselves,

00:01:01.357 --> 00:01:02.434
fascinating questions.

00:01:02.434 --> 00:01:03.740
How do they do this?

00:01:03.740 --> 00:01:04.936
You know, we look around us.

00:01:04.936 --> 00:01:06.782
How do we, you know, eat a muffin?

00:01:06.782 --> 00:01:09.245
And how does that allow us to move around

00:01:09.245 --> 00:01:11.115
and think and do all the things we do?

00:01:11.115 --> 00:01:13.457
Where did the energy from
that muffin come from?

00:01:13.457 --> 00:01:17.236
How are we similar to
a plant or an insect?

00:01:17.236 --> 00:01:21.803
And we are eerily or strangely similar

00:01:21.803 --> 00:01:22.956
to these things.

00:01:22.956 --> 00:01:25.648
We actually have a lot more
in common with, you know,

00:01:25.648 --> 00:01:27.091
that tree outside your window,

00:01:27.091 --> 00:01:30.105
or that insect, that bee,
that might be buzzing around,

00:01:30.105 --> 00:01:31.555
than you realize.

00:01:31.555 --> 00:01:32.689
Even with the bacteria

00:01:32.689 --> 00:01:35.812
that you can only even see

00:01:35.812 --> 00:01:37.809
at a microscopic level.

00:01:37.809 --> 00:01:39.686
In fact, we have so much bacteria

00:01:39.686 --> 00:01:41.930
as part of what makes us, us.

00:01:41.930 --> 00:01:43.672
So these are fascinating questions.

00:01:43.672 --> 00:01:47.099
How did life even emerge?

00:01:47.099 --> 00:01:48.989
And so over the course of what you see

00:01:48.989 --> 00:01:50.776
in Biology on Khan Academy,

00:01:50.776 --> 00:01:53.567
we're going to answer these fundamental,

00:01:53.567 --> 00:01:55.132
fascinating questions.

00:01:55.132 --> 00:01:58.169
We're going to think
about things like energy

00:01:58.169 --> 00:02:00.915
and the role of energy in life.

00:02:00.915 --> 00:02:04.609
We're going to thing about
important molecules in biology.

00:02:04.609 --> 00:02:06.513
And perhaps most importantly,

00:02:06.513 --> 00:02:11.498
DNA and its role in reproduction
and containing information.

00:02:11.498 --> 00:02:14.232
And we're going to study cells,

00:02:14.232 --> 00:02:17.587
which are the basic
building block of life.

00:02:17.587 --> 00:02:19.309
And as we'll see, even
though we view cells

00:02:19.309 --> 00:02:21.923
as these super, super small, small things,

00:02:21.923 --> 00:02:25.202
cells in and of themselves
are incredibly complex.

00:02:25.202 --> 00:02:27.358
And if you compare them
to an atomic scale,

00:02:27.358 --> 00:02:28.852
they're quite large.

00:02:28.852 --> 00:02:31.899
In fact, this entire blue
background that I have there,

00:02:31.899 --> 00:02:34.216
that's the surface of an immune cell.

00:02:34.216 --> 00:02:37.095
And what you see here emerging from it,

00:02:37.095 --> 00:02:39.972
these little yellow things.

00:02:39.972 --> 00:02:44.635
These are HIV viruses,
emerging from an immune cell.

00:02:44.635 --> 00:02:46.091
So even though you imagine cells

00:02:46.091 --> 00:02:48.497
as these very, very
small microscopic things,

00:02:48.497 --> 00:02:50.178
this incredible complexity.

00:02:50.178 --> 00:02:51.102
Even viruses.

00:02:51.102 --> 00:02:53.123
Viruses are one of
these fascinating things

00:02:53.123 --> 00:02:56.687
that kind of are right on the
edge between life and nonlife.

00:02:56.687 --> 00:03:00.513
They definitely reproduce,
and they definitely evolve.

00:03:00.513 --> 00:03:03.403
But they don't necessarily
have a metabolism.

00:03:03.403 --> 00:03:04.729
We'll learn a lot more about that.

00:03:04.729 --> 00:03:07.104
They don't necessarily
use energy and growth

00:03:07.104 --> 00:03:09.738
in the same way that we
would associate with life.

00:03:09.738 --> 00:03:12.419
And then perhaps one of the
biggest questions of all is

00:03:12.419 --> 00:03:13.924
how did life come about?

00:03:13.924 --> 00:03:15.549
And we will study that

00:03:15.549 --> 00:03:19.815
as we look at evolution
and natural selection.

00:03:19.815 --> 00:03:23.800
So welcome to Khan
Academy's Biology section.

00:03:23.800 --> 00:03:26.767
I think you're going
to find it fascinating.

00:03:26.767 --> 00:03:29.625
You're going to realize
that biology, in some way,

00:03:29.625 --> 00:03:32.904
is the most complex of the sciences.

00:03:32.904 --> 00:03:35.786
And in a lot of ways, the one
that we understand the least.

00:03:35.786 --> 00:03:39.470
It's going to be built
on top of chemistry,

00:03:39.470 --> 00:03:43.766
which in turn is built on top of physics,

00:03:44.986 --> 00:03:49.478
which in turn is built
on top of mathematics.

00:03:51.748 --> 00:03:53.674
And biology is one of our

00:03:53.674 --> 00:03:55.288
Frankly, even in the last hundred years,

00:03:55.288 --> 00:03:57.298
we're just starting to scratch the surface

00:03:57.298 --> 00:03:58.250
of understanding it.

00:03:58.250 --> 00:03:59.901
But what's really exciting

00:03:59.901 --> 00:04:02.384
is where the field of biology is going.

00:04:02.384 --> 00:04:04.422
As we understand things at a deeper level,

00:04:04.422 --> 00:04:05.796
at a molecular level,

00:04:05.796 --> 00:04:07.037
we're going to start thinking about

00:04:07.037 --> 00:04:09.365
how can we even do things
like engineer biology,

00:04:09.365 --> 00:04:10.860
or affect the world around us?

00:04:10.860 --> 00:04:12.574
It's going to raise all sorts of

00:04:12.574 --> 00:04:15.486
fascinating and deep
and ethical questions.

00:04:15.486 --> 00:04:17.160
So, hopefully you enjoy this.

00:04:17.160 --> 00:04:19.236
Biology is one of the most, arguably,

00:04:19.236 --> 00:04:21.342
maybe the most fascinating subject of all.

00:04:21.342 --> 00:04:22.015
I don't want to offend

00:04:22.015 --> 00:04:23.474
the chemists and the physicists out there.

00:04:23.474 --> 00:04:25.430
I actually find those
quite fascinating as well.

00:04:25.430 --> 00:04:28.144
But we're going to answer, or
attempt to start to answer,

00:04:28.144 --> 00:04:32.243
some of the most fundamental
questions of our existence.


Transforming exponential graphs (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.447 --> 00:00:01.886
- [Voiceover] We're told the graph of y

00:00:01.886 --> 00:00:03.650
equals two to the x is shown below,

00:00:03.650 --> 00:00:04.766
so that's the graph.

00:00:04.766 --> 00:00:06.205
It's an exponential function.

00:00:06.205 --> 00:00:07.748
Which of the following is a graph of y

00:00:07.748 --> 00:00:09.874
is equal to negative one times two

00:00:09.874 --> 00:00:13.170
to the x plus three plus four?

00:00:13.170 --> 00:00:16.223
They give us four choices down here.

00:00:16.223 --> 00:00:19.184
Before we even look
closely at those choices,

00:00:20.564 --> 00:00:22.817
let's just think about what this

00:00:22.817 --> 00:00:25.187
would look like if it was
transformed into that.

00:00:25.187 --> 00:00:27.264
You might notice that what we have here,

00:00:27.264 --> 00:00:29.179
this y that we wanna find the graph of,

00:00:29.179 --> 00:00:31.408
is a transformation of this original one.

00:00:31.408 --> 00:00:33.000
How do we transform it?

00:00:33.000 --> 00:00:35.345
We've replaced x with x plus three.

00:00:35.345 --> 00:00:37.760
Then we multiplied that by negative one,

00:00:37.760 --> 00:00:39.222
and then we add four.

00:00:39.222 --> 00:00:40.860
Let's take it step by step.

00:00:40.860 --> 00:00:42.682
This is y equals two to the x.

00:00:42.682 --> 00:00:45.202
What I wanna do next is let's graph y

00:00:45.202 --> 00:00:47.522
is equal to two to the x

00:00:47.522 --> 00:00:49.739
plus three power.

00:00:49.739 --> 00:00:52.281
If you replace x with x plus three,

00:00:52.281 --> 00:00:54.127
you're going to shift the graph

00:00:56.111 --> 00:00:58.352
to the left by three.

00:00:58.352 --> 00:01:00.594
That might be a little
bit counterintuitive,

00:01:00.594 --> 00:01:02.229
but when we actually
think about some points,

00:01:02.229 --> 00:01:04.087
it'll hopefully make some sense here.

00:01:04.087 --> 00:01:06.409
For example, over in our original graph,

00:01:06.409 --> 00:01:08.475
when x is equal to zero,

00:01:08.475 --> 00:01:10.750
y is equal to one.

00:01:13.061 --> 00:01:15.780
How do we get y equal
one for our new graph,

00:01:15.780 --> 00:01:17.441
for this thing right over here?

00:01:17.441 --> 00:01:20.076
To get y equals one
here, the exponent here

00:01:20.076 --> 00:01:22.119
still has to be zero, so
that's going to happen

00:01:22.119 --> 00:01:24.232
at x equals negative three.

00:01:24.232 --> 00:01:26.717
That's going to happen at
x equals negative three.

00:01:26.717 --> 00:01:28.620
Y is equal to one.

00:01:28.620 --> 00:01:31.873
Notice, we shifted to the left

00:01:31.873 --> 00:01:33.476
by three.

00:01:33.476 --> 00:01:35.115
Likewise, in our original graph,

00:01:35.115 --> 00:01:38.493
when x is two, y is four.

00:01:38.493 --> 00:01:41.511
How do we get y equals four in this thing

00:01:41.511 --> 00:01:43.009
right over here?

00:01:43.009 --> 00:01:46.664
For y to be equal to
four, this exponent here

00:01:46.664 --> 00:01:49.950
needs to be equal to two,
and so for this exponent

00:01:49.950 --> 00:01:53.338
to be equal to two, 'cause
two squared is four,

00:01:53.338 --> 00:01:54.895
for this exponent to be equal to two,

00:01:54.895 --> 00:01:57.100
x is going to be equal to negative one.

00:01:57.100 --> 00:01:59.712
When x is equal to negative one,

00:01:59.712 --> 00:02:01.523
y is equal to four.

00:02:01.523 --> 00:02:03.218
When x is equal to negative one,

00:02:03.218 --> 00:02:04.542
y is equal to four.

00:02:04.542 --> 00:02:07.023
Notice we shifted to the left

00:02:08.678 --> 00:02:10.140
by three.

00:02:10.140 --> 00:02:12.358
So this thing, which isn't our final graph

00:02:12.358 --> 00:02:14.308
that we're looking for, is gonna look

00:02:18.824 --> 00:02:20.745
something like,

00:02:20.745 --> 00:02:24.704
like that, which shifted,
it's y equals two to the x,

00:02:24.704 --> 00:02:27.065
shifted to the left by three.

00:02:27.065 --> 00:02:29.456
Now let's figure out what the graph of,

00:02:29.456 --> 00:02:31.766
now let's multiply this
expression times negative one.

00:02:31.766 --> 00:02:33.647
Notice we're slowly building up

00:02:33.647 --> 00:02:35.017
to our goal.

00:02:35.017 --> 00:02:36.642
Now let's figure out the graph of y

00:02:36.642 --> 00:02:39.056
is equal to negative one times two

00:02:39.056 --> 00:02:42.098
to the x plus three.

00:02:42.098 --> 00:02:44.745
Here, when y equals two
to the x plus three,

00:02:44.745 --> 00:02:46.417
if we multiply that times negative one,

00:02:46.417 --> 00:02:47.894
whatever y we had, we're gonna have

00:02:47.894 --> 00:02:49.298
the negative of that.

00:02:49.298 --> 00:02:50.726
Instead of when x is
equal to negative three

00:02:50.726 --> 00:02:52.537
having positive one, when
x equals negative three,

00:02:52.537 --> 00:02:53.733
you're gonna have negative one.

00:02:53.733 --> 00:02:55.474
We multiplied by negative one.

00:02:55.474 --> 00:02:57.576
When x is equal to negative
one, instead of having four,

00:02:57.576 --> 00:03:00.525
you're going to have negative four.

00:03:00.525 --> 00:03:03.241
Our graph is gonna be flipped over,

00:03:03.241 --> 00:03:06.318
it's flipped over the x axis.

00:03:06.318 --> 00:03:08.494
It's going to look something

00:03:10.984 --> 00:03:12.157
like this.

00:03:12.157 --> 00:03:14.130
This is not a perfect
drawing, but it'll give us

00:03:14.130 --> 00:03:15.605
a sense of things, and we can look

00:03:15.605 --> 00:03:18.182
at which of these graphs match up to that.

00:03:18.182 --> 00:03:22.048
Then finally, we wanna
add that four there.

00:03:22.048 --> 00:03:23.835
We wanna figure out the graph of y

00:03:23.835 --> 00:03:26.761
equals negative one times two to the x

00:03:26.761 --> 00:03:29.699
plus three plus four.

00:03:29.699 --> 00:03:31.358
We wanna take what we just had

00:03:31.358 --> 00:03:33.401
and shift it up by four.

00:03:33.401 --> 00:03:36.895
Instead of this being a
negative one right over here,

00:03:36.895 --> 00:03:40.413
this is going to be a negative one

00:03:40.413 --> 00:03:43.444
plus four is three.

00:03:43.444 --> 00:03:44.964
Instead of this being a negative four,

00:03:44.964 --> 00:03:48.644
negative four plus four is zero.

00:03:48.644 --> 00:03:50.687
Instead of our horizontal asymptote

00:03:50.687 --> 00:03:53.357
being at y equals zero,
our horizontal asymptote

00:03:53.357 --> 00:03:55.698
is going be at y equals four.

00:03:55.698 --> 00:03:58.322
It's gonna look like, let me draw,

00:03:58.322 --> 00:03:59.981
I can do a better job than that.

00:03:59.981 --> 00:04:01.456
Our horizontal asymptote is gonna

00:04:01.456 --> 00:04:03.522
be right over there, so our graph is going

00:04:03.522 --> 00:04:05.031
to look something like

00:04:08.565 --> 00:04:09.587
this.

00:04:09.587 --> 00:04:12.524
We just shifted that red graph up by four.

00:04:15.181 --> 00:04:16.563
Shifted it up by four.

00:04:16.563 --> 00:04:19.025
We have a horizontal
asymptote at y equals four.

00:04:19.025 --> 00:04:22.507
Let's look at which of
these choices match that.

00:04:22.507 --> 00:04:25.037
Choice A right over here
has a horizontal asymptote

00:04:25.037 --> 00:04:28.590
at y equals four, but it is shifted

00:04:28.590 --> 00:04:31.434
on the horizontal
direction inappropriately.

00:04:31.434 --> 00:04:32.990
In fact, it looks like it might have not

00:04:32.990 --> 00:04:35.033
been shifted to the left.

00:04:35.033 --> 00:04:36.972
We can rule this one out.

00:04:36.972 --> 00:04:38.969
Let's rule that one out.

00:04:38.969 --> 00:04:40.958
This one over here,

00:04:43.355 --> 00:04:46.792
this one approaches our
asymptote as x increases,

00:04:46.792 --> 00:04:48.812
so that's not right.

00:04:48.812 --> 00:04:50.287
It should approach our asymptote

00:04:50.287 --> 00:04:53.177
as x decreases, so we ruled that one out.

00:04:53.177 --> 00:04:57.100
Choice C looks like what we just graphed.

00:04:57.100 --> 00:05:00.270
Horizontal asymptote at x equals four.

00:05:00.270 --> 00:05:04.287
When x equals negative
three, y is equal to three.

00:05:04.287 --> 00:05:05.772
That's what we got.

00:05:05.772 --> 00:05:07.490
When x is equal to negative one,

00:05:07.490 --> 00:05:09.394
y is equal to zero.

00:05:09.394 --> 00:05:10.811
This looks right.

00:05:10.811 --> 00:05:13.435
You could even try those points out.

00:05:13.435 --> 00:05:15.582
We like choice C.

00:05:15.582 --> 00:05:17.741
D is clearly off.


Graphing exponential functions

WEBVTT
Kind: captions
Language: en

00:00:00.248 --> 00:00:02.280
- [Voiceover] We're told, use
the interactive graph below

00:00:02.280 --> 00:00:05.009
to sketch a graph of y
is equal to negative two,

00:00:05.009 --> 00:00:08.109
times three to the x, plus five.

00:00:08.109 --> 00:00:10.861
And so this is clearly
an exponential function

00:00:10.861 --> 00:00:11.522
right over here.

00:00:11.522 --> 00:00:14.200
Let's think about the behavior as x is,

00:00:14.200 --> 00:00:15.742
when x is very negative,

00:00:15.742 --> 00:00:17.693
or when x is very positive.

00:00:17.693 --> 00:00:21.420
When x is very negative, three
to a very negative number

00:00:21.420 --> 00:00:24.730
like let's say you had three
to the negative 3rd power,

00:00:24.730 --> 00:00:28.958
that would be 1/27th, or three
to the negative 4th power,

00:00:28.958 --> 00:00:31.694
that would be 1/81st.

00:00:31.694 --> 00:00:34.375
So this is going to get smaller,
and smaller, and smaller.

00:00:34.375 --> 00:00:38.753
It's going to approach zero
as x becomes more negative.

00:00:38.753 --> 00:00:41.398
And since this is approaching zero,

00:00:41.398 --> 00:00:43.952
this whole thing right over
here is going to approach zero.

00:00:43.952 --> 00:00:46.076
And so this whole expression is,

00:00:46.076 --> 00:00:48.074
if this first part is approaching zero,

00:00:48.074 --> 00:00:50.616
then this whole expression
is going to approach five.

00:00:50.616 --> 00:00:52.417
So we're going to have
a horizontal asymptote

00:00:52.417 --> 00:00:54.899
that we're going to approach
as we go to the left.

00:00:54.899 --> 00:00:58.080
As x gets more and more negative,

00:00:58.080 --> 00:01:01.575
we're going to approach positive five.

00:01:01.575 --> 00:01:04.037
And then as x gets larger,
and larger, and larger,

00:01:04.037 --> 00:01:06.673
three to the x is growing exponentially.

00:01:06.673 --> 00:01:08.656
But then we're multiplying
it times negative two,

00:01:08.656 --> 00:01:10.932
so it's going to become more,
and more, and more negative.

00:01:10.932 --> 00:01:12.591
And then we add a five.

00:01:12.591 --> 00:01:16.957
And so what we have here,
well this looks like a line.

00:01:16.957 --> 00:01:19.012
We want to graph an exponential.

00:01:19.012 --> 00:01:22.598
So let's go pick the
exponential in terms of x.

00:01:22.598 --> 00:01:23.771
There you have it.

00:01:23.771 --> 00:01:25.619
And so we can move three things.

00:01:25.619 --> 00:01:27.528
We can move this point,
it doesn't even need

00:01:27.528 --> 00:01:28.689
to just be the y-intercept,

00:01:28.689 --> 00:01:30.940
although that's a convenient
thing to figure it out.

00:01:30.940 --> 00:01:32.840
We can move this point here.

00:01:32.840 --> 00:01:34.860
And we can move the asymptote.

00:01:34.860 --> 00:01:37.578
And maybe the asymptote's
the first interesting thing.

00:01:37.578 --> 00:01:40.384
We said as x becomes
more, and more, and more,

00:01:40.384 --> 00:01:43.095
and more negative, y is
going to approach five.

00:01:43.095 --> 00:01:45.337
So let me put this up here.

00:01:45.337 --> 00:01:46.858
So that's our asymptote.

00:01:46.858 --> 00:01:48.237
It doesn't look like it quite yet,

00:01:48.237 --> 00:01:50.491
but when we try out some values for x

00:01:50.491 --> 00:01:51.536
and the corresponding y's,

00:01:51.536 --> 00:01:53.369
and we move these points accordingly,

00:01:53.369 --> 00:01:56.272
hopefully our exponential
is going to look right.

00:01:56.272 --> 00:01:58.381
So let's think about, let's
pick some convenient x's.

00:01:58.381 --> 00:02:00.448
So let's think about
when x is equal to zero.

00:02:00.448 --> 00:02:04.045
If x is equal to zero, three
to the zeroth power is one.

00:02:04.045 --> 00:02:06.461
Negative two times one is negative two,

00:02:06.461 --> 00:02:09.142
plus three is three.

00:02:09.142 --> 00:02:12.427
So when x is equal to zero, y is three.

00:02:12.427 --> 00:02:15.407
And let's think about
when x is equal to one,

00:02:15.407 --> 00:02:17.864
and I'm just picking that
'cause it's easy to compute.

00:02:17.864 --> 00:02:21.845
Three to the first power is
three, times negative two

00:02:21.845 --> 00:02:26.056
is negative six, plus
five is negative one.

00:02:26.056 --> 00:02:30.166
So when x is one, y is negative one.

00:02:30.166 --> 00:02:32.104
And so let's see, does
this, is this consistent

00:02:32.104 --> 00:02:33.846
with what we just described?

00:02:33.846 --> 00:02:37.178
When x is very negative,
we should be approaching,

00:02:37.178 --> 00:02:39.488
we should be approaching positive five,

00:02:39.488 --> 00:02:40.567
and that looks like the case.

00:02:40.567 --> 00:02:42.925
As we move to the left, we're
getting closer, and closer,

00:02:42.925 --> 00:02:45.490
and closer to five, in fact,
it looks like they overlap,

00:02:45.490 --> 00:02:46.514
but it's really, we're
just getting closer,

00:02:46.514 --> 00:02:49.579
and closer, and closer 'cause this term,

00:02:49.579 --> 00:02:50.972
this term right over
here is getting smaller,

00:02:50.972 --> 00:02:52.675
and smaller, and smaller
as x becomes more,

00:02:52.675 --> 00:02:54.148
and more, and more negative.

00:02:54.148 --> 00:02:56.773
But then as x becomes
more and more positive,

00:02:56.773 --> 00:02:58.687
this term becomes really negative

00:02:58.687 --> 00:03:00.684
'cause we're multiplying
it times a negative two,

00:03:00.684 --> 00:03:02.588
and we see that it
becomes really negative,

00:03:02.588 --> 00:03:05.492
so I feel pretty good about
what we've just graphed.

00:03:05.492 --> 00:03:07.313
We've graphed the horizontal
asymptote, it makes sense,

00:03:07.313 --> 00:03:09.484
and we've picked two
points that sit on this,

00:03:09.484 --> 00:03:11.341
on the graph of this exponential.

00:03:11.341 --> 00:03:14.072
So I can check my answer,
and we got it right.