Weak acid equilibria

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:01.520
- [Instructor] Before
we get into the topic

00:00:01.520 --> 00:00:06.520
of weak acid equilibria, let's
look at a strong acid first.

00:00:06.710 --> 00:00:10.380
So let's say that HA is a strong
acid and reacts with water

00:00:10.380 --> 00:00:13.800
to produce the hydronium ion and A-minus,

00:00:13.800 --> 00:00:16.140
the conjugate base to HA.

00:00:16.140 --> 00:00:18.840
Technically, the reaction
comes to an equilibrium.

00:00:18.840 --> 00:00:22.310
However, the equilibrium
favors the products so much

00:00:22.310 --> 00:00:24.110
that we don't draw an equilibrium arrow.

00:00:24.110 --> 00:00:26.410
We simply draw an arrow going to the right

00:00:26.410 --> 00:00:28.850
indicating at equilibrium,
we have essentially

00:00:28.850 --> 00:00:32.690
all H30-plus and A-minus, and no HA.

00:00:32.690 --> 00:00:37.690
So we assume that a strong
acid ionizes 100% in solution,

00:00:38.020 --> 00:00:41.600
and we can see that in
our particulate diagrams.

00:00:41.600 --> 00:00:43.820
And keep in mind that
these particulate diagrams

00:00:43.820 --> 00:00:46.430
are not meant to represent
the entire solution,

00:00:46.430 --> 00:00:48.730
just a small portion of
it so we can get an idea

00:00:48.730 --> 00:00:51.380
about what's happening
in the entire solution.

00:00:51.380 --> 00:00:54.310
We can think about the
particulate diagram on the left

00:00:54.310 --> 00:00:57.773
representing the initial
concentration of our acid, HA.

00:00:58.790 --> 00:01:03.060
So there are five HA particles in solution

00:01:03.060 --> 00:01:06.160
and also five water particles.

00:01:06.160 --> 00:01:08.540
Since the reaction goes to completion,

00:01:08.540 --> 00:01:12.060
the HA particles are going to
react with the water particles

00:01:12.060 --> 00:01:17.060
to produce five hydronium,
H3O-plus, and five A-minus.

00:01:17.100 --> 00:01:20.220
We can see that in our
second particulate diagram,

00:01:20.220 --> 00:01:22.920
which represents the equilibrium mixture

00:01:22.920 --> 00:01:25.670
and there are five hydroniums present

00:01:25.670 --> 00:01:28.786
and also five A-minuses.

00:01:28.786 --> 00:01:31.630
We can write the equilibrium
constant expression

00:01:31.630 --> 00:01:34.430
for this reaction from
the balanced equation.

00:01:34.430 --> 00:01:36.920
we look at our products and
say it'd be the concentration

00:01:36.920 --> 00:01:39.450
of H3O-plus, and since
there's a coefficient of one,

00:01:39.450 --> 00:01:41.110
it'd be the concentration of H30-plus

00:01:41.110 --> 00:01:43.550
raised to the first power,
times the concentration

00:01:43.550 --> 00:01:46.700
of A-minus also raised to
the first power, divided by

00:01:46.700 --> 00:01:49.810
the concentration of HA
raised to the first power,

00:01:49.810 --> 00:01:51.590
and pure water is left out

00:01:51.590 --> 00:01:54.170
of the equilibrium constant expression.

00:01:54.170 --> 00:01:57.180
So that's equal to the
equilibrium constant K,

00:01:57.180 --> 00:02:00.180
and since this shows the
ionization of an acid,

00:02:00.180 --> 00:02:02.330
we could call this Ka value,

00:02:02.330 --> 00:02:04.400
the acid ionization constant,

00:02:04.400 --> 00:02:07.160
or the acid dissociation constant.

00:02:07.160 --> 00:02:09.840
Since the equilibrium
lies so far to the right

00:02:09.840 --> 00:02:13.100
for a strong acid, at
equilibrium, there's going to be

00:02:13.100 --> 00:02:17.180
essentially all products and
very, very little reactants.

00:02:17.180 --> 00:02:20.560
That means the Ka value for a strong acid

00:02:20.560 --> 00:02:23.310
will be much greater than one.

00:02:23.310 --> 00:02:25.360
Now let's look at a weak acid.

00:02:25.360 --> 00:02:27.350
We're gonna use the same
general equation here.

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HA plus H2O yields H3O-plus plus A-minus.

00:02:31.890 --> 00:02:35.090
However, if HA is a generic weak acid,

00:02:35.090 --> 00:02:38.260
we're going to include an
equilibrium arrow here.

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The equilibrium arrow needs to be included

00:02:40.450 --> 00:02:43.560
because weak acids only partially ionize,

00:02:43.560 --> 00:02:47.340
and we can see that in
the particulate diagrams.

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In the particulate diagram on the left,

00:02:49.280 --> 00:02:52.220
this is supposed to represent
the initial concentration

00:02:52.220 --> 00:02:56.686
of our weak acid HA and there
are five particles of HA

00:02:56.686 --> 00:02:59.640
and five particles of water.

00:02:59.640 --> 00:03:02.430
Since weak acids only
partially ionize, let's say

00:03:02.430 --> 00:03:07.430
that only one HA particle
reacts with one water molecule

00:03:07.550 --> 00:03:12.470
to produce one hydronium
ion and one A-minus anion.

00:03:12.470 --> 00:03:17.060
So if one HA particle reacts
with one H2O particle,

00:03:17.060 --> 00:03:21.230
that gives us one hydronium,
H30-plus and one A-minus.

00:03:21.230 --> 00:03:25.780
So out of the original five HA particles,

00:03:25.780 --> 00:03:30.640
only one of them ionized,
leaving four HA particles

00:03:30.640 --> 00:03:34.400
still in solution at
equilibrium, therefore HA

00:03:34.400 --> 00:03:38.500
only partially ionized and is a weak acid,

00:03:38.500 --> 00:03:41.330
the equilibrium constant
expression for a weak acid

00:03:41.330 --> 00:03:43.530
looks the same as it
did for our strong acid,

00:03:43.530 --> 00:03:46.050
because we use the same general equation.

00:03:46.050 --> 00:03:49.610
However, since a weak acid
only partially ionizes

00:03:49.610 --> 00:03:52.270
at equilibrium, there's
gonna be a small amount

00:03:52.270 --> 00:03:56.200
of products and a large
amount of reactants.

00:03:56.200 --> 00:03:59.516
Therefore, the Ka value for a weak acid

00:03:59.516 --> 00:04:02.460
will be less than one.

00:04:02.460 --> 00:04:03.950
Let's look at some Ka values

00:04:03.950 --> 00:04:06.970
for some weak acids at 25 degrees Celsius.

00:04:06.970 --> 00:04:09.150
Hydrofluoric acid has a Ka value

00:04:09.150 --> 00:04:11.600
of 6.3 times 10 to the negative fourth,

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and acetic acid has a Ka value

00:04:13.710 --> 00:04:16.950
of 1.8 times 10 to the negative fifth.

00:04:16.950 --> 00:04:19.920
Since both of these have
Ka values less than one,

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they're both considered to be weak acids.

00:04:22.580 --> 00:04:26.730
However, the higher the Ka
value, the stronger the acid.

00:04:26.730 --> 00:04:31.060
So out of these two weak
acids, since hydrofluoric acid

00:04:31.060 --> 00:04:33.960
has the higher Ka value, hydrofluoric acid

00:04:33.960 --> 00:04:36.143
is the stronger acid of the two.

00:04:37.040 --> 00:04:40.420
Now let's do a calculation
using a weak acid.

00:04:40.420 --> 00:04:44.460
Let's say we have a 0.10 molar
solution of benzoic acid,

00:04:44.460 --> 00:04:46.910
which is a weak acid, and
the pH of the solution

00:04:46.910 --> 00:04:50.070
is 2.60 at 25 degrees Celsius.

00:04:50.070 --> 00:04:52.590
Our goal is to calculate the Ka value

00:04:52.590 --> 00:04:56.060
for benzoic acid at 25 degrees Celsius.

00:04:56.060 --> 00:04:58.090
Since we're given the pH in the problem,

00:04:58.090 --> 00:05:02.940
we can go ahead and plug the
pH of 2.60 into this equation.

00:05:02.940 --> 00:05:07.730
So 2.60 is equal to the negative
log of the concentration

00:05:07.730 --> 00:05:11.160
of hydronium ions, and to
solve for the concentration

00:05:11.160 --> 00:05:13.740
of hydronium ions, first
moved the negative sign

00:05:13.740 --> 00:05:16.100
to the left, which gives us negative 2.60,

00:05:16.100 --> 00:05:20.720
is equal to log of the
concentration of hydronium ions.

00:05:20.720 --> 00:05:25.330
And to get rid of the log,
we take 10 to both sides.

00:05:25.330 --> 00:05:28.590
This gives us the
concentration of hydronium ions

00:05:28.590 --> 00:05:33.590
is equal to 2.5 times 10 to
the negative third molar.

00:05:35.740 --> 00:05:37.420
It's also important to recognize this is

00:05:37.420 --> 00:05:41.820
the equilibrium concentration
of hydronium ions.

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The next step is to write out the equation

00:05:43.610 --> 00:05:46.640
showing benzoic acid reacting with water.

00:05:46.640 --> 00:05:49.580
So here's the chemical
formula for benzoic acid

00:05:49.580 --> 00:05:52.430
reacting with water to
form the hydronium ion

00:05:52.430 --> 00:05:54.620
and the conjugate base to benzoic acid,

00:05:54.620 --> 00:05:56.860
which is the benzoate anion.

00:05:56.860 --> 00:05:59.470
To help us find the Ka
value for benzoic acid,

00:05:59.470 --> 00:06:01.190
we're gonna use an ICE table,

00:06:01.190 --> 00:06:03.640
where I stands for the
initial concentration,

00:06:03.640 --> 00:06:05.610
C is the change in concentration,

00:06:05.610 --> 00:06:08.240
and E is the equilibrium concentration.

00:06:08.240 --> 00:06:11.190
In our problem, we were told
the initial concentration

00:06:11.190 --> 00:06:15.550
of benzoic acid was 0.10
molar, and if we assume

00:06:15.550 --> 00:06:17.910
that benzoic acid hasn't reacted yet,

00:06:17.910 --> 00:06:21.050
the initial concentration of
hydronium ion would be zero,

00:06:21.050 --> 00:06:23.430
and the initial concentration of benzoate

00:06:23.430 --> 00:06:25.170
would also be zero.

00:06:25.170 --> 00:06:28.100
We just calculated the
equilibrium concentration

00:06:28.100 --> 00:06:31.100
of hydronium ions, which we found to be

00:06:31.100 --> 00:06:34.960
2.5 times 10 to the negative third molar.

00:06:34.960 --> 00:06:38.023
So we're going to go
ahead and put that under

00:06:38.023 --> 00:06:41.570
our hydronium ion in for the
equilibrium concentration.

00:06:41.570 --> 00:06:44.890
So if the initial concentration
of hydronium ion was zero

00:06:44.890 --> 00:06:46.620
and the equilibrium concentration

00:06:46.620 --> 00:06:49.700
is 2.5 times 10 to the
negative third, there was

00:06:49.700 --> 00:06:53.730
an increase in 2.5 times
10 to the negative third,

00:06:53.730 --> 00:06:55.660
so we're going to write
that under hydronium

00:06:55.660 --> 00:06:58.600
for the change in the concentration.

00:06:58.600 --> 00:07:01.030
And looking at the mole
ratio of hydronium ion

00:07:01.030 --> 00:07:04.510
to benzoate, it's a one-to-one mole ratio.

00:07:04.510 --> 00:07:06.700
So if hydronium ion increased

00:07:06.700 --> 00:07:09.330
by 2.5 times 10 to the negative third,

00:07:09.330 --> 00:07:12.600
so did the benzoate anions,
I can go ahead and write here

00:07:12.600 --> 00:07:15.770
plus 2.5 times 10 to the negative third,

00:07:15.770 --> 00:07:18.040
which means the equilibrium concentration

00:07:18.040 --> 00:07:20.770
of benzoate anion would also be

00:07:20.770 --> 00:07:23.770
2.5 times 10 to the negative third.

00:07:23.770 --> 00:07:25.800
Since we saw an increase
in the concentration

00:07:25.800 --> 00:07:29.140
of hydronium ion and
benzoate, that increase

00:07:29.140 --> 00:07:30.390
in the concentration of the products

00:07:30.390 --> 00:07:34.290
came from the ionization
of our weak acids.

00:07:34.290 --> 00:07:36.000
In the balanced equation, there's a one

00:07:36.000 --> 00:07:38.570
in front of benzoic acid, so if we write

00:07:38.570 --> 00:07:42.440
plus 2.5 times 10 to the
negative third for hydronium,

00:07:42.440 --> 00:07:45.490
and it's a one-to-one
mole ratio from hydronium

00:07:45.490 --> 00:07:47.980
to benzoic acid, we would have to write

00:07:47.980 --> 00:07:51.520
minus 2.5 times 10 to the negative third,

00:07:51.520 --> 00:07:55.140
since we lost some of the
benzoic acid when it ionized,

00:07:55.140 --> 00:07:58.400
therefore the equilibrium
concentration of benzoic acid

00:07:58.400 --> 00:08:03.400
would be 0.10 minus 2.5 times
10 to the negative third.

00:08:06.890 --> 00:08:09.180
Since our goal is to
calculate the Ka value

00:08:09.180 --> 00:08:11.550
for benzoic acid, the
next step is to write out

00:08:11.550 --> 00:08:13.830
the equilibrium constant expression,

00:08:13.830 --> 00:08:15.870
which we get from the balanced equation.

00:08:15.870 --> 00:08:18.160
It would be the concentration
of hydronium ions

00:08:18.160 --> 00:08:21.280
times the concentration of benzoate anions

00:08:21.280 --> 00:08:24.420
divided by the concentration
of benzoic acid.

00:08:24.420 --> 00:08:27.050
Since these are
equilibrium concentrations,

00:08:27.050 --> 00:08:29.290
we can get the equilibrium concentrations

00:08:29.290 --> 00:08:31.340
from the ICE table and plug them in

00:08:31.340 --> 00:08:34.350
to our equilibrium constant expression.

00:08:34.350 --> 00:08:37.160
So we can plug in the
equilibrium concentration

00:08:37.160 --> 00:08:39.360
for hydronium ions, we can plug in

00:08:39.360 --> 00:08:43.200
the equilibrium concentration
for benzoate anions,

00:08:43.200 --> 00:08:45.857
and we can plug in the
equilibrium concentration

00:08:45.857 --> 00:08:48.434
for benzoic acid.

00:08:48.434 --> 00:08:50.930
Here we have the
equilibrium concentrations

00:08:50.930 --> 00:08:53.760
plugged in to the equilibrium
constant expression,

00:08:53.760 --> 00:08:56.626
and when we solve for Ka, we find that Ka

00:08:56.626 --> 00:08:59.000
for benzoic acid at 25 degrees Celsius

00:08:59.000 --> 00:09:02.420
is equal to 6.4 times 10
to the negative fifth.

00:09:02.420 --> 00:09:04.870
Notice that Ka is less than one

00:09:04.870 --> 00:09:07.970
because benzoic acid is a weak acid.

00:09:07.970 --> 00:09:12.470
Finally, let's talk about the
idea of percent ionization.

00:09:12.470 --> 00:09:14.620
And to help us think
about percent ionization.

00:09:14.620 --> 00:09:18.040
Let's go back to our
example with a strong acid.

00:09:18.040 --> 00:09:21.370
We started off with five HA particles

00:09:21.370 --> 00:09:24.930
and all five of those HA particles ionized

00:09:24.930 --> 00:09:28.070
to produce five hydronium ions in solution

00:09:28.070 --> 00:09:31.310
and five A-minus anions in solution.

00:09:31.310 --> 00:09:35.720
Therefore 100% of our acid ionized.

00:09:35.720 --> 00:09:39.350
So we can write 100% ionization

00:09:39.350 --> 00:09:42.940
for this hypothetical strong acid HA.

00:09:42.940 --> 00:09:45.390
Now let's go back to our weak acid.

00:09:45.390 --> 00:09:49.150
For our weak acid, we started
off with five HA particles,

00:09:49.150 --> 00:09:51.750
but only one of them ionized to produce

00:09:51.750 --> 00:09:54.500
one H3O-plus and one A-minus.

00:09:54.500 --> 00:09:58.330
So four remain unionized at equilibrium,

00:09:58.330 --> 00:10:03.120
therefore one out of five
ionized, so the percent ionization

00:10:03.120 --> 00:10:08.120
for this hypothetical
weak acid would be 20%.

00:10:08.132 --> 00:10:11.840
Notice that the one
particle of HA that ionized

00:10:11.840 --> 00:10:16.030
made one hydronium ion, so we can also use

00:10:16.030 --> 00:10:19.430
the concentration of
hydronium ion at equilibrium

00:10:19.430 --> 00:10:22.610
to calculate percent ionization.

00:10:22.610 --> 00:10:25.350
So the equation for percent Ionization

00:10:25.350 --> 00:10:28.470
is equal to the equilibrium concentration

00:10:28.470 --> 00:10:32.540
of hydronium ions divided
by the initial concentration

00:10:32.540 --> 00:10:37.200
of the acid HA times 100%.

00:10:37.200 --> 00:10:40.900
So looking at our ICE table
for our benzoic acid problem,

00:10:40.900 --> 00:10:44.320
the equilibrium concentration
of hydronium ions

00:10:44.320 --> 00:10:47.740
was 2.5 times 10 to the
negative third molar,

00:10:47.740 --> 00:10:52.740
and the initial concentration
of our acid was 0.10 molar,

00:10:53.210 --> 00:10:56.040
so we can plug those into our equation.

00:10:56.040 --> 00:10:59.770
So we plug in our concentrations
and molar cancels,

00:10:59.770 --> 00:11:03.850
and we find for the final
answer, the percent ionization

00:11:03.850 --> 00:11:08.850
of benzoic acid is equal to 2.5%.


Strong base solutions

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:01.740
- [Instructor] When dissolved in water,

00:00:01.740 --> 00:00:04.670
a strong base like potassium hydroxide

00:00:04.670 --> 00:00:07.020
will dissociate completely in solution

00:00:07.020 --> 00:00:09.020
to form hydroxide ions.

00:00:09.020 --> 00:00:11.040
Potassium hydroxide is an example

00:00:11.040 --> 00:00:14.390
of a group 1A metal hydroxide.

00:00:14.390 --> 00:00:16.950
Other examples include lithium hydroxide

00:00:16.950 --> 00:00:19.100
and sodium hydroxide.

00:00:19.100 --> 00:00:21.840
Group 2A metal hydroxides
are also considered

00:00:21.840 --> 00:00:23.510
to be strong bases.

00:00:23.510 --> 00:00:25.790
For example, calcium hydroxide

00:00:25.790 --> 00:00:27.660
is a group 2A metal hydroxide,

00:00:27.660 --> 00:00:30.320
and so is strontium hydroxide.

00:00:30.320 --> 00:00:33.640
Let's do a problem with a
group 1A metal hydroxide,

00:00:33.640 --> 00:00:35.520
sodium hydroxide.

00:00:35.520 --> 00:00:38.930
Let's say the pH of the solution is 13.00

00:00:38.930 --> 00:00:42.230
and our goal is to calculate
the initial concentration

00:00:42.230 --> 00:00:44.330
of sodium hydroxide.

00:00:44.330 --> 00:00:46.780
First, we brought out
the disillusion equation.

00:00:46.780 --> 00:00:50.390
Solid sodium hydroxide
dissociates completely in water

00:00:50.390 --> 00:00:55.390
to form sodium cations and
hydroxide anions in solution.

00:00:55.550 --> 00:00:56.850
Looking at the balanced equation,

00:00:56.850 --> 00:00:59.310
there's a one in front of sodium hydroxide

00:00:59.310 --> 00:01:02.130
and a one in front of hydroxide ions,

00:01:02.130 --> 00:01:05.260
therefore the concentration
of hydroxide ions

00:01:05.260 --> 00:01:09.680
is equal to the initial
concentration of sodium hydroxide.

00:01:09.680 --> 00:01:12.200
And we can find the
concentration of hydroxide ions

00:01:12.200 --> 00:01:14.980
in solution from the pH.

00:01:14.980 --> 00:01:16.720
At 25 degrees Celsius,

00:01:16.720 --> 00:01:21.523
the pH plus the pOH is equal to 14.00.

00:01:24.470 --> 00:01:28.100
So we can plug the pH into our equation,

00:01:28.100 --> 00:01:31.340
which gives us 13.00,

00:01:31.340 --> 00:01:36.340
plus the pOH is equal to 14.00.

00:01:36.490 --> 00:01:40.943
So the pOH of the
solution is equal to 1.00,

00:01:43.290 --> 00:01:47.360
and the pOH is equal to the negative log

00:01:47.360 --> 00:01:51.410
of the concentration of hydroxide ions.

00:01:51.410 --> 00:01:55.890
So we can plug the pOH into this equation,

00:01:55.890 --> 00:02:00.890
which gives us 1.00 is
equal to the negative log

00:02:01.100 --> 00:02:04.053
of the concentration of hydroxide ions.

00:02:05.160 --> 00:02:06.530
To solve for the concentration

00:02:06.530 --> 00:02:08.670
of hydroxide ions in solution,

00:02:08.670 --> 00:02:11.960
first we move the negative
sign to the left side,

00:02:11.960 --> 00:02:15.570
which gives us negative
1.00 is equal to the log

00:02:15.570 --> 00:02:18.540
of the concentration of hydroxide ions.

00:02:18.540 --> 00:02:23.100
To get rid of the log,
we take 10 to both sides.

00:02:23.100 --> 00:02:27.870
So the concentration of
hydroxide ions in this solution

00:02:27.870 --> 00:02:31.160
is equal to 10 to the negative 1.00,

00:02:31.160 --> 00:02:35.810
which is equal to .10 molar.

00:02:35.810 --> 00:02:38.220
Because sodium hydroxide is a strong base

00:02:38.220 --> 00:02:40.670
that dissociates completely in solution

00:02:40.670 --> 00:02:42.520
to form hydroxide ions,

00:02:42.520 --> 00:02:44.390
if the concentration of hydroxide ions

00:02:44.390 --> 00:02:46.590
in solution is .10 molar,

00:02:46.590 --> 00:02:50.563
so is the initial concentration
of sodium hydroxide.

00:02:52.110 --> 00:02:55.980
Now let's do a problem with
a group 2A metal hydroxide.

00:02:55.980 --> 00:02:58.360
Let's say the initial
concentration of a solution

00:02:58.360 --> 00:03:02.570
of calcium hydroxide is .0010 molar,

00:03:02.570 --> 00:03:05.040
and our goal is to find
the pH of the solution

00:03:05.040 --> 00:03:07.550
at 25 degrees Celsius.

00:03:07.550 --> 00:03:09.730
Calcium hydroxide is a strong base

00:03:09.730 --> 00:03:11.920
that dissociates completely in solution

00:03:11.920 --> 00:03:16.150
to form calcium two plus
ions and hydroxide anions,

00:03:16.150 --> 00:03:19.280
and looking at the mole ratios
in this disillusion equation,

00:03:19.280 --> 00:03:22.070
there's a one in front
of calcium hydroxide,

00:03:22.070 --> 00:03:24.160
a one in front of calcium two plus,

00:03:24.160 --> 00:03:27.060
and a two in front of hydroxide ions.

00:03:27.060 --> 00:03:30.790
Since the mole ratio of
calcium hydroxide to calcium

00:03:30.790 --> 00:03:32.290
is one to one,

00:03:32.290 --> 00:03:34.760
if the initial concentration
of calcium hydroxide

00:03:34.760 --> 00:03:37.190
is .0010 molar,

00:03:37.190 --> 00:03:40.070
that's also the concentration
of calcium ions.

00:03:40.070 --> 00:03:44.600
So it's .0010 molar in solution.

00:03:44.600 --> 00:03:48.340
The mole ratio of calcium
hydroxide to hydroxide ions

00:03:48.340 --> 00:03:50.590
is one to two,

00:03:50.590 --> 00:03:53.130
so if the initial concentration
of calcium hydroxide

00:03:53.130 --> 00:03:55.560
is .0010 molar,

00:03:55.560 --> 00:03:58.440
the concentration of
hydroxide ions in solution

00:03:58.440 --> 00:04:00.800
is twice that concentration,

00:04:00.800 --> 00:04:05.453
so two times .0010 molar
is equal to .0020 molar.

00:04:08.630 --> 00:04:11.140
Now that we know the
concentration of hydroxide ions,

00:04:11.140 --> 00:04:14.440
we can calculate the pH of the solution.

00:04:14.440 --> 00:04:18.360
One way to calculate the
pH is to first find the pOH

00:04:18.360 --> 00:04:19.420
of the solution,

00:04:19.420 --> 00:04:23.060
and pOH is equal to the negative log

00:04:23.060 --> 00:04:27.110
of the concentration of hydroxide ions.

00:04:27.110 --> 00:04:30.490
So we can plug in the
concentration of hydroxide ions

00:04:30.490 --> 00:04:32.020
into our equation,

00:04:32.020 --> 00:04:34.730
which gives us the pOH of the solution

00:04:34.730 --> 00:04:38.263
is equal to the negative log of .0020,

00:04:42.140 --> 00:04:43.650
and when you do the calculation,

00:04:43.650 --> 00:04:48.590
the pOH is equal to 2.70.

00:04:48.590 --> 00:04:51.970
Notice, since we have
two significant figures

00:04:51.970 --> 00:04:53.200
for the concentration,

00:04:53.200 --> 00:04:56.563
we need two decimal places for our pOH.

00:04:57.480 --> 00:04:58.730
So to find the pH,

00:04:58.730 --> 00:05:03.730
we know that pH plus pOH is equal to 14.00

00:05:06.020 --> 00:05:08.760
at 25 degrees Celsius.

00:05:08.760 --> 00:05:13.140
So we can plug in the pOH of 2.70,

00:05:13.140 --> 00:05:18.140
and that gives us pH plus
2.70 is equal to 14.00.

00:05:20.380 --> 00:05:24.113
So the pH of the solution
is equal to 11.30.


Strong acid solutions

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.100
- [Instructor] A strong acid is an acid

00:00:02.100 --> 00:00:05.340
that ionizes 100% in solution.

00:00:05.340 --> 00:00:09.230
For example, hydrochloric
acid, HCl, as a strong acid

00:00:09.230 --> 00:00:11.960
it donates a proton to water, H2O,

00:00:11.960 --> 00:00:14.810
to form the hydronium ion, H3O plus,

00:00:14.810 --> 00:00:16.650
and the conjugate base to HCl

00:00:16.650 --> 00:00:19.750
which is the chloride anion, Cl minus.

00:00:19.750 --> 00:00:22.300
In reality, this reaction
reaches an equilibrium.

00:00:22.300 --> 00:00:25.040
However, the equilibrium
lies so far to the right

00:00:25.040 --> 00:00:28.060
and favors the product so
much that we don't draw

00:00:28.060 --> 00:00:30.520
an equilibrium arrow,
we simply draw an arrow

00:00:30.520 --> 00:00:33.210
going to the right,
indicating the reaction

00:00:33.210 --> 00:00:35.750
essentially goes to completion.

00:00:35.750 --> 00:00:38.300
And if the reaction
essentially goes to completion,

00:00:38.300 --> 00:00:41.890
we can say that hydrochloric
acid ionizes 100%

00:00:41.890 --> 00:00:45.800
and forms hydronium ions
and chloride anions.

00:00:45.800 --> 00:00:47.980
So essentially there is no more HCl left,

00:00:47.980 --> 00:00:52.460
it's all turned into
H3O plus and Cl minus.

00:00:52.460 --> 00:00:54.860
It's also acceptable to
completely leave water

00:00:54.860 --> 00:00:59.070
out of the equation and to
show hydrochloric acid, HCl,

00:00:59.070 --> 00:01:02.420
turning into H plus and Cl minus.

00:01:02.420 --> 00:01:04.740
Once again, since HCl is a strong acid,

00:01:04.740 --> 00:01:06.950
there's only an arrow going to the right

00:01:06.950 --> 00:01:10.350
indicating HCl ionizes 100%.

00:01:10.350 --> 00:01:12.830
And since there's only one
water molecule difference

00:01:12.830 --> 00:01:17.370
between H plus and H3O
plus, H plus and H3O plus

00:01:17.370 --> 00:01:19.490
are used interchangeably.

00:01:19.490 --> 00:01:21.200
Hydrochloric acid is an example

00:01:21.200 --> 00:01:23.350
of a monoprotic strong acid.

00:01:23.350 --> 00:01:26.490
Monoprotic means, hydrochloric
acid has one proton

00:01:26.490 --> 00:01:28.890
that it can donate in solution.

00:01:28.890 --> 00:01:31.730
Other examples of monoprotic
strong acids include

00:01:31.730 --> 00:01:34.233
hydrobromic acid, HBr,
hydroiodic acid, HI,

00:01:36.300 --> 00:01:40.627
nitric acid, HNO3, and
perchloric acid, HClO4.

00:01:41.980 --> 00:01:45.680
Sulfuric acid is H2SO4
and it's a strong acid,

00:01:45.680 --> 00:01:49.050
but it's a diprotic acid,
meaning it has two protons

00:01:49.050 --> 00:01:52.550
that it can donate, however,
only the first ionization

00:01:52.550 --> 00:01:55.310
for sulfuric acid is strong.

00:01:55.310 --> 00:01:58.810
Let's calculate the pH of
a strong acid solution.

00:01:58.810 --> 00:02:03.220
In this case, we're gonna
do a 0.040M solution

00:02:03.220 --> 00:02:05.320
of nitric acid.

00:02:05.320 --> 00:02:10.180
Nitric acid is HNO3, and
nitric acid reacts with water

00:02:10.180 --> 00:02:14.690
to form hydronium, H3O plus,
and nitrate, NO3 minus,

00:02:14.690 --> 00:02:17.910
which is the conjugate base 2HNO3.

00:02:17.910 --> 00:02:20.060
Because nitric acid is a strong acid,

00:02:20.060 --> 00:02:22.670
we assume the reaction goes to completion.

00:02:22.670 --> 00:02:25.740
Therefore, if the initial
concentration of nitric acid

00:02:25.740 --> 00:02:30.310
is 0.040M, looking at our mole ratios

00:02:30.310 --> 00:02:32.210
in the balanced equation,
there's a one in front

00:02:32.210 --> 00:02:35.560
of nitric acid and there's also
a one in front of hydronium

00:02:35.560 --> 00:02:37.630
and a one in front of nitrate.

00:02:37.630 --> 00:02:39.740
Therefore, if the reaction
goes to completion,

00:02:39.740 --> 00:02:43.897
the concentration of
hydronium would also be 0.040M

00:02:45.250 --> 00:02:47.100
and the same with the nitrate anion,

00:02:47.100 --> 00:02:51.510
that would also have a
concentration of 0.040M.

00:02:51.510 --> 00:02:55.000
Since our goal is to calculate
the pH of this solution,

00:02:55.000 --> 00:02:57.950
we know that the equation
for pH is pH is equal

00:02:57.950 --> 00:03:02.950
to the negative log of the
concentration of hydronium ions.

00:03:03.010 --> 00:03:05.580
Therefore, we just need to
plug in the concentration

00:03:05.580 --> 00:03:09.000
of hydronium ions into our equation.

00:03:09.000 --> 00:03:10.580
This gives us the pH is equal

00:03:10.580 --> 00:03:14.073
to the negative log of 0.040,

00:03:17.400 --> 00:03:22.360
which is equal to 1.40.

00:03:22.360 --> 00:03:24.900
So even though this is
a pretty dilute solution

00:03:24.900 --> 00:03:28.560
of nitric acid, because
nitric acid is a strong acid,

00:03:28.560 --> 00:03:30.760
the pH is pretty low.

00:03:30.760 --> 00:03:33.220
Also note since we have
two significant figures

00:03:33.220 --> 00:03:35.490
for the concentration of hydronium ions,

00:03:35.490 --> 00:03:39.420
we need two decimal places
for our final answer.

00:03:39.420 --> 00:03:42.430
Let's do another problem
with a strong acid solution.

00:03:42.430 --> 00:03:45.510
Let's say we have 100 ml
of an aqueous solution

00:03:45.510 --> 00:03:48.400
of hydroiodic acid and
the pH of the solution

00:03:48.400 --> 00:03:50.170
is equal to 1.50.

00:03:50.170 --> 00:03:52.740
And our goal is to find the mass of HI

00:03:52.740 --> 00:03:55.340
that's present in solution.

00:03:55.340 --> 00:03:58.470
Hydroiodic acid reacts with
water to form the hydronium ion

00:03:58.470 --> 00:04:00.090
and the iodide anion.

00:04:00.090 --> 00:04:05.090
And the mole ratio of HI
to H3O plus is one-to-one.

00:04:05.500 --> 00:04:08.380
So if we can find the
concentration of hydronium ion

00:04:08.380 --> 00:04:10.600
and solution, that should also be

00:04:12.118 --> 00:04:15.070
the initial concentration
of hydroiodic acid.

00:04:15.070 --> 00:04:15.903
And once we find

00:04:15.903 --> 00:04:17.780
the initial concentration
of hydroiodic acid,

00:04:17.780 --> 00:04:20.620
we can find the mass of HI that's present.

00:04:20.620 --> 00:04:22.610
Since we are given the pH in the problem,

00:04:22.610 --> 00:04:25.680
we can plug that directly
into our equation,

00:04:25.680 --> 00:04:30.680
which gives us 1.50 is
equal to the negative log

00:04:31.960 --> 00:04:35.473
of the concentration of hydronium ions.

00:04:36.730 --> 00:04:38.980
To solve for the concentration
of hydronium ions,

00:04:38.980 --> 00:04:41.680
we can first move the negative
sign to the left side,

00:04:41.680 --> 00:04:45.440
which gives us negative
1.50 is equal to the log

00:04:45.440 --> 00:04:48.380
of the concentration of hydronium ions.

00:04:48.380 --> 00:04:53.380
And to get rid of the log,
we can take 10 to both sides.

00:04:53.380 --> 00:04:55.190
So the concentration of hydronium ions

00:04:55.190 --> 00:04:58.830
is equal to 10 to the negative 1.50,

00:04:58.830 --> 00:05:03.730
which is equal to 0.032.

00:05:03.730 --> 00:05:07.373
So the concentration of
hydronium ions is 0.032M.

00:05:08.390 --> 00:05:11.960
And because the mole ratio
of hydronium ion to HI

00:05:11.960 --> 00:05:15.320
is one-to-one, the initial
concentration of HI

00:05:15.320 --> 00:05:17.600
is also 0.032M.

00:05:19.660 --> 00:05:21.820
Now that we know the
initial concentration of HI,

00:05:21.820 --> 00:05:24.800
we're ready to find
the mass of HI present.

00:05:24.800 --> 00:05:26.370
Molarity is moles per liter,

00:05:26.370 --> 00:05:28.040
so let's go ahead and rewrite this

00:05:28.040 --> 00:05:33.040
as 0.032 moles per liter.

00:05:34.630 --> 00:05:38.180
The volume of the solution
is 100 milliliters,

00:05:38.180 --> 00:05:42.180
which is equal to 0.100 liters.

00:05:42.180 --> 00:05:45.810
So if we multiply moles
per liter by the volume,

00:05:45.810 --> 00:05:50.810
which is 0.100 liters, liters
will cancel and give us moles.

00:05:51.600 --> 00:05:52.960
So this is equal to

00:05:52.960 --> 00:05:57.683
0.0032 moles of HI.

00:05:58.640 --> 00:06:01.140
Since our goal is to find
the mass of HI present,

00:06:01.140 --> 00:06:03.300
the final step is to
multiply the moles of HI

00:06:03.300 --> 00:06:04.950
by the molar mass which is

00:06:04.950 --> 00:06:09.950
128 grams per one mole of HI.

00:06:10.400 --> 00:06:15.400
So most of HI would cancel out
and this gives us 0.41 grams

00:06:16.370 --> 00:06:17.553
as our final answer.


Worked examples: Calculating [H₃O⁺] and pH

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.290
- [Instructor] Here are some equations

00:00:01.290 --> 00:00:04.270
that are often used in pH calculations.

00:00:04.270 --> 00:00:06.200
For example, let's say a solution

00:00:06.200 --> 00:00:08.770
is formed at 25 degrees Celsius

00:00:08.770 --> 00:00:12.310
and the solution has a pOH of 4.75,

00:00:12.310 --> 00:00:14.870
and our goal is to
calculate the concentration

00:00:14.870 --> 00:00:18.320
of hydronium ions in solution, H3O+.

00:00:18.320 --> 00:00:20.900
One way to start this problem
is to use this equation,

00:00:20.900 --> 00:00:25.170
pH plus pOH is equal to 14.00.

00:00:25.170 --> 00:00:28.670
And we have the pOH equal to 4.75,

00:00:28.670 --> 00:00:31.700
so we can plug that into our equation.

00:00:31.700 --> 00:00:34.880
That gives us pH plus 4.75

00:00:36.540 --> 00:00:40.380
is equal to 14.00.

00:00:40.380 --> 00:00:45.380
And solving for the pH, we get
that the pH is equal to 9.25.

00:00:47.770 --> 00:00:49.820
So we have the pH and our goal is to solve

00:00:49.820 --> 00:00:52.020
for the concentration of hydronium ions,

00:00:52.020 --> 00:00:54.630
and pH is equal to the negative log

00:00:54.630 --> 00:00:57.760
of the concentration of hydronium ions.

00:00:57.760 --> 00:01:02.260
So we can plug our pH
right into this equation.

00:01:02.260 --> 00:01:05.680
So that would give us the pH which is 9.25

00:01:07.100 --> 00:01:09.770
is equal to the negative log

00:01:09.770 --> 00:01:13.573
of the concentration of hydronium ions.

00:01:14.920 --> 00:01:16.750
Next, we need to solve
for the concentration

00:01:16.750 --> 00:01:18.010
of hydronium ions.

00:01:18.010 --> 00:01:20.620
So we could move the negative
sign over to the left side,

00:01:20.620 --> 00:01:22.910
which gives us negative 9.25

00:01:22.910 --> 00:01:26.410
is equal to the log of the
concentration of hydronium ions.

00:01:26.410 --> 00:01:30.550
And to get rid of that log,
we can take 10 to both sides.

00:01:30.550 --> 00:01:34.330
So that's gonna give us the
concentration of hydronium ions,

00:01:34.330 --> 00:01:39.330
H3O+, is equal to 10 to the negative 9.25.

00:01:42.337 --> 00:01:44.277
And 10 to the negative 9.25

00:01:44.277 --> 00:01:47.944
is equal to 5.6 times
10 to the negative 10.

00:01:49.970 --> 00:01:52.990
So the concentration of hydronium ions

00:01:52.990 --> 00:01:56.450
in our solution at 25 degrees Celsius

00:01:56.450 --> 00:02:01.330
is equal to 5.6 times 10
to the negative 10th molar.

00:02:01.330 --> 00:02:04.450
Also notice that because
we had two decimal places

00:02:04.450 --> 00:02:08.860
for our pH, we have the
concentration of hydronium ions

00:02:08.860 --> 00:02:11.860
to two significant figures.

00:02:11.860 --> 00:02:14.230
There's another way to
do the same problem.

00:02:14.230 --> 00:02:16.060
Since we have the pOH,

00:02:16.060 --> 00:02:20.200
we could use the pOH equation
to find the concentration

00:02:20.200 --> 00:02:23.000
of hydroxide ions in solution.

00:02:23.000 --> 00:02:28.000
So we would just need to plug
in the pOH into this equation

00:02:28.330 --> 00:02:32.600
which gives us 4.75,

00:02:32.600 --> 00:02:35.470
which is the pOH, is
equal to the negative log

00:02:35.470 --> 00:02:39.690
of the concentration of hydroxide ions.

00:02:39.690 --> 00:02:42.260
Next, we can move the negative
sign over to the left side.

00:02:42.260 --> 00:02:47.260
So negative 4.75 is equal to
the log of the concentration

00:02:47.930 --> 00:02:50.510
of hydroxide ions.

00:02:50.510 --> 00:02:54.653
And to get rid of the log, we
just take 10 to both sides.

00:02:55.570 --> 00:02:57.630
10 to the negative 4.75

00:02:57.630 --> 00:03:02.630
is equal to 1.8 times 10
to the negative fifth.

00:03:03.440 --> 00:03:06.320
So the concentration of hydroxide ions

00:03:06.320 --> 00:03:10.010
is equal to 1.8 times 10 to
the negative fifth molar.

00:03:10.010 --> 00:03:11.540
So our goal is to find the concentration

00:03:11.540 --> 00:03:13.150
of hydronium ions in solution,

00:03:13.150 --> 00:03:16.900
and we have the concentration
of hydroxide ions in solution,

00:03:16.900 --> 00:03:19.570
so we can use the Kw equation

00:03:19.570 --> 00:03:21.880
because the concentration
of hydronium ions

00:03:21.880 --> 00:03:25.440
times the concentration of
hydroxide ions is equal to Kw,

00:03:25.440 --> 00:03:29.420
which is equal to 1.0 times
10 to the negative 14th

00:03:29.420 --> 00:03:32.010
at 25 degrees Celsius.

00:03:32.010 --> 00:03:35.130
So we can plug in the
concentration of hydroxide ions

00:03:35.130 --> 00:03:37.680
into our equation.

00:03:37.680 --> 00:03:42.440
That gives us 1.8 times
10 to the negative fifth

00:03:42.440 --> 00:03:44.770
times the concentration of hydronium ions,

00:03:44.770 --> 00:03:47.710
which we'll just write
as x in our equation

00:03:47.710 --> 00:03:48.890
is equal to Kw

00:03:48.890 --> 00:03:53.063
which is equal to 1.0 times
10 to the negative 14.

00:03:54.370 --> 00:03:55.300
Solving for x,

00:03:55.300 --> 00:04:00.300
we find the x is equal to 5.6
times 10 to the negative 10th.

00:04:02.130 --> 00:04:04.360
So the concentration of hydronium ions

00:04:04.360 --> 00:04:08.230
is equal to 5.6 times 10
to the negative 10th molar.

00:04:08.230 --> 00:04:10.320
So even though we used
two different equations

00:04:10.320 --> 00:04:11.930
from the first time we did this problem,

00:04:11.930 --> 00:04:14.610
we ended up with the same answer
that we did the first time,

00:04:14.610 --> 00:04:17.560
5.6 times 10 to the negative 10th molar.

00:04:17.560 --> 00:04:20.220
So it doesn't really matter
which approach you take.

00:04:20.220 --> 00:04:21.750
Finally, let's look at an example

00:04:21.750 --> 00:04:25.950
where the temperature is
not 25 degrees Celsius.

00:04:25.950 --> 00:04:28.250
Let's say, we have a sample of pure water

00:04:28.250 --> 00:04:30.090
at 50 degrees Celsius,

00:04:30.090 --> 00:04:32.803
and our goal is to calculate the pH.

00:04:33.800 --> 00:04:36.620
Pure water is a neutral substance

00:04:36.620 --> 00:04:41.620
which means the concentration
of hydronium ions, H3O+,

00:04:41.910 --> 00:04:46.260
is equal to the concentration
of hydroxide ions, OH-.

00:04:46.260 --> 00:04:49.290
Right now, we don't know what
those concentrations are,

00:04:49.290 --> 00:04:51.860
but we know that the
concentration of hydronium ions

00:04:51.860 --> 00:04:53.910
times the concentration of hydroxide ions

00:04:53.910 --> 00:04:55.840
is equal to Kw.

00:04:55.840 --> 00:04:58.470
However, we have to be careful because Kw

00:04:58.470 --> 00:05:01.670
is only equal to 1.0
times 10 to negative 14th

00:05:01.670 --> 00:05:03.870
at 25 degrees Celsius.

00:05:03.870 --> 00:05:05.700
And at 50 degrees Celsius,

00:05:05.700 --> 00:05:09.350
Kw is equal to 5.5 times
10 to the negative 14th.

00:05:10.350 --> 00:05:11.183
So let's go ahead and write

00:05:11.183 --> 00:05:16.183
Kw is equal to 5.5 times
10 to the negative 14th.

00:05:16.630 --> 00:05:20.110
And if we make the concentration
of hydronium ions x,

00:05:20.110 --> 00:05:21.970
then the concentration of hydroxide ions

00:05:21.970 --> 00:05:25.560
would also have to be x
since the two are equal.

00:05:25.560 --> 00:05:28.400
So we would have x times x

00:05:28.400 --> 00:05:32.460
is equal to 5.5 times
10 to the negative 14th.

00:05:32.460 --> 00:05:34.230
So that gives us x squared

00:05:34.230 --> 00:05:38.660
is equal to 5.5 times
10 to the negative 14th.

00:05:38.660 --> 00:05:41.200
And if we take the square
root of both sides,

00:05:41.200 --> 00:05:42.790
we find that x is equal

00:05:42.790 --> 00:05:47.790
to 2.3 times 10 to the negative seventh.

00:05:48.380 --> 00:05:50.480
And since x is equal to the concentration

00:05:50.480 --> 00:05:52.530
of hydronium ions in solution,

00:05:52.530 --> 00:05:54.310
the concentration of hydronium ions

00:05:54.310 --> 00:05:57.660
is 2.3 times 10 to the
negative seventh molar.

00:05:57.660 --> 00:05:59.220
Now that we know the concentration

00:05:59.220 --> 00:06:01.160
of hydronium ions in solution,

00:06:01.160 --> 00:06:05.260
we can use our pH equation
to find the pH of water

00:06:05.260 --> 00:06:07.500
at 50 degrees Celsius.

00:06:07.500 --> 00:06:10.370
So we plug our concentration
of hydronium ions

00:06:10.370 --> 00:06:11.960
into our equation,

00:06:11.960 --> 00:06:13.750
and we take the negative log of that,

00:06:13.750 --> 00:06:17.340
and we get that the pH is equal to 6.64.

00:06:17.340 --> 00:06:21.380
Notice with two significant
figures for the concentration,

00:06:21.380 --> 00:06:24.800
we get two decimal places for our answer.

00:06:24.800 --> 00:06:26.180
If we had used Kw

00:06:26.180 --> 00:06:30.010
is equal to 1.0 times
10 to the negative 14th,

00:06:30.010 --> 00:06:34.170
we would've gotten a pH of 7.00,

00:06:34.170 --> 00:06:37.960
but that's only true
at 25 degrees Celsius.

00:06:37.960 --> 00:06:41.310
Since Kw is temperature dependent,

00:06:41.310 --> 00:06:43.210
if the temperature is something other

00:06:43.210 --> 00:06:48.210
than 25 degrees Celsius, the
pH of water will not be seven.

00:06:48.250 --> 00:06:51.530
So in this case, we
calculated it to be 6.64.

00:06:51.530 --> 00:06:55.540
The pH of pure water is
6.64 at 50 degrees Celsius.

00:06:55.540 --> 00:06:58.740
However, water is still
a neutral substance.

00:06:58.740 --> 00:07:00.572
The concentration of hydronium ions

00:07:00.572 --> 00:07:04.364
is equal to the concentration
of hydroxide ions.


The pH scale

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:01.610
- [Instructor] For a sample of pure water

00:00:01.610 --> 00:00:05.490
at 25 degrees Celsius, the
concentration of hydronium ion

00:00:05.490 --> 00:00:09.210
is equal to 1.0 times 10 to
the negative seventh molar.

00:00:09.210 --> 00:00:12.070
Because the concentrations
are often very small,

00:00:12.070 --> 00:00:14.720
it's much more convenient
to express the concentration

00:00:14.720 --> 00:00:17.320
of hydronium ion in terms of pH.

00:00:17.320 --> 00:00:20.470
And pH is defined as the negative log

00:00:20.470 --> 00:00:23.320
of the concentration of hydronium ion.

00:00:23.320 --> 00:00:28.030
Since H+ and H3O+ are used
interchangeably in chemistry,

00:00:28.030 --> 00:00:30.580
sometimes you'll see the
pH equation written out

00:00:30.580 --> 00:00:32.580
as pH is equal to the negative log

00:00:32.580 --> 00:00:35.540
of the concentration of H+ ion.

00:00:35.540 --> 00:00:37.720
So to find the pH of pure water,

00:00:37.720 --> 00:00:40.070
we just need to plug in the concentration

00:00:40.070 --> 00:00:43.450
of hydronium ions into our equation.

00:00:43.450 --> 00:00:45.820
So the pH is equal to the negative log

00:00:45.820 --> 00:00:48.650
of 1.0 times 10 to the negative seventh

00:00:48.650 --> 00:00:51.310
which is equal to 7.00.

00:00:51.310 --> 00:00:55.210
So the pH of pure water
at 25 degrees Celsius

00:00:55.210 --> 00:00:56.930
is equal to seven.

00:00:56.930 --> 00:00:59.160
Notice that there are
two significant figures

00:00:59.160 --> 00:01:00.520
for the concentration

00:01:00.520 --> 00:01:04.070
and there are two decimal
places for the final answer.

00:01:04.070 --> 00:01:06.270
And that's because for a logarithm,

00:01:06.270 --> 00:01:09.640
only the numbers to the
right of the decimal point

00:01:09.640 --> 00:01:11.560
are significant figures.

00:01:11.560 --> 00:01:14.710
Therefore, two significant
figures for a concentration

00:01:14.710 --> 00:01:17.023
means two decimal places.

00:01:18.340 --> 00:01:20.830
Let's say we have a sample of lemon juice

00:01:20.830 --> 00:01:23.170
and the measured concentration
of hydronium ions

00:01:23.170 --> 00:01:27.390
in solution is 3.6 times 10
to negative fourth molar.

00:01:27.390 --> 00:01:29.890
Since we have the concentration
of hydronium ions,

00:01:29.890 --> 00:01:32.460
we can simply plug in that concentration

00:01:32.460 --> 00:01:34.890
into the equation for a pH.

00:01:34.890 --> 00:01:37.720
So the pH is equal to the negative log

00:01:37.720 --> 00:01:41.530
of the concentration of 3.6
times 10 to the negative fourth

00:01:41.530 --> 00:01:43.850
which is 3.44.

00:01:43.850 --> 00:01:45.960
Notice that we have
two significant figures

00:01:45.960 --> 00:01:47.200
for the concentration

00:01:47.200 --> 00:01:50.763
therefore we have two decimal
places in our final answer.

00:01:51.940 --> 00:01:54.840
So plugging in negative
log of 3.6 times 10

00:01:54.840 --> 00:01:56.520
to the negative fourth on your calculator,

00:01:56.520 --> 00:01:59.450
it gives you 3.44 for the answer.

00:01:59.450 --> 00:02:01.640
However, there's a way
of estimating the pH

00:02:01.640 --> 00:02:03.940
without using a calculator.

00:02:03.940 --> 00:02:08.110
The first step is to say 3.6
times 10 to the negative fourth

00:02:08.110 --> 00:02:11.350
is between one times 10
to the negative fourth

00:02:11.350 --> 00:02:14.380
and 10 times 10 to the negative fourth.

00:02:14.380 --> 00:02:16.050
10 times 10 to the negative fourth

00:02:16.050 --> 00:02:19.330
is the same thing as one times
10 to the negative third.

00:02:19.330 --> 00:02:21.140
If the concentration of hydronium ions

00:02:21.140 --> 00:02:23.450
is one times 10 to the negative third,

00:02:23.450 --> 00:02:26.410
you can find the pH of that
without using a calculator

00:02:26.410 --> 00:02:28.270
if you know your logarithms.

00:02:28.270 --> 00:02:31.740
The pH would be equal to three.

00:02:31.740 --> 00:02:33.520
And if the concentration of hydronium ions

00:02:33.520 --> 00:02:35.700
is one times 10 to the negative fourth,

00:02:35.700 --> 00:02:37.640
this is a log base 10 system,

00:02:37.640 --> 00:02:41.290
so the negative log of one
times 10 to the negative fourth

00:02:41.290 --> 00:02:45.390
would be equal to a pH of four.

00:02:45.390 --> 00:02:48.310
Since 3.6 times 10 to the negative fourth

00:02:48.310 --> 00:02:51.220
is between one times 10
to the negative fourth

00:02:51.220 --> 00:02:53.880
and one times 10 to the negative third,

00:02:53.880 --> 00:02:58.360
the pH of this concentration
must be between a pH of four

00:02:58.360 --> 00:03:00.140
and a pH three.

00:03:00.140 --> 00:03:02.100
And we saw that with our calculator.

00:03:02.100 --> 00:03:04.043
The pH came out to be 3.44.

00:03:05.750 --> 00:03:08.120
Let's say we have some cleaning
solution at room temperature

00:03:08.120 --> 00:03:09.860
which is 25 degrees Celsius

00:03:09.860 --> 00:03:12.750
and the cleaning solution
has some ammonia in it.

00:03:12.750 --> 00:03:16.140
The concentration of hydroxide
ions in solution is measured

00:03:16.140 --> 00:03:19.340
to be 2.0 times 10 to
the negative third molar.

00:03:19.340 --> 00:03:22.810
And our goal is to calculate
the pH of the solution.

00:03:22.810 --> 00:03:25.680
The first step is to use the Kw equation

00:03:25.680 --> 00:03:27.970
which says that the
concentration of hydronium ions

00:03:27.970 --> 00:03:31.730
times the concentration of
hydroxide ions is equal to Kw

00:03:31.730 --> 00:03:33.560
which at 25 degrees Celsius

00:03:33.560 --> 00:03:37.170
is equal to 1.0 times
10 to the negative 14th.

00:03:37.170 --> 00:03:39.340
We can go ahead and plug
in the concentration

00:03:39.340 --> 00:03:43.120
of hydroxide ions into the Kw equation.

00:03:43.120 --> 00:03:48.120
So that's 2.0 times 10
to the negative third.

00:03:48.160 --> 00:03:51.240
And we don't know the
concentration of hydronium ions is,

00:03:51.240 --> 00:03:53.800
we'll make that X and so
we're gonna solve for X.

00:03:53.800 --> 00:03:58.760
And this is equal to one
times 10 to the negative 14th.

00:03:58.760 --> 00:04:02.210
Solving for X, X is equal to 5.0

00:04:02.210 --> 00:04:04.380
times 10 to the negative 12

00:04:04.380 --> 00:04:06.420
and this is the equal to the concentration

00:04:06.420 --> 00:04:08.010
of hydronium ions.

00:04:08.010 --> 00:04:11.280
So the concentration of
hydronium ions is equal to 5.0

00:04:11.280 --> 00:04:13.800
times 10 to the negative 12 molar.

00:04:13.800 --> 00:04:16.070
Now that we know the
concentration of hydronium ions

00:04:16.070 --> 00:04:20.030
in solution, we can plug that
into our equation for pH.

00:04:20.030 --> 00:04:23.300
So the pH is equal to
the negative log of 5.0

00:04:23.300 --> 00:04:28.300
times 10 to the negative
12 which gives a 11.30.

00:04:28.470 --> 00:04:31.190
Notice since we have
two significant figures

00:04:31.190 --> 00:04:32.380
for the concentration,

00:04:32.380 --> 00:04:36.440
we need two decimal places
for our final answer.

00:04:36.440 --> 00:04:37.860
Now that we've calculated the pH

00:04:37.860 --> 00:04:39.370
of three different substances,

00:04:39.370 --> 00:04:42.870
let's find where they rank on
what's called the pH scale.

00:04:42.870 --> 00:04:46.980
The pH scale normally
goes from zero to 14.

00:04:46.980 --> 00:04:51.660
However, it is possible to go
below zero or to go above 14.

00:04:51.660 --> 00:04:53.950
We calculate that a sample of pure water

00:04:53.950 --> 00:04:58.950
at 25 degrees Celsius has a pH of 7.00.

00:04:59.260 --> 00:05:02.520
That puts it right in the
middle of the pH scale.

00:05:02.520 --> 00:05:06.200
And we say that water
is a neutral substance.

00:05:06.200 --> 00:05:08.480
An aqueous solution has a pH of seven

00:05:08.480 --> 00:05:12.290
is considered to be a neutral solution.

00:05:12.290 --> 00:05:14.640
An aqueous solution with
a pH less than seven

00:05:14.640 --> 00:05:17.200
is considered to be an acidic solution.

00:05:17.200 --> 00:05:22.010
So for lemon juice, we
calculated the pH to be 3.44

00:05:22.010 --> 00:05:24.530
which is right about here on our pH scale

00:05:24.530 --> 00:05:26.580
so lemon juice is acidic.

00:05:26.580 --> 00:05:29.360
And as you go to the left on the pH scale,

00:05:29.360 --> 00:05:32.080
you increase in acidity.

00:05:32.080 --> 00:05:34.800
For example, if we had a
solution with a pH of six

00:05:34.800 --> 00:05:37.060
and another solution with a pH of five,

00:05:37.060 --> 00:05:40.520
the solution with a pH
of five is more acidic.

00:05:40.520 --> 00:05:42.700
And going back to our equation for pH,

00:05:42.700 --> 00:05:44.320
pH is equal to the negative log

00:05:44.320 --> 00:05:46.500
of the concentration of hydronium ions.

00:05:46.500 --> 00:05:48.170
This is log based 10.

00:05:48.170 --> 00:05:50.880
Therefore the solution with the pH of five

00:05:50.880 --> 00:05:55.680
is 10 times more acidic than
a solution with a pH of six.

00:05:55.680 --> 00:05:58.040
And because of the way
the equation written,

00:05:58.040 --> 00:06:03.040
the higher the concentration
of hydronium ions in solution,

00:06:03.800 --> 00:06:07.710
the lower the value for the pH.

00:06:07.710 --> 00:06:11.350
And the lower the concentration
for the hydronium ions

00:06:11.350 --> 00:06:15.473
in solution, the higher
the value for them pH.

00:06:17.090 --> 00:06:20.030
And if an aqueous solution
has a pH greater than seven,

00:06:20.030 --> 00:06:22.740
we say that aqueous solution is basic.

00:06:22.740 --> 00:06:26.530
For example, our cleaning
solution with some ammonia in it

00:06:26.530 --> 00:06:29.310
had a pH of 11.30

00:06:29.310 --> 00:06:32.160
so that's right about
here on the pH scale.

00:06:32.160 --> 00:06:34.550
So we would consider that
cleaning solution with ammonia

00:06:34.550 --> 00:06:36.510
to be a basic solution.

00:06:36.510 --> 00:06:38.920
As you move to the right on the pH scale,

00:06:38.920 --> 00:06:43.030
you increase in the
basicity of the solution.

00:06:43.030 --> 00:06:45.330
So we've seen that pH is
equal to the negative log

00:06:45.330 --> 00:06:47.230
of the concentration of hydronium ions

00:06:47.230 --> 00:06:49.230
which you could also write pH is equal

00:06:49.230 --> 00:06:51.690
to the negative log of H+.

00:06:51.690 --> 00:06:54.830
Running it this way on the
right makes it easier to see

00:06:54.830 --> 00:06:57.890
how the pattern works
for other situations.

00:06:57.890 --> 00:07:02.630
For example, pOH is
defined as the negative log

00:07:02.630 --> 00:07:05.080
of the concentration of hydroxide ions.

00:07:05.080 --> 00:07:06.773
So we have the pOH here

00:07:06.773 --> 00:07:08.582
and then we have the
concentration of hydroxide ions

00:07:08.582 --> 00:07:10.810
over here, the same way we wrote pH

00:07:10.810 --> 00:07:13.730
with the concentration
of H+ ions over here.

00:07:13.730 --> 00:07:18.573
And also pKw would be equal
to the negative log of Kw.

00:07:19.430 --> 00:07:21.570
Let's go back to the Kw equation

00:07:21.570 --> 00:07:23.750
which says that the
concentration of hydronium ions

00:07:23.750 --> 00:07:25.800
times the concentration of hydroxide ions

00:07:25.800 --> 00:07:28.060
is equal to Kw.

00:07:28.060 --> 00:07:31.890
If we take the negative log
of both sides of this equation

00:07:31.890 --> 00:07:33.830
and we use our log properties,

00:07:33.830 --> 00:07:36.080
we get that the negative
log of the concentration

00:07:36.080 --> 00:07:39.410
of H3O+ plus the negative
log of the concentration

00:07:39.410 --> 00:07:42.663
of OH- is equal to the negative log of Kw.

00:07:43.740 --> 00:07:46.790
The negative log of the
concentration of H3O+

00:07:46.790 --> 00:07:48.710
is just equal to the pH,

00:07:48.710 --> 00:07:50.920
so we can write the pH down here,

00:07:50.920 --> 00:07:53.030
plus the negative log of the concentration

00:07:53.030 --> 00:07:56.390
of hydroxide ions,
that's equal to the pOH.

00:07:56.390 --> 00:08:01.390
So we have pH plus pOH is
equal to the negative log of Kw

00:08:02.580 --> 00:08:04.803
is equal to pKw.

00:08:07.340 --> 00:08:11.340
At 25 degrees Celsius, Kw is equal to 1.0

00:08:11.340 --> 00:08:13.760
times 10 to the negative 14.

00:08:13.760 --> 00:08:18.550
Therefore the negative log of
Kw is the negative log of 1.0

00:08:18.550 --> 00:08:23.403
times 10 to the negative
14 which is equal to 14.00.

00:08:26.420 --> 00:08:29.297
So we can plug in 14.00 in for pKw

00:08:31.880 --> 00:08:34.700
which gives us a very useful equation

00:08:34.700 --> 00:08:39.700
that says that the pH plus
the pOH is equal to 14.00.

00:08:42.430 --> 00:08:44.550
And this equation is true

00:08:44.550 --> 00:08:47.653
when the temperature
is 25 degrees Celsius.


Autoionization of water

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:02.610
- [Instructor] The
autoionization of water refers

00:00:02.610 --> 00:00:06.130
to the reaction of water
molecules to form two ions,

00:00:06.130 --> 00:00:08.860
the hydronium ion, which is H3O+,

00:00:08.860 --> 00:00:11.237
and the hydroxide ion, which is OH-.

00:00:12.290 --> 00:00:14.810
Water can function as an acid or a base,

00:00:14.810 --> 00:00:15.930
and in this reaction,

00:00:15.930 --> 00:00:19.330
one water molecule functions
as a Bronsted-Lowry acid

00:00:19.330 --> 00:00:21.070
and donates a proton

00:00:21.070 --> 00:00:23.370
and another water molecule functions

00:00:23.370 --> 00:00:25.330
as a Bronsted-Lowry base

00:00:25.330 --> 00:00:27.640
and accepts a proton.

00:00:27.640 --> 00:00:28.473
In the reaction,

00:00:28.473 --> 00:00:32.160
the base takes an H+ ion from the acid

00:00:32.160 --> 00:00:36.180
and these two electrons are
left behind on this oxygen.

00:00:36.180 --> 00:00:40.963
Adding an H+ to H2O gives
the hydronium ion H3O+,

00:00:40.963 --> 00:00:45.963
and taking away an H+ from H2O
gives the hydroxide ion OH-.

00:00:46.700 --> 00:00:49.510
We can write an equilibrium
constant expression

00:00:49.510 --> 00:00:50.940
for this reaction.

00:00:50.940 --> 00:00:55.120
So we would write the equilibrium
constant K is equal to,

00:00:55.120 --> 00:00:56.490
we would start with our products.

00:00:56.490 --> 00:00:59.700
We'd have the concentration
of hydronium ions.

00:00:59.700 --> 00:01:03.900
And since we have a coefficient
of one in front of hydronium

00:01:03.900 --> 00:01:05.070
in the balanced equation,

00:01:05.070 --> 00:01:07.050
it'd be the concentration
of hydronium ions

00:01:07.050 --> 00:01:08.730
raised to the first power

00:01:08.730 --> 00:01:12.620
times the concentration of hydroxide ions.

00:01:12.620 --> 00:01:14.360
And once again, there's
a coefficient of one,

00:01:14.360 --> 00:01:15.390
the balanced equation.

00:01:15.390 --> 00:01:17.470
So it's the concentration
of hydroxide ions

00:01:17.470 --> 00:01:19.060
raised to the first power.

00:01:19.060 --> 00:01:20.250
For the reactants,

00:01:20.250 --> 00:01:21.470
liquid water is left out

00:01:21.470 --> 00:01:23.930
of the equilibrium constant expression.

00:01:23.930 --> 00:01:25.900
Normally we would write KC,

00:01:25.900 --> 00:01:27.780
where the C stands for concentration

00:01:27.780 --> 00:01:29.380
for the equilibrium constant,

00:01:29.380 --> 00:01:31.460
since we're dealing with concentrations.

00:01:31.460 --> 00:01:34.710
However, this is a special
equilibrium constant expression

00:01:34.710 --> 00:01:36.850
for the autoionization of water,

00:01:36.850 --> 00:01:38.410
so instead of writing KC,

00:01:38.410 --> 00:01:39.770
we're gonna write KW,

00:01:39.770 --> 00:01:42.190
where W stands for water.

00:01:42.190 --> 00:01:45.660
KW is equal to 1.0 times
10 to the negative 14

00:01:45.660 --> 00:01:47.730
at 25 degrees Celsius,

00:01:47.730 --> 00:01:50.500
And with such a low value for KW,

00:01:50.500 --> 00:01:52.710
so this value is much less than one,

00:01:52.710 --> 00:01:54.560
that tells us at equilibrium,

00:01:54.560 --> 00:01:57.600
we have an extremely small
concentration of hydronium

00:01:57.600 --> 00:02:00.000
and hydroxide ions.

00:02:00.000 --> 00:02:05.000
So mostly we have H2O
molecules at equilibrium.

00:02:05.070 --> 00:02:06.799
Let's go ahead and solve
for the concentration

00:02:06.799 --> 00:02:10.670
of hydronium ions and
hydroxide ions at equilibrium.

00:02:10.670 --> 00:02:11.740
In the balanced equation,

00:02:11.740 --> 00:02:12.960
there's a coefficient of one

00:02:12.960 --> 00:02:15.710
in front of both hydronium and hydroxide.

00:02:15.710 --> 00:02:17.450
Therefore, at equilibrium,

00:02:17.450 --> 00:02:20.280
these two concentrations are equal.

00:02:20.280 --> 00:02:22.840
Since we don't know what
those concentrations are,

00:02:22.840 --> 00:02:25.900
we're gonna represent
it by writing in here X.

00:02:25.900 --> 00:02:30.900
So this would be X times X is equal to 1.0

00:02:31.060 --> 00:02:34.090
times 10 to the negative 14.

00:02:34.090 --> 00:02:37.709
So we would have X squared is equal to 1.0

00:02:37.709 --> 00:02:40.250
times 10 to the negative 14.

00:02:40.250 --> 00:02:43.760
And taking the square root of both sides,

00:02:43.760 --> 00:02:47.380
we would find that X is equal to 1.0

00:02:47.380 --> 00:02:50.840
times 10 to the negative seven.

00:02:50.840 --> 00:02:52.860
Therefore, if we had
a sample of pure water

00:02:52.860 --> 00:02:55.000
at 25 degrees Celsius,

00:02:55.000 --> 00:02:56.980
the concentration of hydronium ions

00:02:56.980 --> 00:02:58.794
and the concentration of hydroxide ions

00:02:58.794 --> 00:03:00.329
would be equal to 1.0

00:03:00.329 --> 00:03:03.023
times 10 to the negative seventh molar.

00:03:04.080 --> 00:03:05.860
Instead of using two water molecules

00:03:05.860 --> 00:03:08.580
to show the autoionization of water,

00:03:08.580 --> 00:03:12.880
it's also possible to show it
using only one water molecule.

00:03:12.880 --> 00:03:16.717
So H2O could break up to form H+ and OH-.

00:03:18.235 --> 00:03:20.840
So H+, which is the hydrogen ion,

00:03:20.840 --> 00:03:24.360
is sometimes used
interchangeably with H3O+,

00:03:24.360 --> 00:03:26.630
which is the hydronium ion.

00:03:26.630 --> 00:03:28.920
We've just seen that pure
water has a concentration

00:03:28.920 --> 00:03:31.460
of hydronium ions equal
to the concentration

00:03:31.460 --> 00:03:33.160
of hydroxide ions.

00:03:33.160 --> 00:03:36.620
Therefore, pure water
is a neutral substance,

00:03:36.620 --> 00:03:39.280
and for any aqueous solution
where the concentration

00:03:39.280 --> 00:03:40.450
of hydronium ion is equal

00:03:40.450 --> 00:03:42.650
to the concentration of hydroxide ion,

00:03:42.650 --> 00:03:45.690
we would classify that
as a neutral solution.

00:03:45.690 --> 00:03:47.810
If an aqueous solution has a concentration

00:03:47.810 --> 00:03:50.153
of hydronium ions that's
greater than the concentration

00:03:50.153 --> 00:03:51.970
of hydroxide ions,

00:03:51.970 --> 00:03:55.830
we would classify the solution
as an acidic solution.

00:03:55.830 --> 00:03:57.970
And if an aqueous solution
has a concentration

00:03:57.970 --> 00:04:00.510
of hydronium ions that's
less than the concentration

00:04:00.510 --> 00:04:01.760
of hydroxide ions,

00:04:01.760 --> 00:04:03.930
or you could say the
concentration of hydroxide ions

00:04:03.930 --> 00:04:05.900
is greater than that of hydronium,

00:04:05.900 --> 00:04:09.210
the solution would be
considered a basic solution.

00:04:09.210 --> 00:04:11.560
In the equation that we've
already talked about,

00:04:11.560 --> 00:04:13.290
the concentration of hydronium ions

00:04:13.290 --> 00:04:15.210
times the concentration of hydroxide ions

00:04:15.210 --> 00:04:16.680
is equal to KW,

00:04:16.680 --> 00:04:20.070
which is equal to 1.0
times 10 to the negative 14

00:04:20.070 --> 00:04:22.400
at 25 degrees Celsius.

00:04:22.400 --> 00:04:24.150
This equation is true

00:04:24.150 --> 00:04:25.870
if you're dealing with an acidic solution,

00:04:25.870 --> 00:04:27.370
a neutral solution,

00:04:27.370 --> 00:04:29.090
or a basic solution.

00:04:29.090 --> 00:04:33.550
And I'll call this equation
the KW equation from now on.

00:04:33.550 --> 00:04:35.430
Let's say we have an aqueous solution

00:04:35.430 --> 00:04:38.240
and the concentration of
hydronium ions in the solution

00:04:38.240 --> 00:04:41.470
is equal to 4.0 times 10
to the negative six molar

00:04:41.470 --> 00:04:43.470
at 25 degrees Celsius,

00:04:43.470 --> 00:04:45.730
and our goal is to
calculate the concentration

00:04:45.730 --> 00:04:50.370
of hydroxide ions in the
solution at 25 degrees Celsius.

00:04:50.370 --> 00:04:52.960
To solve for the concentration
of hydroxide ion,

00:04:52.960 --> 00:04:56.250
we can use our KW equation.

00:04:56.250 --> 00:04:58.760
So we need to plug in
for the concentration

00:04:58.760 --> 00:05:00.940
of hydronium ion.

00:05:00.940 --> 00:05:05.940
So that gives us 4.0 times
10 to the negative six

00:05:06.030 --> 00:05:08.140
times the concentration of hydroxide ion,

00:05:08.140 --> 00:05:10.100
which we'll just go ahead and make X here,

00:05:10.100 --> 00:05:11.940
and all that's equal to KW,

00:05:11.940 --> 00:05:16.140
which is equal to 1.0 times
10 to the negative 14.

00:05:16.140 --> 00:05:17.300
Solving for X,

00:05:17.300 --> 00:05:22.300
we find the X is equal to 2.5
times 10 to the negative nine.

00:05:23.350 --> 00:05:27.770
And since X is equal to the
concentration of hydroxide ion,

00:05:27.770 --> 00:05:30.570
the concentration of
hydroxide ion is equal to 2.5

00:05:30.570 --> 00:05:33.350
times 10 to the negative ninth molar.

00:05:33.350 --> 00:05:34.810
For this aqueous solution,

00:05:34.810 --> 00:05:38.560
the concentration of
hydronium ion is greater

00:05:38.560 --> 00:05:42.150
than the concentration of hydroxide ion.

00:05:42.150 --> 00:05:45.180
Therefore, this is an acidic solution.

00:05:45.180 --> 00:05:46.510
So let me go ahead and write that in here.

00:05:46.510 --> 00:05:49.210
This is an acidic solution.

00:05:49.210 --> 00:05:51.380
An equilibrium constant is only constant

00:05:51.380 --> 00:05:52.990
at a specific temperature.

00:05:52.990 --> 00:05:55.690
For example, at 25 degrees Celsius,

00:05:55.690 --> 00:05:59.750
KW is equal to 1.0 times
10 to the negative 14.

00:05:59.750 --> 00:06:01.130
But if you change the temperature,

00:06:01.130 --> 00:06:03.530
you change the value for KW.

00:06:03.530 --> 00:06:05.000
At 50 degrees Celsius,

00:06:05.000 --> 00:06:09.310
KW is equal to 5.5 times
10 to the negative 14.

00:06:09.310 --> 00:06:12.340
So an increase in temperature
from 25 degrees Celsius

00:06:12.340 --> 00:06:15.840
to 50 degrees Celsius causes an increase

00:06:15.840 --> 00:06:18.480
in the value for KW.

00:06:18.480 --> 00:06:23.480
So increase in temperature
causes an increase in KW.

00:06:23.650 --> 00:06:26.210
And we can use Le Chatelier's
principle to predict

00:06:26.210 --> 00:06:29.730
if the autoionization of water
is an endothermic reaction

00:06:29.730 --> 00:06:31.900
or an exothermic reaction.

00:06:31.900 --> 00:06:35.360
An increase in KW means
an increased concentration

00:06:35.360 --> 00:06:39.240
of hydronium ion and
hydroxide ion at equilibrium.

00:06:39.240 --> 00:06:42.610
Therefore, the net reaction
must have gone to the right

00:06:42.610 --> 00:06:45.730
to increase the amount of our products.

00:06:45.730 --> 00:06:48.440
And if we treat heat as a reactant

00:06:49.400 --> 00:06:51.080
and we increase the temperature,

00:06:51.080 --> 00:06:52.360
it's as if we've increased

00:06:52.360 --> 00:06:54.420
the amount of one of our reactants.

00:06:54.420 --> 00:06:57.450
Therefore, according to
Le Chatelier's principle,

00:06:57.450 --> 00:06:59.740
the net reaction is
gonna shift to the right

00:06:59.740 --> 00:07:01.550
to make more of the product.

00:07:01.550 --> 00:07:02.850
Since that's what we observed

00:07:02.850 --> 00:07:04.700
by increasing the value for KW,

00:07:04.700 --> 00:07:06.990
we know that the autoionization of water

00:07:06.990 --> 00:07:09.840
is an endothermic reaction.

00:07:09.840 --> 00:07:11.610
If we had put heat on the product side

00:07:11.610 --> 00:07:14.090
and treated this like
an exothermic reaction,

00:07:14.090 --> 00:07:16.350
we would've gotten a shift
in the wrong direction.

00:07:16.350 --> 00:07:17.950
We would've got a shift back to the left.

00:07:17.950 --> 00:07:20.450
So we know it's not exothermic.

00:07:20.450 --> 00:07:23.510
Finally, let's calculate the
concentration of hydronium ions

00:07:23.510 --> 00:07:27.530
and hydroxide ions in
a sample of pure water

00:07:27.530 --> 00:07:29.940
at 50 degrees Celsius.

00:07:29.940 --> 00:07:32.070
We can still use the KW equation.

00:07:32.070 --> 00:07:34.810
So KW is equal to the
concentration of hydronium ions

00:07:34.810 --> 00:07:37.270
times the concentration of hydroxide ions.

00:07:37.270 --> 00:07:40.680
However, since the temperature
is now 50 degrees Celsius,

00:07:40.680 --> 00:07:43.980
we can't use 1.0 times
10 to the negative 14

00:07:43.980 --> 00:07:46.770
because that's the KW
at 25 degrees Celsius.

00:07:46.770 --> 00:07:49.500
We need to use the KW
at 50 degrees Celsius,

00:07:49.500 --> 00:07:52.970
which is 5.5 times 10 to the negative 14.

00:07:52.970 --> 00:07:54.540
For the autoionization of water,

00:07:54.540 --> 00:07:56.070
the mole ratio of hydronium ion

00:07:56.070 --> 00:07:58.400
to hydroxide ion is one-to-one.

00:07:58.400 --> 00:08:00.520
Therefore the concentration
of hydronium ion

00:08:00.520 --> 00:08:03.490
is equal to the concentration
of hydroxide ion.

00:08:03.490 --> 00:08:05.330
So when we plug in for hydronium,

00:08:05.330 --> 00:08:07.090
if we say that concentration is X,

00:08:07.090 --> 00:08:09.790
then the concentration of
hydroxide would also be X.

00:08:09.790 --> 00:08:12.510
So we have X times X is equal to KW,

00:08:12.510 --> 00:08:15.923
which is equal to 5.5 times
10 to the negative 14.

00:08:16.800 --> 00:08:21.800
So X squared is equal to 5.5
times 10 to the negative 14.

00:08:21.900 --> 00:08:23.230
And to solve for X,

00:08:23.230 --> 00:08:26.980
we simply take the square
root of both sides.

00:08:26.980 --> 00:08:31.980
So X is equal to 2.3 times
10 to the negative seven.

00:08:33.510 --> 00:08:35.550
So the concentration of hydronium ions

00:08:35.550 --> 00:08:38.460
is equal to the concentration
of hydroxide ions,

00:08:38.460 --> 00:08:42.100
which is 2.3 times 10 to
the negative seventh molar.

00:08:42.100 --> 00:08:44.640
Notice that this is a higher concentration

00:08:44.640 --> 00:08:47.040
than we got at 25 degrees Celsius,

00:08:47.040 --> 00:08:50.680
which makes sense because the
value for KW has increased.

00:08:50.680 --> 00:08:52.598
However, since the
concentration of hydronium

00:08:52.598 --> 00:08:57.598
is still equal to the
concentration of hydroxide ions,

00:08:58.140 --> 00:09:00.303
pure water is still neutral.


Digital and analog information

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:01.640
- [Instructor] In this
video, we're going to talk

00:00:01.640 --> 00:00:04.050
about analog versus digital.

00:00:04.050 --> 00:00:06.300
Something that's analog can be any value

00:00:06.300 --> 00:00:08.940
within a given range
while something digital

00:00:08.940 --> 00:00:13.070
is represented by a number of
discreet or separate levels.

00:00:13.070 --> 00:00:14.390
To distinguish these two ideas,

00:00:14.390 --> 00:00:16.320
I like to think about clocks.

00:00:16.320 --> 00:00:18.550
An analog clock has the
numbers in the hands

00:00:18.550 --> 00:00:20.400
and it's analog because the motion

00:00:20.400 --> 00:00:22.430
of those hands is continuous.

00:00:22.430 --> 00:00:23.950
They can sweep across the circle

00:00:23.950 --> 00:00:27.530
representing any of infinite
times on that clock.

00:00:27.530 --> 00:00:30.140
For example, between 3:06 and 3:07,

00:00:30.140 --> 00:00:32.250
the minute hand is actually
going to be at some point

00:00:32.250 --> 00:00:34.290
between those marks on the clock

00:00:34.290 --> 00:00:36.520
showing one of the
infinitely possible times

00:00:36.520 --> 00:00:38.280
that the clock can represent.

00:00:38.280 --> 00:00:40.460
Compare that to a digital clock.

00:00:40.460 --> 00:00:44.530
A digital clock is only going
to show you 3:06 or 3:07.

00:00:44.530 --> 00:00:47.600
It will never display any of
the many fractional seconds

00:00:47.600 --> 00:00:49.330
between those two times.

00:00:49.330 --> 00:00:52.440
Digital only takes on
certain discrete values

00:00:52.440 --> 00:00:55.520
and it has a finite
number of those values.

00:00:55.520 --> 00:00:58.190
So an analog wave or
signal will smoothly sweep

00:00:58.190 --> 00:01:01.300
across the infinitely many
possible values it has

00:01:01.300 --> 00:01:02.960
while a digital wave or signal

00:01:02.960 --> 00:01:05.870
will only be at one of a
number of discrete values.

00:01:05.870 --> 00:01:09.560
So the shape of the wave will
be more square or step like.

00:01:09.560 --> 00:01:10.520
Let's check out an example

00:01:10.520 --> 00:01:12.290
so this makes a little more sense.

00:01:12.290 --> 00:01:13.440
I like music.

00:01:13.440 --> 00:01:15.410
So we're gonna talk about sound.

00:01:15.410 --> 00:01:18.270
Sound is an analog signal or wave.

00:01:18.270 --> 00:01:21.120
So if we look at a graph
of sound, volume over time,

00:01:21.120 --> 00:01:24.750
it's going to have a smooth
continuous analog wave form,

00:01:24.750 --> 00:01:28.140
both the amplitude or the
volume, and the frequency,

00:01:28.140 --> 00:01:31.340
what we hear as pitch
are changing continuously

00:01:31.340 --> 00:01:33.490
between infinite possible values.

00:01:33.490 --> 00:01:35.460
All right, and that's because sound waves,

00:01:35.460 --> 00:01:38.390
the vibration of particles
propagating through the air

00:01:38.390 --> 00:01:40.550
actually changes continuously.

00:01:40.550 --> 00:01:43.300
The very first sound recording
and reproduction technology

00:01:43.300 --> 00:01:46.690
imprinted that analog wave
directly onto a material.

00:01:46.690 --> 00:01:50.260
For example, records imprint
that sound wave into vinyl

00:01:50.260 --> 00:01:53.780
and cassettes imprint
the sound wave onto tape.

00:01:53.780 --> 00:01:55.790
A major drawback of this technology

00:01:55.790 --> 00:01:59.190
is after the sound to play back
exactly as it was recorded,

00:01:59.190 --> 00:02:02.100
that wave form needs to
stay untouched, right?

00:02:02.100 --> 00:02:05.700
So think about scratching vinyl
or smudging a cassette tape,

00:02:05.700 --> 00:02:07.940
that's directly deforming the wave.

00:02:07.940 --> 00:02:10.080
So you'll never be able
to reproduce the sound

00:02:10.080 --> 00:02:11.920
exactly as it was recorded.

00:02:11.920 --> 00:02:16.020
So, technology advanced and
sound waves became digitized.

00:02:16.020 --> 00:02:16.853
Here's how.

00:02:17.850 --> 00:02:20.050
All right, so recall
our analog sound wave.

00:02:20.050 --> 00:02:22.980
We have a smooth analog wave
that's taking on any number

00:02:22.980 --> 00:02:25.700
of infinitely possible
values within this range.

00:02:25.700 --> 00:02:27.880
In order to digitize this wave,

00:02:27.880 --> 00:02:30.510
we're going to ascribe
numbers to the amplitude

00:02:30.510 --> 00:02:32.280
at different points, all right?

00:02:32.280 --> 00:02:33.740
Watch this magic.

00:02:33.740 --> 00:02:35.200
So we go over here and make a scale.

00:02:35.200 --> 00:02:36.780
So we're breaking up the amplitudes

00:02:36.780 --> 00:02:39.170
into discrete possibilities.

00:02:39.170 --> 00:02:41.810
Then we can go through the
wave and at specific points

00:02:41.810 --> 00:02:44.550
of the wave measure what is the amplitude

00:02:44.550 --> 00:02:46.010
based on that scale.

00:02:46.010 --> 00:02:48.470
So over here, we're at the
first point of the scale.

00:02:48.470 --> 00:02:51.120
At this peak, we're at the
second point of our scale.

00:02:51.120 --> 00:02:55.200
Then the first, the third,
the second, the fourth,

00:02:55.200 --> 00:02:56.483
back down to the first.

00:02:57.370 --> 00:03:00.020
Now that we have this way
broken up into levels, right?

00:03:00.020 --> 00:03:02.450
We can ascribe the
numbers and we effectively

00:03:02.450 --> 00:03:05.730
turn this analog wave
into a set of numbers,

00:03:05.730 --> 00:03:09.740
one, two, one, three, two, four, one.

00:03:09.740 --> 00:03:12.083
Our wave has been digitized.

00:03:12.930 --> 00:03:15.310
Now that digitize wave can be played back

00:03:15.310 --> 00:03:18.350
through a speaker to
recreate the analog wave.

00:03:18.350 --> 00:03:20.980
As long as the sampling
happens at a quick enough rate,

00:03:20.980 --> 00:03:22.950
humans can't tell the difference.

00:03:22.950 --> 00:03:24.650
All right, so the digitization of waves

00:03:24.650 --> 00:03:27.200
is all about ascribing specific numbers

00:03:27.200 --> 00:03:30.130
to some of those mechanical
properties of the wave.

00:03:30.130 --> 00:03:32.270
The important thing here
is that now that the wave

00:03:32.270 --> 00:03:34.890
has been digitized, the
digitized sound wave

00:03:34.890 --> 00:03:37.320
can be reliably stored, processed,

00:03:37.320 --> 00:03:39.550
and communicated with computers.

00:03:39.550 --> 00:03:41.830
So some information is
lost in translation,

00:03:41.830 --> 00:03:43.790
but once the wave is digitized,

00:03:43.790 --> 00:03:46.470
it's quality will never degrade, okay?

00:03:46.470 --> 00:03:49.300
And that allows for a lot
more reliable technology

00:03:49.300 --> 00:03:51.220
because the wave is
represented with numbers

00:03:51.220 --> 00:03:54.670
instead of it being physically
imprinted on some material.

00:03:54.670 --> 00:03:58.200
So humans prefer to store
information like sound digitally,

00:03:58.200 --> 00:04:00.720
and there are ways to turn analog signals,

00:04:00.720 --> 00:04:03.250
which can represent any of
infinite possible values

00:04:03.250 --> 00:04:05.100
into digital information,

00:04:05.100 --> 00:04:07.220
which is useful because
information is stored

00:04:07.220 --> 00:04:10.353
only at a number of
discreet or separate levels.


Wave properties

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:02.080
- [Instructor] Imagine
that I'm standing here

00:00:02.080 --> 00:00:03.640
holding the end of a rope.

00:00:03.640 --> 00:00:05.980
I'm over here on the left end,
and while holding the rope,

00:00:05.980 --> 00:00:08.610
I rapidly moved my hand up, down,

00:00:08.610 --> 00:00:10.820
and back to the starting position.

00:00:10.820 --> 00:00:12.700
If we were to take a snapshot of the rope

00:00:12.700 --> 00:00:15.150
immediately after I finished my motion,

00:00:15.150 --> 00:00:17.220
we're going to see something like this.

00:00:17.220 --> 00:00:19.220
The rope has a squiggly disturbance

00:00:19.220 --> 00:00:21.880
that mirrors the motion
I made with my hand,

00:00:21.880 --> 00:00:24.780
up, down, and back to the middle.

00:00:24.780 --> 00:00:27.340
And the rest of the rope is still flat.

00:00:27.340 --> 00:00:28.580
You might've seen something like this

00:00:28.580 --> 00:00:30.080
if you've ever played with a jump rope

00:00:30.080 --> 00:00:31.770
and wiggled it back and forth,

00:00:31.770 --> 00:00:33.900
or a slinky and you've seen that oscillate

00:00:33.900 --> 00:00:35.340
back and forth on the ground,

00:00:35.340 --> 00:00:36.530
or if you've been in the gym

00:00:36.530 --> 00:00:39.520
and seen somebody doing exercises
with large battle ropes,

00:00:39.520 --> 00:00:42.350
slamming them up and down repeatedly.

00:00:42.350 --> 00:00:44.360
And we know that over
time, this disturbance

00:00:44.360 --> 00:00:47.340
is actually going to make
its way through the rope.

00:00:47.340 --> 00:00:49.980
If this is what we observe
right after my hand motion,

00:00:49.980 --> 00:00:51.410
at some later point in time,

00:00:51.410 --> 00:00:53.040
we will observe that the
beginning of the rope

00:00:53.040 --> 00:00:56.170
is back to its original shape,
the squiggly disturbance

00:00:56.170 --> 00:00:58.420
has made its way further down the rope,

00:00:58.420 --> 00:01:00.150
and it will keep traveling
in this direction

00:01:00.150 --> 00:01:02.510
until it reaches the end of the rope.

00:01:02.510 --> 00:01:05.630
This is exactly what a wave is in physics.

00:01:05.630 --> 00:01:08.110
A wave is a disturbance, in this case,

00:01:08.110 --> 00:01:10.870
the squiggle in the rope
caused by my hand motion,

00:01:10.870 --> 00:01:12.960
and that disturbance can propagate,

00:01:12.960 --> 00:01:16.300
it can travel or move in
a particular direction.

00:01:16.300 --> 00:01:19.773
So a wave is a disturbance
that can propagate.

00:01:20.970 --> 00:01:24.210
This particular example is
called a mechanical wave.

00:01:24.210 --> 00:01:26.180
It's called a mechanical
wave because the disturbance

00:01:26.180 --> 00:01:29.440
is traveling through a medium,
in this case, the rope.

00:01:29.440 --> 00:01:31.783
So mechanical waves
travel through a medium.

00:01:32.670 --> 00:01:34.300
One important point about waves

00:01:34.300 --> 00:01:36.130
that is worth noting right now

00:01:36.130 --> 00:01:40.340
is that waves transfer energy
without transferring matter.

00:01:40.340 --> 00:01:41.173
So what that means

00:01:41.173 --> 00:01:42.720
is that the disturbance
that is moving here,

00:01:42.720 --> 00:01:45.450
this squiggly shape is
moving through the rope,

00:01:45.450 --> 00:01:48.580
but it isn't moving the rope
to a different position.

00:01:48.580 --> 00:01:50.280
Any part of the rope might go up and down

00:01:50.280 --> 00:01:52.530
as the wave travels through that section,

00:01:52.530 --> 00:01:55.340
but the rope itself is not going anywhere,

00:01:55.340 --> 00:01:57.020
rather it's the kinetic energy

00:01:57.020 --> 00:01:59.230
imparted to the rope by my hand

00:01:59.230 --> 00:02:01.680
that is transferring
from particle to particle

00:02:01.680 --> 00:02:03.800
and making its way through the rope.

00:02:03.800 --> 00:02:07.703
So waves transfer energy, but not matter.

00:02:09.510 --> 00:02:10.470
So in my first example,

00:02:10.470 --> 00:02:12.600
I only jerked my hand up and down once,

00:02:12.600 --> 00:02:14.290
which created a single wave pulse

00:02:14.290 --> 00:02:16.040
that moved through my rope.

00:02:16.040 --> 00:02:18.250
If instead, I were to keep moving my hand

00:02:18.250 --> 00:02:19.900
up and down consistently,

00:02:19.900 --> 00:02:22.660
I would see a wave form that
looks something like this.

00:02:22.660 --> 00:02:24.210
And when we model a wave,

00:02:24.210 --> 00:02:26.730
there are a few key characteristics
that we need to know

00:02:26.730 --> 00:02:28.180
about that wave.

00:02:28.180 --> 00:02:30.270
First is the period.

00:02:30.270 --> 00:02:32.710
Period is measured in
seconds and it tells us

00:02:32.710 --> 00:02:35.523
how long it takes for one
wave cycle to complete.

00:02:36.400 --> 00:02:37.930
Next is the wavelength.

00:02:37.930 --> 00:02:40.430
Measured in units of distance like meters,

00:02:40.430 --> 00:02:43.440
the wavelength is the distance
between identical points

00:02:43.440 --> 00:02:45.290
of adjacent waves.

00:02:45.290 --> 00:02:47.520
And finally there's frequency.

00:02:47.520 --> 00:02:50.650
So if the wave point that we've
drawn here takes one second,

00:02:50.650 --> 00:02:53.080
there are four cycles in that one second,

00:02:53.080 --> 00:02:55.610
that means it has a frequency of 4hz

00:02:55.610 --> 00:02:57.870
or four cycles per second.

00:02:57.870 --> 00:03:00.930
So the frequency measured
in cycles per second

00:03:00.930 --> 00:03:03.963
tells us how many wave cycles
there are every second.

00:03:05.130 --> 00:03:08.870
Now using just these basic
anatomical properties of a wave,

00:03:08.870 --> 00:03:09.703
we can start to pick out

00:03:09.703 --> 00:03:11.770
more interesting physical characteristics,

00:03:11.770 --> 00:03:14.980
like speed or distance over time.

00:03:14.980 --> 00:03:17.370
If we want to know how
fast a wave is traveling,

00:03:17.370 --> 00:03:18.960
we can take its wavelength,

00:03:18.960 --> 00:03:21.590
which is the distance
covered by a single cycle,

00:03:21.590 --> 00:03:23.770
and multiply that by the frequency,

00:03:23.770 --> 00:03:26.870
which is how many cycles
are completed in a second,

00:03:26.870 --> 00:03:28.560
a given amount of time.

00:03:28.560 --> 00:03:29.770
The cycles cancel out

00:03:29.770 --> 00:03:32.010
and we're left with units
of distance over time,

00:03:32.010 --> 00:03:33.630
the same as speed.

00:03:33.630 --> 00:03:35.860
And that's our equation
for the speed of the wave.

00:03:35.860 --> 00:03:38.660
Wavelength times frequency.

00:03:38.660 --> 00:03:41.210
The standard units for
speed are meters per second.

00:03:42.430 --> 00:03:43.263
There are a couple of factors

00:03:43.263 --> 00:03:45.330
that can affect the speed of a wave.

00:03:45.330 --> 00:03:47.230
The first is the wave type.

00:03:47.230 --> 00:03:50.370
So different types of waves
move at different speeds.

00:03:50.370 --> 00:03:52.210
A relatable example of different waves

00:03:52.210 --> 00:03:54.430
moving at different speeds is lightning.

00:03:54.430 --> 00:03:55.710
Have you ever seen lightning strike

00:03:55.710 --> 00:03:57.160
or been in a thunderstorm?

00:03:57.160 --> 00:03:58.680
You know that the first thing you see

00:03:58.680 --> 00:04:01.880
is the flash of lightning,
and then you hear the thunder

00:04:01.880 --> 00:04:04.800
associated with that lightning flash.

00:04:04.800 --> 00:04:07.723
So the lightning comes first
and the thunder comes second.

00:04:08.610 --> 00:04:10.330
That's because those
are two different waves

00:04:10.330 --> 00:04:12.380
that are part of the same phenomenon.

00:04:12.380 --> 00:04:15.170
When the lightning strike
hits, you see the flash first

00:04:15.170 --> 00:04:18.220
because that's an
electromagnetic wave, light.

00:04:18.220 --> 00:04:19.590
It travels much faster

00:04:19.590 --> 00:04:22.770
than the sound associated
with the lightning strike.

00:04:22.770 --> 00:04:24.370
Electromagnetic waves are special

00:04:24.370 --> 00:04:26.800
not only because they travel really fast,

00:04:26.800 --> 00:04:29.930
but they also don't need a
medium to travel through.

00:04:29.930 --> 00:04:32.390
The thunder on the other
hand is a sound wave

00:04:32.390 --> 00:04:34.260
traveling slower than the light.

00:04:34.260 --> 00:04:37.900
So you'll see the lightning
before you hear the thunder.

00:04:37.900 --> 00:04:40.993
Different wave types
move at different speeds.

00:04:42.180 --> 00:04:44.860
The second key factor that
can affect the speed of a wave

00:04:44.860 --> 00:04:47.620
is the medium through
which the wave travels

00:04:47.620 --> 00:04:50.240
and we'll consider sound
as an example here.

00:04:50.240 --> 00:04:51.610
So when someone is talking, right,

00:04:51.610 --> 00:04:53.770
we have this talking head
creating some vibrations

00:04:53.770 --> 00:04:55.430
of the particles in front of their mouth,

00:04:55.430 --> 00:04:56.770
that's the sound wave.

00:04:56.770 --> 00:04:58.680
It's the vibration of those particles

00:04:58.680 --> 00:05:00.500
propagating through the air.

00:05:00.500 --> 00:05:03.520
When you speak, your
vocal cords exert force

00:05:03.520 --> 00:05:05.610
on the particles just in front of you.

00:05:05.610 --> 00:05:08.710
They vibrate back and forth,
creating a compression

00:05:08.710 --> 00:05:11.630
that transfers to the
surrounding particles.

00:05:11.630 --> 00:05:15.580
As the vibrations continue to
propagate, the sound travels.

00:05:15.580 --> 00:05:16.760
You can imagine

00:05:16.760 --> 00:05:19.470
that if these particles
are packed closer together,

00:05:19.470 --> 00:05:22.640
those vibrations are going to
transfer a lot more quickly

00:05:22.640 --> 00:05:24.680
because the particles
are colliding much faster

00:05:24.680 --> 00:05:26.420
than if they're further apart.

00:05:26.420 --> 00:05:30.000
So sound travels much
faster in water, a liquid,

00:05:30.000 --> 00:05:32.860
than it does in air,
for that exact reason.

00:05:32.860 --> 00:05:35.630
The particles in the
liquid are closer together.

00:05:35.630 --> 00:05:38.380
Since they're closer
compacted, they collide more,

00:05:38.380 --> 00:05:41.530
and the propagation of
the wave happens faster.

00:05:41.530 --> 00:05:43.750
So different ways move at different speeds

00:05:43.750 --> 00:05:46.160
and the medium through
which a wave travels

00:05:46.160 --> 00:05:48.123
can also affect the speed of a wave.

00:05:49.300 --> 00:05:51.630
All right, so let's try to
summarize all this information.

00:05:51.630 --> 00:05:55.490
We have waves, a wave, a
disturbance that can propagate,

00:05:55.490 --> 00:05:57.930
and it has a few key characteristics.

00:05:57.930 --> 00:05:59.120
There's the period

00:05:59.120 --> 00:06:01.520
or how long it takes
one cycle to complete,

00:06:01.520 --> 00:06:02.830
there's the wavelength,

00:06:02.830 --> 00:06:04.710
the distance between identical points

00:06:04.710 --> 00:06:07.170
on two waves that are next to each other,

00:06:07.170 --> 00:06:08.740
and the frequency,

00:06:08.740 --> 00:06:11.860
which is how many waves
cycles complete in one second.

00:06:11.860 --> 00:06:14.590
In this case, we have
two cycles in one second

00:06:14.590 --> 00:06:15.917
for a frequency of 2hz.

00:06:17.170 --> 00:06:20.750
Wave speed is found by multiplying
wavelength and frequency,

00:06:20.750 --> 00:06:23.800
and that wave speed is
affected by the type of wave

00:06:23.800 --> 00:06:26.660
and the medium through
which the wave travels.

00:06:26.660 --> 00:06:29.150
Mechanical waves are ways
that travel through a medium,

00:06:29.150 --> 00:06:33.270
so sound, a slinky or rope, ocean waves,

00:06:33.270 --> 00:06:36.440
and electromagnetic waves
like light are special

00:06:36.440 --> 00:06:38.680
because they can travel through a vacuum.

00:06:38.680 --> 00:06:42.050
They don't have to have a
medium in order to propagate.

00:06:42.050 --> 00:06:44.440
But all waves, no matter what type,

00:06:44.440 --> 00:06:47.313
transfer energy, not matter.


Calculating height using energy

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:02.400
- [Instructor] So I have
an uncompressed spring here

00:00:02.400 --> 00:00:04.860
and the spring has a spring constant

00:00:04.860 --> 00:00:06.960
of four Newtons per meter.

00:00:06.960 --> 00:00:10.600
Then I take a 10 gram
mass, a 10 gram ball,

00:00:10.600 --> 00:00:13.040
and I put it at the top of
the spring and I push down

00:00:13.040 --> 00:00:15.980
to compress that spring by 10 centimeters.

00:00:15.980 --> 00:00:18.350
And so let's call that
scenario one right over there,

00:00:18.350 --> 00:00:21.890
where our mass is on top
of this compressed spring.

00:00:21.890 --> 00:00:23.600
And then let's say, we let go.

00:00:23.600 --> 00:00:26.490
And then the spring launches
this mass into the air,

00:00:26.490 --> 00:00:29.670
and then this mass is gonna
hit some maximum height.

00:00:29.670 --> 00:00:31.330
And let's call that scenario two

00:00:31.330 --> 00:00:33.550
when we are hitting that maximum height.

00:00:33.550 --> 00:00:37.310
And my question to you is,
what is that maximum height

00:00:37.310 --> 00:00:40.080
based on all of the information
that I have given you here?

00:00:40.080 --> 00:00:41.750
And I'll give you a hint.

00:00:41.750 --> 00:00:43.520
What we have to think about is the idea

00:00:43.520 --> 00:00:45.410
that energy is conserved.

00:00:45.410 --> 00:00:48.160
The total energy in scenario
one is gonna be equal

00:00:48.160 --> 00:00:50.820
to the total energy in scenario two.

00:00:50.820 --> 00:00:53.240
So pause this video and
see if you can figure out

00:00:53.240 --> 00:00:55.240
what that maximum height is going to be.

00:00:56.370 --> 00:00:59.700
All right, now let's work
through this together.

00:00:59.700 --> 00:01:02.570
I told you that the total
energy is gonna be conserved,

00:01:02.570 --> 00:01:05.290
but what's that total energy
going to be made up of?

00:01:05.290 --> 00:01:07.870
Well, it's gonna have
some potential energy

00:01:07.870 --> 00:01:11.670
and it's going to have
some kinetic energy.

00:01:11.670 --> 00:01:13.760
And so another way to think about it is,

00:01:13.760 --> 00:01:16.210
our potential energy in scenario one

00:01:16.210 --> 00:01:19.220
plus our kinetic energy in scenario one

00:01:19.220 --> 00:01:24.220
needs to be equal to our
potential energy in scenario two,

00:01:25.340 --> 00:01:30.070
plus our kinetic energy in scenario two.

00:01:30.070 --> 00:01:32.180
And there might be other energies here

00:01:32.180 --> 00:01:34.320
that you could think
about heat due to friction

00:01:34.320 --> 00:01:36.020
with the air, but we're not going to,

00:01:36.020 --> 00:01:38.240
we're going to ignore those
to just simplify the problem.

00:01:38.240 --> 00:01:40.250
We can assume that this
is happening in a vacuum,

00:01:40.250 --> 00:01:42.320
that might help us a little bit.

00:01:42.320 --> 00:01:44.170
So if we think about
the potential energies,

00:01:44.170 --> 00:01:46.690
there's actually two types of
potential energies at play.

00:01:46.690 --> 00:01:48.490
There's gravitational potential energy,

00:01:48.490 --> 00:01:50.730
and there is elastic potential energy

00:01:50.730 --> 00:01:52.520
due to the fact that this mass is sitting

00:01:52.520 --> 00:01:54.150
on a compressed spring.

00:01:54.150 --> 00:01:56.530
So our gravitational potential energy

00:01:56.530 --> 00:01:59.310
is going to be our mass times the strength

00:01:59.310 --> 00:02:01.050
of our gravitational field,

00:02:01.050 --> 00:02:04.020
times our height in position one,

00:02:04.020 --> 00:02:07.120
and our elastic potential energy

00:02:07.120 --> 00:02:12.120
is going to be equal to one
half times our spring constant,

00:02:12.430 --> 00:02:15.780
times how much that spring
is compressed, squared.

00:02:15.780 --> 00:02:19.250
And so this is all of our
potential energy in scenario one,

00:02:19.250 --> 00:02:20.083
right over here.

00:02:20.083 --> 00:02:21.750
And then we add our kinetic energy.

00:02:21.750 --> 00:02:25.920
So plus, we have one half times our mass,

00:02:25.920 --> 00:02:29.590
times our velocity in
scenario one, squared.

00:02:29.590 --> 00:02:31.590
And so the sum of this is gonna be the sum

00:02:31.590 --> 00:02:33.600
of all of this in scenario two.

00:02:33.600 --> 00:02:38.160
And so that is going to
be equal to mass times G

00:02:38.160 --> 00:02:40.229
times height in scenario two,

00:02:40.229 --> 00:02:44.300
plus one half times our spring constant

00:02:44.300 --> 00:02:46.910
times how much that's spring is compressed

00:02:46.910 --> 00:02:49.300
in scenario two squared,

00:02:49.300 --> 00:02:53.570
plus one half times our mass,

00:02:53.570 --> 00:02:56.680
times our velocity in
scenario two, squared.

00:02:56.680 --> 00:03:00.540
And just as a reminder, what
we have right over here,

00:03:00.540 --> 00:03:03.520
this is our potential
energy in scenario two.

00:03:03.520 --> 00:03:05.810
And then our kinetic
energy is right over there.

00:03:05.810 --> 00:03:08.250
Now this might look
really hairy and daunting,

00:03:08.250 --> 00:03:10.690
but there's a lot of simplification here.

00:03:10.690 --> 00:03:13.760
We can define our starting
point right over here,

00:03:13.760 --> 00:03:16.540
H one as being equal to zero,

00:03:16.540 --> 00:03:18.080
which will simplify this dramatically.

00:03:18.080 --> 00:03:20.940
Cause if H one is zero, then
this term right over here,

00:03:20.940 --> 00:03:23.870
zero, we also know that
our velocity is zero

00:03:23.870 --> 00:03:25.350
in our starting scenario.

00:03:25.350 --> 00:03:27.620
So that would make our
kinetic energy zero.

00:03:27.620 --> 00:03:29.634
So the left-hand side is really

00:03:29.634 --> 00:03:32.330
all about our elastic potential energy.

00:03:32.330 --> 00:03:35.030
So it's gonna be one half
times our spring constant,

00:03:35.030 --> 00:03:36.990
times how much we have
compressed the spring

00:03:36.990 --> 00:03:39.010
in scenario one, squared.

00:03:39.010 --> 00:03:41.870
And then on the right-hand
side, what's going on?

00:03:41.870 --> 00:03:45.160
Well, in this scenario, our
spring is no longer compressed.

00:03:45.160 --> 00:03:48.970
So our elastic potential
energy is now zero.

00:03:48.970 --> 00:03:51.060
And what about our kinetic energy?

00:03:51.060 --> 00:03:53.750
Well, at maximum height right over here,

00:03:53.750 --> 00:03:57.450
your ball is actually
stationary for an instant,

00:03:57.450 --> 00:03:58.840
for a moment.

00:03:58.840 --> 00:04:01.260
It's right at that
moment where it is going

00:04:01.260 --> 00:04:04.340
from moving up to starting to move down.

00:04:04.340 --> 00:04:07.820
And so our velocity is
actually zero right over here.

00:04:07.820 --> 00:04:12.040
So V2 is actually zero,
just like V1 was zero.

00:04:12.040 --> 00:04:13.550
So that's gonna be zero.

00:04:13.550 --> 00:04:16.750
And so you have a scenario
where our initial elastic

00:04:16.750 --> 00:04:21.180
potential energy is going to
be equal to our scenario two,

00:04:21.180 --> 00:04:24.090
gravitational potential energy.

00:04:24.090 --> 00:04:27.840
And so we just need to solve
for this right over here.

00:04:27.840 --> 00:04:30.340
That is going to be our maximum height.

00:04:30.340 --> 00:04:33.570
To do that, we can divide both sides by mg

00:04:33.570 --> 00:04:38.570
and we get h2 is equal to
1/2 k delta x, one squared,

00:04:42.740 --> 00:04:44.683
all of that over mg.

00:04:46.910 --> 00:04:48.850
And we know what all of these things are,

00:04:48.850 --> 00:04:50.780
and I'll write it out with the units.

00:04:50.780 --> 00:04:54.820
This is going to be equal to one half

00:04:54.820 --> 00:04:56.130
times our spring constant,

00:04:56.130 --> 00:05:00.223
which is four Newtons per meter.

00:05:01.080 --> 00:05:03.120
Now our change in x is 10 centimeters,

00:05:03.120 --> 00:05:04.320
but we have to be very careful.

00:05:04.320 --> 00:05:07.440
We can't just put a 10 centimeters
here and then square it.

00:05:07.440 --> 00:05:09.300
We want the units to match up.

00:05:09.300 --> 00:05:10.980
So we're not dealing with
centimeters and grams.

00:05:10.980 --> 00:05:13.030
We're dealing with meters and kilograms.

00:05:13.030 --> 00:05:15.680
So I wanna convert this
10 centimeters to meters.

00:05:15.680 --> 00:05:17.760
Well, that's gonna be 0.1 meters.

00:05:17.760 --> 00:05:21.580
So I will write this as times 0.1.

00:05:21.580 --> 00:05:23.150
That's how much the spring is compressed.

00:05:23.150 --> 00:05:25.390
That's our delta x in terms of meters.

00:05:25.390 --> 00:05:27.100
And so that is gonna be squared.

00:05:27.100 --> 00:05:30.350
And then all of that
over, what is our mass?

00:05:30.350 --> 00:05:32.960
And once again, we want
to express our mass

00:05:32.960 --> 00:05:34.710
in terms of kilograms.

00:05:34.710 --> 00:05:39.710
So our mass is 0.01 kilograms.

00:05:41.180 --> 00:05:46.180
G is 9.8 meters per second squared.

00:05:47.140 --> 00:05:51.750
And when you calculate it, this
is approximately 0.2 meters.

00:05:51.750 --> 00:05:53.910
And the units actually all do work out.

00:05:53.910 --> 00:05:55.940
Because if you look at
Newtons as being the same

00:05:55.940 --> 00:05:59.980
as kilogram meters per second squared,

00:05:59.980 --> 00:06:04.120
this kilograms cancels with
that kilograms right over there.

00:06:04.120 --> 00:06:06.850
You can see that this per second squared

00:06:06.850 --> 00:06:08.860
is gonna cancel with
that per second squared

00:06:08.860 --> 00:06:09.840
right over there.

00:06:09.840 --> 00:06:12.780
And then this meter is going
to cancel out with that meter.

00:06:12.780 --> 00:06:14.700
And so what you're left
with is a meter squared

00:06:14.700 --> 00:06:15.880
divided by meters,

00:06:15.880 --> 00:06:17.820
which is just going to
leave you with meters,

00:06:17.820 --> 00:06:18.780
just like that.

00:06:18.780 --> 00:06:19.800
And we're done.

00:06:19.800 --> 00:06:22.700
We figured out the maximum
height using just our knowledge

00:06:22.700 --> 00:06:27.700
of the conservation of energy
is approximately 0.2 meters.