Sn1 reactions intro

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:02.460
So what do we have right here?

00:00:02.460 --> 00:00:05.731
Well, this is just
a water molecule.

00:00:05.731 --> 00:00:07.230
And this right over
here, let's see,

00:00:07.230 --> 00:00:11.720
we have 1, 2, 3 carbons
in our backbone.

00:00:11.720 --> 00:00:15.130
So this would be
meth, ethy, propane.

00:00:15.130 --> 00:00:16.760
We all have single
bonds right here.

00:00:16.760 --> 00:00:18.110
So this is propane.

00:00:18.110 --> 00:00:19.700
Let me write that out.

00:00:19.700 --> 00:00:23.960
This is-- give myself
some space-- propane.

00:00:23.960 --> 00:00:25.760
And then on the
number 2 carbon, I

00:00:25.760 --> 00:00:28.490
have both a methyl
group and a bromo group.

00:00:28.490 --> 00:00:31.680
And bromo comes before
methyl in alphabetical order.

00:00:31.680 --> 00:00:33.740
So I could write this as
2-bromo-2-methylpropane.

00:00:45.350 --> 00:00:47.764
So that's this molecule that
we have right over here.

00:00:47.764 --> 00:00:49.180
Now, what we're
going to imagine--

00:00:49.180 --> 00:00:50.730
it's sitting in
some water, and I

00:00:50.730 --> 00:00:53.490
have one water molecule
drawn right over here.

00:00:53.490 --> 00:00:56.230
Let's imagine that we have
this bromine right over here.

00:00:56.230 --> 00:00:58.920
It's pretty neutral.

00:00:58.920 --> 00:01:02.702
It is neutral because it
has seven valence electrons.

00:01:02.702 --> 00:01:04.660
Or you could say, it has
the charge equivalent.

00:01:04.660 --> 00:01:06.930
It has 1, 2, 3, 4, 5, 6.

00:01:06.930 --> 00:01:09.110
And then it has half
of this bond that's

00:01:09.110 --> 00:01:11.000
composed of two
electrons, so that's

00:01:11.000 --> 00:01:14.210
what's keeping this
neutral right over here.

00:01:14.210 --> 00:01:17.680
But we do know that
bromine-- well,

00:01:17.680 --> 00:01:22.312
it's not completely unstable
if it were to take an electron,

00:01:22.312 --> 00:01:24.520
because that would get it
to eight valence electrons.

00:01:24.520 --> 00:01:26.890
We know that it's more
electronegative than carbon,

00:01:26.890 --> 00:01:28.730
which means that, if
it's sharing a bond,

00:01:28.730 --> 00:01:32.620
it's more likely to take the
entire bond than carbon is.

00:01:32.620 --> 00:01:34.740
So let's imagine a scenario.

00:01:34.740 --> 00:01:37.680
Let's imagine a scenario
where bromine takes these two

00:01:37.680 --> 00:01:41.100
carbons in this bond
right over here.

00:01:41.100 --> 00:01:44.100
And this isn't going to be a
super-fast thing that happens,

00:01:44.100 --> 00:01:45.690
because this molecule
right over here,

00:01:45.690 --> 00:01:48.429
the 2 bromo-2 methylpropane,
is actually reasonably stable.

00:01:48.429 --> 00:01:50.970
So I'm not saying this is going
to just happen automatically.

00:01:50.970 --> 00:01:52.500
It's going to happen super-fast.

00:01:52.500 --> 00:01:53.970
But it could happen.

00:01:53.970 --> 00:01:55.770
And so to show that
this isn't going

00:01:55.770 --> 00:01:58.050
to be only a one-way
reaction, I'll

00:01:58.050 --> 00:02:00.110
show that it's in equilibrium.

00:02:00.110 --> 00:02:03.530
So the equilibrium, I'll
draw two arrows like that.

00:02:03.530 --> 00:02:06.570
And so once this
happens, the bromine

00:02:06.570 --> 00:02:08.460
would turn into a bromide anion.

00:02:08.460 --> 00:02:10.449
So let me draw it right over.

00:02:10.449 --> 00:02:16.460
So the bromide anion-- so it's
going to have 1, 2, 3, 4, 5, 6.

00:02:16.460 --> 00:02:24.060
And then both electrons
from that shared bond,

00:02:24.060 --> 00:02:27.120
it now has a negative 1 charge.

00:02:27.120 --> 00:02:30.390
Now, if that happened, what
would happen to this left over,

00:02:30.390 --> 00:02:35.820
to what's now going to be the
thing that the bromine was

00:02:35.820 --> 00:02:37.760
connected to, I
guess you could say?

00:02:37.760 --> 00:02:39.060
Well, let's draw that.

00:02:39.060 --> 00:02:42.590
So we have this carbon,
our number 2 carbon.

00:02:42.590 --> 00:02:45.640
We have the CH3
right over there.

00:02:45.640 --> 00:02:49.090
And we go back to CH3.

00:02:49.090 --> 00:02:51.500
And then I had this
blue methyl group.

00:02:51.500 --> 00:02:53.570
So I had this blue
methyl group right here.

00:02:53.570 --> 00:02:56.705
And the way I'm drawing, it
kind of pops out of the page.

00:02:56.705 --> 00:03:00.610
My attempt to draw it popping
out of the page-- so this

00:03:00.610 --> 00:03:05.130
is it popping out of the page.

00:03:05.130 --> 00:03:07.750
And now, what's its
charge going to be?

00:03:07.750 --> 00:03:09.950
Well, it just gave
up half of this bond,

00:03:09.950 --> 00:03:12.950
the equivalent of an
electron's charge.

00:03:12.950 --> 00:03:14.540
So it just gave
up-- you can kind of

00:03:14.540 --> 00:03:16.570
think of it as giving
up an electron.

00:03:16.570 --> 00:03:19.165
So this is going to
have a positive charge.

00:03:22.970 --> 00:03:24.970
And a positive charged
carbon like this,

00:03:24.970 --> 00:03:27.520
we would call this
a carbocation.

00:03:27.520 --> 00:03:34.610
So this right over here is a
carbocation, positive charge.

00:03:34.610 --> 00:03:37.930
And the reason why this is
not a crazy thing to happen

00:03:37.930 --> 00:03:39.970
is that this is a
tertiary carbon, which

00:03:39.970 --> 00:03:43.720
means it is connected to
three other carbons-- 1, 2, 3.

00:03:43.720 --> 00:03:45.800
A tertiary carbon, one
way to think about it

00:03:45.800 --> 00:03:48.660
is that this positive
charge can kind of

00:03:48.660 --> 00:03:51.660
be shared with its
brothers a little bit more.

00:03:51.660 --> 00:03:55.545
So a tertiary carbocation is
more stable than a secondary

00:03:55.545 --> 00:03:57.862
and definitely more
stable than a primary.

00:03:57.862 --> 00:04:00.320
Secondary would be if it was
only connected to two carbons.

00:04:00.320 --> 00:04:02.361
But this is tertiary-- so
let me write this down.

00:04:02.361 --> 00:04:05.180
This is a tertiary carbocation.

00:04:05.180 --> 00:04:08.640
This carbon is connected
to three other carbons.

00:04:08.640 --> 00:04:10.530
That's where the
tertiary comes from.

00:04:10.530 --> 00:04:12.602
So once again, it can happen.

00:04:12.602 --> 00:04:14.310
I'm not saying that
this reaction is just

00:04:14.310 --> 00:04:16.100
going to go superfast
in that direction.

00:04:16.100 --> 00:04:20.839
that's why we did these
equilibrium arrows.

00:04:20.839 --> 00:04:22.830
Now, let's bring this
oxygen into play.

00:04:26.970 --> 00:04:30.580
Sorry, not the oxygen, let's
bring this water into play.

00:04:30.580 --> 00:04:34.410
So this water, as you can
imagine, the water is neutral.

00:04:34.410 --> 00:04:36.610
It seems pretty
reasonable in the state.

00:04:36.610 --> 00:04:40.840
But the oxygen end
of the water does

00:04:40.840 --> 00:04:44.260
have a partial negative charge.

00:04:44.260 --> 00:04:49.430
We've seen in the past that,
because oxygen is a lot more

00:04:49.430 --> 00:04:51.170
electronegative
than hydrogen, you

00:04:51.170 --> 00:04:54.310
have a partial negative side
on the side of the oxygens,

00:04:54.310 --> 00:04:56.700
and you have a partial
positive charge,

00:04:56.700 --> 00:04:58.990
on the side the hydrogens.

00:05:01.720 --> 00:05:03.970
And so you have this
positive carbocation.

00:05:03.970 --> 00:05:06.100
This has a partially
negative charge.

00:05:06.100 --> 00:05:08.900
So you could imagine
that this molecule might

00:05:08.900 --> 00:05:12.032
be attracted to this is nucleus.

00:05:12.032 --> 00:05:13.740
Now, it's not going
to be super attracted

00:05:13.740 --> 00:05:15.720
the way that a
hydroxide anion would.

00:05:15.720 --> 00:05:18.104
A hydroxide anion has
a full negative charge.

00:05:18.104 --> 00:05:20.270
This only just has a little
partial negative charge.

00:05:20.270 --> 00:05:22.730
But it could still be attracted
to something very positive

00:05:22.730 --> 00:05:23.600
like this.

00:05:23.600 --> 00:05:28.440
So in this situation, the water
molecule is a weak nucleophile.

00:05:28.440 --> 00:05:34.150
So this is a weak nucleophile,
not as strong as a hydroxide

00:05:34.150 --> 00:05:35.790
anion that has a
full negative charge.

00:05:35.790 --> 00:05:37.581
But if it sees something
positive like this

00:05:37.581 --> 00:05:39.330
and it bumps in
the right way, you

00:05:39.330 --> 00:05:43.350
can imagine that one of
these pairs of this oxygen

00:05:43.350 --> 00:05:55.130
would then go and essentially
form a bond, making

00:05:55.130 --> 00:05:57.802
this carbon right
over here neutral.

00:05:57.802 --> 00:05:59.260
So what is that
going to look like?

00:05:59.260 --> 00:06:01.560
And this would happen
faster, so I'm not

00:06:01.560 --> 00:06:03.430
going to do this
as in equilibrium.

00:06:03.430 --> 00:06:06.080
This could happen like this.

00:06:06.080 --> 00:06:09.790
And let me just copy
and paste the bromine,

00:06:09.790 --> 00:06:13.000
so I don't have to
redraw the whole thing.

00:06:13.000 --> 00:06:17.110
So let me copy, and
let me paste it.

00:06:17.110 --> 00:06:21.115
So that's our bromine right
over there, draw it right there.

00:06:21.115 --> 00:06:24.450
And actually, let me
erase this little smudge

00:06:24.450 --> 00:06:26.940
and make sure it still
has a negative charge.

00:06:26.940 --> 00:06:31.490
So negative charge--
it's a bromide anion now.

00:06:31.490 --> 00:06:33.920
And now, let me draw
the everything else.

00:06:33.920 --> 00:06:37.290
So I have this carbon.

00:06:37.290 --> 00:06:39.540
Let me scroll down a little
bit, so I have some space.

00:06:39.540 --> 00:06:43.070
Carbon, I have the
carbon in the back-- CH3.

00:06:43.070 --> 00:06:46.320
I have the carbon on top-- CH3.

00:06:46.320 --> 00:06:53.940
I have the carbon in
front, like that-- CH3.

00:06:53.940 --> 00:06:57.130
And now my oxygen, or I should
say, this molecule right over,

00:06:57.130 --> 00:07:00.420
the water molecule, has
attached to the carbon.

00:07:00.420 --> 00:07:04.530
So these two electrons,
you could say,

00:07:04.530 --> 00:07:05.800
they've attacked the carbon.

00:07:05.800 --> 00:07:09.460
They now form a bond
with the carbon.

00:07:09.460 --> 00:07:12.770
So these are those two
electrons in this covalent bond.

00:07:12.770 --> 00:07:14.650
And now, let's draw
the rest of this.

00:07:14.650 --> 00:07:15.890
So you have oxygen.

00:07:15.890 --> 00:07:18.660
You have these two electrons
over here, these two

00:07:18.660 --> 00:07:20.200
valence electrons.

00:07:20.200 --> 00:07:25.170
And then you have one hydrogen
and then another hydrogen.

00:07:25.170 --> 00:07:29.240
Now, what is going
to be the charge

00:07:29.240 --> 00:07:31.200
right over here on the oxygen?

00:07:31.200 --> 00:07:33.735
Well, it had a partial
negative charge,

00:07:33.735 --> 00:07:35.360
just because it's
more electronegative.

00:07:35.360 --> 00:07:42.360
But it just gave essentially
half of a bond, half of a pair.

00:07:42.360 --> 00:07:44.740
Or it's now sharing a
pair with this carbon,

00:07:44.740 --> 00:07:47.420
so it's equivalent of giving
up one electron's equivalent

00:07:47.420 --> 00:07:48.130
of charge.

00:07:48.130 --> 00:07:50.360
So you would now say
that this thing right now

00:07:50.360 --> 00:07:54.380
has a positive charge.

00:07:54.380 --> 00:07:58.480
So you could imagine maybe
another water molecule

00:07:58.480 --> 00:08:01.640
comes along.

00:08:01.640 --> 00:08:04.114
Let me draw another water
molecule right over here.

00:08:04.114 --> 00:08:05.530
We're sitting in
water, so there's

00:08:05.530 --> 00:08:07.470
plenty of water molecules.

00:08:07.470 --> 00:08:12.430
So another water molecule might
come along and say, OK, well,

00:08:12.430 --> 00:08:15.840
this thing right over
here is positive.

00:08:15.840 --> 00:08:20.750
But this oxygen is hogging
these bonds a little bit more

00:08:20.750 --> 00:08:24.150
than the hydrogens, especially
now that it's become positive.

00:08:24.150 --> 00:08:30.070
And so this might give its
pair to a hydrogen proton.

00:08:30.070 --> 00:08:31.570
And then this bond
with the hydrogen

00:08:31.570 --> 00:08:35.980
will go entirely to the oxygen.

00:08:35.980 --> 00:08:38.230
Let me draw that in
a different color.

00:08:38.230 --> 00:08:44.420
So this bond right over here
goes entirely to the oxygen.

00:08:44.420 --> 00:08:47.220
Electronegativity means if
this is more electronegative

00:08:47.220 --> 00:08:48.990
than that, that
if a bond breaks,

00:08:48.990 --> 00:08:53.870
that pair is more likely to
go to the more electronegative

00:08:53.870 --> 00:08:58.440
molecule, or I should say,
the more electronegative atom.

00:08:58.440 --> 00:09:01.190
So what will this look like
when all is said and done?

00:09:01.190 --> 00:09:04.350
So let's draw an
arrow like this.

00:09:04.350 --> 00:09:07.530
So let me draw my
main molecules.

00:09:07.530 --> 00:09:09.310
So I have the carbon.

00:09:09.310 --> 00:09:12.480
It bonded to another
carbon right up there.

00:09:12.480 --> 00:09:15.385
It bonded to another
carbon right over there.

00:09:18.220 --> 00:09:21.630
We have our carbon in
front just like that.

00:09:21.630 --> 00:09:24.860
We have the bond
with our oxygen.

00:09:24.860 --> 00:09:28.910
So we have our bond with
our oxygen right over there.

00:09:28.910 --> 00:09:30.850
So that's our oxygen.

00:09:30.850 --> 00:09:34.340
We have this bond
with a hydrogen.

00:09:34.340 --> 00:09:36.050
We have this pair.

00:09:36.050 --> 00:09:39.100
And now, it has
gained this pair.

00:09:39.100 --> 00:09:41.920
It has gained this pair
that was with that hydrogen,

00:09:41.920 --> 00:09:43.500
so just like this.

00:09:43.500 --> 00:09:44.765
This is now neutral.

00:09:44.765 --> 00:09:47.440
It went from being
positive-- over here,

00:09:47.440 --> 00:09:49.150
it only had half of this bond.

00:09:49.150 --> 00:09:52.120
So it's kind of the equivalent
of one electron charge.

00:09:52.120 --> 00:09:53.342
Now, it has a whole bond.

00:09:53.342 --> 00:09:54.550
So now it has the equivalent.

00:09:54.550 --> 00:09:56.850
It literally has both electrons.

00:09:56.850 --> 00:09:59.400
So now this thing is neutral.

00:09:59.400 --> 00:10:01.510
And now this molecule
right over here

00:10:01.510 --> 00:10:04.150
becomes a hydronium cation.

00:10:04.150 --> 00:10:05.440
Let me draw that.

00:10:05.440 --> 00:10:06.460
So this is water.

00:10:06.460 --> 00:10:07.360
So let me see.

00:10:07.360 --> 00:10:09.990
So that's the two
hydrogens right over there.

00:10:09.990 --> 00:10:13.600
This is that pair
that is on top.

00:10:13.600 --> 00:10:19.150
And now it is sharing this
pair with the hydrogen,

00:10:19.150 --> 00:10:23.850
essentially with a proton,
so with this hydrogen

00:10:23.850 --> 00:10:24.530
right over here.

00:10:27.680 --> 00:10:31.850
And now, this will
have a positive charge.

00:10:31.850 --> 00:10:33.510
This is a hydronium
cation and maybe

00:10:33.510 --> 00:10:35.137
this bumps into
another water molecule

00:10:35.137 --> 00:10:36.470
and does the same thing with it.

00:10:36.470 --> 00:10:38.780
And it just keeps
going on and on and on.

00:10:38.780 --> 00:10:45.700
And of course, we can't forget
our bromide anion friend.

00:10:45.700 --> 00:10:48.620
So there's our
bromide anion friend.

00:10:48.620 --> 00:10:50.920
We've got a negative
charge right over there.

00:10:50.920 --> 00:10:52.930
And so what are we left with?

00:10:52.930 --> 00:10:58.080
Well, we are left with-- well,
there's this hydronium cation.

00:10:58.080 --> 00:10:59.480
It's a positive charge.

00:10:59.480 --> 00:11:01.920
You have this bromide anion.

00:11:01.920 --> 00:11:05.050
And now, what is this
molecule now called?

00:11:05.050 --> 00:11:07.200
Well, we'll go into more
detail in the future.

00:11:07.200 --> 00:11:10.170
But when you have a hydroxyl
group attached to the carbon

00:11:10.170 --> 00:11:12.840
backbone, it is an alcohol.

00:11:12.840 --> 00:11:17.040
And so our main backbone, once
again, has 1, 2, 3 carbons.

00:11:17.040 --> 00:11:18.470
So that tells us prop.

00:11:18.470 --> 00:11:22.490
But instead of just saying
propane, it's propanol.

00:11:22.490 --> 00:11:24.110
So let's write that down.

00:11:24.110 --> 00:11:32.200
This is propan-- and some people
say just propanol, generally.

00:11:32.200 --> 00:11:34.605
But if we want to be a
little bit particular,

00:11:34.605 --> 00:11:37.922
the hydroxyl group is attached
to the number 2 carbon.

00:11:37.922 --> 00:11:39.130
So we would say, propan-2-ol.

00:11:43.710 --> 00:11:46.750
And of course, we still
have this blue methyl group.

00:11:46.750 --> 00:11:53.990
So we could say 2-methyl--
let me just make this--

00:11:53.990 --> 00:11:59.630
2-methylpropan-2-ol is
what we are left with.

00:11:59.630 --> 00:12:01.260
Now, let's think
about what we can

00:12:01.260 --> 00:12:05.620
name this reaction we just saw.

00:12:05.620 --> 00:12:08.210
Well, we've just
substituted a bromo group

00:12:08.210 --> 00:12:09.680
with a hydroxyl group.

00:12:09.680 --> 00:12:11.290
So I think it's
reasonable to say

00:12:11.290 --> 00:12:13.570
that this is going to be
a substitution reaction.

00:12:17.400 --> 00:12:20.470
And we substituted with a--
it was a weak nucleophile,

00:12:20.470 --> 00:12:23.072
but it was a
nucleophile nonetheless.

00:12:23.072 --> 00:12:24.280
So it involved a nucleophile.

00:12:28.170 --> 00:12:31.600
But this one, unlike
the Sn2 reaction

00:12:31.600 --> 00:12:33.240
in the rate-determining
step-- and this

00:12:33.240 --> 00:12:34.448
is the rate-determining step.

00:12:34.448 --> 00:12:36.450
This is the slowest
step of the process.

00:12:36.450 --> 00:12:38.210
Everything else
will just kind of

00:12:38.210 --> 00:12:40.090
start happening once
this thing happens.

00:12:40.090 --> 00:12:41.690
This thing kind of
is in equilibrium.

00:12:41.690 --> 00:12:43.490
This is the
rate-determining step.

00:12:43.490 --> 00:12:46.740
Only one reactant is involved
in the rate-determining step.

00:12:55.900 --> 00:12:58.080
In this case, it
was the bromo group.

00:12:58.080 --> 00:13:00.540
So you might guess what
we're going to call it.

00:13:00.540 --> 00:13:04.860
It's substitution involving a
nucleophile, so nucleophilic

00:13:04.860 --> 00:13:05.740
substitution.

00:13:05.740 --> 00:13:08.020
You can imagine, this
stands for nucleophilic.

00:13:08.020 --> 00:13:11.450
And only one
reactant is involved

00:13:11.450 --> 00:13:14.140
in the rate-determining step.

00:13:14.140 --> 00:13:17.230
So this right over here, this
whole reaction that we've now

00:13:17.230 --> 00:13:22.490
drawn is an Sn1 reaction.

00:13:22.490 --> 00:13:24.300
We're now going to
explore one of the most

00:13:24.300 --> 00:13:27.600
fundamental reactions
in organic chemistry.

00:13:27.600 --> 00:13:29.370
And first, let's
look at what are

00:13:29.370 --> 00:13:31.340
the things that
are going to react.

00:13:31.340 --> 00:13:34.590
So over here, I have
a hydroxide anion.

00:13:34.590 --> 00:13:37.980
It has a negative 1 charge,
and that negative charge

00:13:37.980 --> 00:13:40.180
comes from the fact
that oxygen is neutral

00:13:40.180 --> 00:13:41.850
if it has six valence electrons.

00:13:41.850 --> 00:13:46.870
But this has 1, 2,
3, 4, 5, 6 plus half

00:13:46.870 --> 00:13:48.790
of this bond with hydrogen.

00:13:48.790 --> 00:13:51.830
So you could think of that
as another electron or it's

00:13:51.830 --> 00:13:54.220
share-- it's got
half of these two,

00:13:54.220 --> 00:13:55.640
so that's another
electron, which

00:13:55.640 --> 00:13:59.380
gives it a negative 1 charge.

00:13:59.380 --> 00:14:01.860
Now, this right over
here, it has one carbon.

00:14:01.860 --> 00:14:03.650
I guess, you could
say, in its backbone.

00:14:03.650 --> 00:14:06.290
So it's a methane.

00:14:06.290 --> 00:14:09.860
And it has a group right over
here consisting of bromine.

00:14:09.860 --> 00:14:13.967
So this is bromomethane.

00:14:13.967 --> 00:14:15.550
So let's think a
little bit about what

00:14:15.550 --> 00:14:18.380
might happen under
the right conditions.

00:14:18.380 --> 00:14:21.840
So this hydroxide anion, we've
already gone over the fact

00:14:21.840 --> 00:14:24.580
that you can kind of view it as
having an extra electron, which

00:14:24.580 --> 00:14:26.300
gives us its negative charge.

00:14:26.300 --> 00:14:29.610
Well, if you have a negative
charged anion like this,

00:14:29.610 --> 00:14:32.460
it will be attracted
to positive things.

00:14:32.460 --> 00:14:37.730
And one example of a
positive thing is a nucleus.

00:14:37.730 --> 00:14:39.760
And so in this
reaction, we are going

00:14:39.760 --> 00:14:44.930
to refer to this hydroxide
anion as the nucleophile.

00:14:48.124 --> 00:14:51.670
Phile literally refers to
things that something likes.

00:14:51.670 --> 00:14:54.430
This thing likes--
it's a nucleus lover,

00:14:54.430 --> 00:14:55.690
I guess you could say.

00:14:55.690 --> 00:15:01.330
And so it might want to share
maybe a pair of electrons

00:15:01.330 --> 00:15:03.710
with a nucleus.

00:15:03.710 --> 00:15:06.740
Now, over here, I
have this bromo group

00:15:06.740 --> 00:15:08.860
in the bromomethane.

00:15:08.860 --> 00:15:12.490
And bromine is quite
electronegative.

00:15:12.490 --> 00:15:14.430
It is more electronegative
than carbon.

00:15:14.430 --> 00:15:16.520
So even in this bond--
we have two electrons

00:15:16.520 --> 00:15:18.590
in this bond right
over here-- bromine

00:15:18.590 --> 00:15:22.150
is going to hog the electrons
a little bit more than carbon.

00:15:22.150 --> 00:15:26.055
And also, electronegativity
says that if this bond were

00:15:26.055 --> 00:15:28.820
to break, it's more likely
that this pair, the pair that's

00:15:28.820 --> 00:15:30.980
in this bond, is going
to go to the bromine

00:15:30.980 --> 00:15:32.950
than it's going to
go to the carbon.

00:15:32.950 --> 00:15:35.680
So the way that this reaction
works-- and it's really

00:15:35.680 --> 00:15:37.430
kind of happening in
one step, and this

00:15:37.430 --> 00:15:39.120
is key to the name
of this reaction,

00:15:39.120 --> 00:15:42.690
as we'll see in a second-- is
that this nucleophile right

00:15:42.690 --> 00:15:44.400
over here might say,
hey, let me share.

00:15:44.400 --> 00:15:45.740
I've got this extra charge.

00:15:45.740 --> 00:15:50.120
Let me share some of that
charge with a nucleus,

00:15:50.120 --> 00:15:51.410
in this case, the carbon.

00:15:51.410 --> 00:15:53.090
Let me draw it like this.

00:15:53.090 --> 00:15:56.590
In this case, the
carbon, so it would

00:15:56.590 --> 00:15:59.100
share this pair, which
would form a bond.

00:15:59.100 --> 00:16:00.930
Carbon would have
half of the pair.

00:16:00.930 --> 00:16:02.620
Oxygen would have
half of the pair.

00:16:02.620 --> 00:16:05.100
So it would share this pair.

00:16:05.100 --> 00:16:07.080
It would share this
pair with carbon.

00:16:07.080 --> 00:16:09.390
Now, if carbon is now
sharing half of this,

00:16:09.390 --> 00:16:13.710
it's essentially
getting an electron.

00:16:13.710 --> 00:16:16.240
It doesn't need this electron,
which bromine is already

00:16:16.240 --> 00:16:17.500
hogging a little bit.

00:16:17.500 --> 00:16:20.830
And so bromine would take
this entire pair back.

00:16:20.830 --> 00:16:24.330
Whenever we draw full arrows
like this, just normal arrows

00:16:24.330 --> 00:16:26.080
that you're used to
seeing, it's referring

00:16:26.080 --> 00:16:27.590
to the action of pairs.

00:16:27.590 --> 00:16:29.440
So this whole pair
is going to bromine.

00:16:29.440 --> 00:16:31.390
This pair right over
here is now going

00:16:31.390 --> 00:16:33.000
to be shared with the carbon.

00:16:33.000 --> 00:16:35.540
It's going to form a
bond with the carbon.

00:16:35.540 --> 00:16:37.840
So when we do both
of these steps,

00:16:37.840 --> 00:16:41.830
after both of these steps,
it would look like this.

00:16:41.830 --> 00:16:44.120
Let me draw my
bromine first, which

00:16:44.120 --> 00:16:47.050
is now going to become
a bromide anion.

00:16:47.050 --> 00:16:55.950
So bromine 1, 2, 3,
4, 5, 6, and now it

00:16:55.950 --> 00:17:01.540
gets both of these
electrons from this bond.

00:17:01.540 --> 00:17:04.810
So it has both of those
electrons right over there.

00:17:04.810 --> 00:17:07.867
Now, this will have
a negative 1 charge.

00:17:07.867 --> 00:17:08.700
How do we know that?

00:17:08.700 --> 00:17:13.099
Well, bromine is neutral, when
it has seven valence electrons.

00:17:13.099 --> 00:17:14.569
This thing now has eight.

00:17:14.569 --> 00:17:18.845
So this now has a
negative charge.

00:17:18.845 --> 00:17:20.470
Now, once again,
let's think about what

00:17:20.470 --> 00:17:22.530
happens to all of this business.

00:17:22.530 --> 00:17:25.900
So let me draw my
carbon first, my carbon.

00:17:25.900 --> 00:17:29.010
Now, let me draw this
hydrogen that's popping out.

00:17:29.010 --> 00:17:32.770
So this hydrogen that's popping
out, let me draw it like that.

00:17:32.770 --> 00:17:34.200
So that's the bond,
trying to show

00:17:34.200 --> 00:17:36.130
that's popping out of the page.

00:17:36.130 --> 00:17:37.384
So that's that hydrogen.

00:17:37.384 --> 00:17:38.800
I just made them
different colors,

00:17:38.800 --> 00:17:40.800
so we can keep track of
the different hydrogens.

00:17:40.800 --> 00:17:43.490
Then the hydrogen that's
going into the page, that's

00:17:43.490 --> 00:17:45.050
that one right over there.

00:17:45.050 --> 00:17:47.350
And you have this
hydrogen that's going up,

00:17:47.350 --> 00:17:49.570
so that hydrogen
right over there.

00:17:49.570 --> 00:17:53.380
And now the hydroxide anion
has bonded to this carbon.

00:17:53.380 --> 00:17:56.420
Now it will be a hydroxyl group.

00:17:56.420 --> 00:17:59.480
So this thing has now bonded.

00:17:59.480 --> 00:18:02.440
So let me draw the bond in the
color of these two electrons.

00:18:02.440 --> 00:18:04.110
It has now bonded.

00:18:04.110 --> 00:18:06.780
So those two electrons
now make up this blur,

00:18:06.780 --> 00:18:08.810
this bond right over here.

00:18:08.810 --> 00:18:14.790
And now we have our oxygen,
but with those two, these two

00:18:14.790 --> 00:18:20.022
right over here, and then
the bond to the hydrogen.

00:18:20.022 --> 00:18:21.230
Now, what just happened here?

00:18:21.230 --> 00:18:23.115
The oxygen is now neutral.

00:18:23.115 --> 00:18:24.290
Why is it neutral?

00:18:24.290 --> 00:18:26.220
Well, it has 1, 2, 3, 4.

00:18:26.220 --> 00:18:27.830
And then it has
half of this bond,

00:18:27.830 --> 00:18:30.030
so you could view
that as an electron

00:18:30.030 --> 00:18:31.950
or the equivalent of
an electron charge.

00:18:31.950 --> 00:18:33.450
And then it has
half of this bond.

00:18:33.450 --> 00:18:35.710
So it has the electron
charge equivalent

00:18:35.710 --> 00:18:38.480
of six valence electrons,
which makes oxygen neutral.

00:18:38.480 --> 00:18:39.730
And now bromine is negative.

00:18:39.730 --> 00:18:41.938
So one way to think about
is that the negative charge

00:18:41.938 --> 00:18:45.500
has been transferred to the
bromine in this reaction.

00:18:45.500 --> 00:18:48.300
Now, in this reaction,
bromine has left.

00:18:48.300 --> 00:18:51.340
And that's why we call
this the leaving group.

00:18:51.340 --> 00:18:58.326
We call this right over here the
leaving group of the reaction.

00:18:58.326 --> 00:18:59.950
This thing right over
here is the thing

00:18:59.950 --> 00:19:01.790
that is going to leave.

00:19:01.790 --> 00:19:03.592
That is the leaving group.

00:19:03.592 --> 00:19:05.800
Now, the other thing that
you might find interesting,

00:19:05.800 --> 00:19:08.510
it looks like kind of
this hydroxide anion--

00:19:08.510 --> 00:19:11.330
some people even say--
attacked this bromomethane.

00:19:11.330 --> 00:19:12.630
The bromomethane left.

00:19:16.870 --> 00:19:19.900
The bromine left, not
bromomethane left.

00:19:19.900 --> 00:19:21.950
It attacked the bromomethane.

00:19:21.950 --> 00:19:23.524
The bromine left.

00:19:23.524 --> 00:19:25.440
So in some ways, you'd
say that there's really

00:19:25.440 --> 00:19:27.140
two reactants over here.

00:19:27.140 --> 00:19:31.180
You have the hydroxide anion,
and you have the bromide anion.

00:19:31.180 --> 00:19:33.440
And this carbon right over
here with the hydrogens,

00:19:33.440 --> 00:19:35.750
this was kind of the thing
that just to some degree

00:19:35.750 --> 00:19:37.360
facilitated the reaction.

00:19:37.360 --> 00:19:39.340
This thing bumped
into it and attached,

00:19:39.340 --> 00:19:40.760
and then this thing left.

00:19:40.760 --> 00:19:43.874
And so in this context,
we would call the carbon

00:19:43.874 --> 00:19:46.290
with a three hydrogens-- we
would call this the substrate.

00:19:50.800 --> 00:19:54.510
Now, what should we
call this reaction?

00:19:54.510 --> 00:19:58.080
Well one way to think about it
is, we substituted this bromo

00:19:58.080 --> 00:20:00.720
group with a hydroxyl group.

00:20:00.720 --> 00:20:04.410
So it might make sense to call
this a substitution reaction.

00:20:04.410 --> 00:20:05.695
Substitution has occurred.

00:20:08.690 --> 00:20:10.414
Now, what did we
substitute with?

00:20:10.414 --> 00:20:12.080
Well, we substituted
with a nucleophile.

00:20:18.250 --> 00:20:20.310
And how many reactants
were involved

00:20:20.310 --> 00:20:21.980
in the rate-determining step?

00:20:21.980 --> 00:20:24.188
And I know that might not
make a lot of sense to you,

00:20:24.188 --> 00:20:27.610
but this had essentially
one determining step.

00:20:27.610 --> 00:20:29.680
This had to want to
bump in the right way.

00:20:29.680 --> 00:20:33.240
And this would want to have to
leave right at the same time,

00:20:33.240 --> 00:20:35.180
or roughly the
same time, in order

00:20:35.180 --> 00:20:37.020
for this reaction to occur.

00:20:37.020 --> 00:20:39.070
So this reactant
and this reactant

00:20:39.070 --> 00:20:41.400
were involved in
figuring out, well,

00:20:41.400 --> 00:20:43.640
how fast is this
reaction going to happen?

00:20:43.640 --> 00:20:47.202
And we're going to see future
reactions, where you don't have

00:20:47.202 --> 00:20:49.410
both reactants involved in
the rate-determining step.

00:20:49.410 --> 00:20:51.000
We'll see only one
reactant involved

00:20:51.000 --> 00:20:52.440
in the rate determining step.

00:20:52.440 --> 00:20:57.790
But I'll just write this right
over here-- two reactants

00:20:57.790 --> 00:21:00.620
in rate determining step.

00:21:00.620 --> 00:21:03.700
There's a garbage truck outside.

00:21:03.700 --> 00:21:06.440
I think the garbage truck
has left now, safe to resume.

00:21:06.440 --> 00:21:08.570
So as we were saying,
there are two reactants

00:21:08.570 --> 00:21:18.160
involved in the rate
determining step.

00:21:18.160 --> 00:21:20.270
And so the shorthand
for this reaction

00:21:20.270 --> 00:21:27.720
is to call it an Sn2 reaction.

00:21:30.560 --> 00:21:32.970
It's substitution with
a nucleophile, where

00:21:32.970 --> 00:21:37.796
both reactants are involved
in the rate-determining step.


Steric hindrance

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.930
We've got four scenarios
right over here.

00:00:02.930 --> 00:00:07.960
In each scenario, we
have a hydroxide anion.

00:00:07.960 --> 00:00:10.060
And then, we have
another molecule.

00:00:10.060 --> 00:00:11.720
And right over
here, this molecule,

00:00:11.720 --> 00:00:13.990
this is a bromomethane.

00:00:13.990 --> 00:00:15.960
So it's a bromo group
attached to a carbon.

00:00:15.960 --> 00:00:18.620
And this carbon isn't
attached to any other carbons.

00:00:18.620 --> 00:00:20.690
It's only attached
to other hydrogens.

00:00:20.690 --> 00:00:24.690
So carbon like this we
call a primer-- actually,

00:00:24.690 --> 00:00:25.250
not a primer.

00:00:25.250 --> 00:00:32.310
We call this a
methylcarbon because it

00:00:32.310 --> 00:00:34.927
is attached to no other carbons.

00:00:34.927 --> 00:00:36.760
Now, let's think about
this right over here.

00:00:36.760 --> 00:00:38.630
What's the difference between
this scenario and that one?

00:00:38.630 --> 00:00:40.046
We still have our
hydroxide anion.

00:00:40.046 --> 00:00:41.790
We still have the bromo group.

00:00:41.790 --> 00:00:46.180
But now this carbon is
attached to another carbon.

00:00:46.180 --> 00:00:49.555
So this right over here
is a primary carbon.

00:00:56.880 --> 00:00:58.740
It's attached to
one other carbon.

00:00:58.740 --> 00:01:00.380
And I think you
see a pattern here.

00:01:00.380 --> 00:01:01.440
What about this one?

00:01:01.440 --> 00:01:03.989
Well, this carbon right over
here is a secondary carbon.

00:01:03.989 --> 00:01:05.363
It's attached to
2 other carbons.

00:01:08.970 --> 00:01:11.590
And then, finally, this
carbon right over here

00:01:11.590 --> 00:01:13.130
is a tertiary carbon.

00:01:13.130 --> 00:01:14.875
It's attached to
3 other carbons.

00:01:20.090 --> 00:01:22.350
And as we'll see in a
few seconds, the fact

00:01:22.350 --> 00:01:24.850
that it has these other carbons
and that these other carbons

00:01:24.850 --> 00:01:28.810
kind of might block some
or all of the reaction,

00:01:28.810 --> 00:01:34.380
it will change what
actually goes on.

00:01:34.380 --> 00:01:37.540
So let's go, first, to the
first scenario right over here.

00:01:37.540 --> 00:01:39.650
We have a hydroxide anion.

00:01:39.650 --> 00:01:42.860
We've already seen
that a hydroxide anion

00:01:42.860 --> 00:01:44.890
is a decent nucleophile.

00:01:44.890 --> 00:01:46.870
So this right over
here is a nucleophile.

00:01:51.260 --> 00:01:54.400
And bromine is a
decent leaving group.

00:01:59.330 --> 00:02:02.720
And so assuming that it's
in a appropriate solution--

00:02:02.720 --> 00:02:04.670
we'll cover that
in future videos--

00:02:04.670 --> 00:02:10.669
you could imagine
that one of these,

00:02:10.669 --> 00:02:15.430
if this hydroxide kind of
bumps into this bromomethane

00:02:15.430 --> 00:02:19.970
in just the right way, that
this pair right over here

00:02:19.970 --> 00:02:24.450
could form a bond
with this carbon.

00:02:24.450 --> 00:02:26.660
And just as that is happening,
essentially the carbon

00:02:26.660 --> 00:02:28.760
is gaining half of
a bond, which is

00:02:28.760 --> 00:02:31.820
kind of the equivalent
of a electron charge.

00:02:31.820 --> 00:02:37.920
So then, this bromine could
take away this entire pair.

00:02:37.920 --> 00:02:39.840
Instead of just kind of
sharing it and having

00:02:39.840 --> 00:02:42.320
the equivalent of 1 electron
charge from this pair,

00:02:42.320 --> 00:02:43.830
it has both of them.

00:02:43.830 --> 00:02:45.840
And then, it'll become
the bromide anion.

00:02:45.840 --> 00:02:47.560
And we've already
seen that before.

00:02:47.560 --> 00:02:50.940
That is an Sn2 reaction.

00:02:50.940 --> 00:02:56.015
It's a nucleophilic substitution
where both reactants--

00:02:56.015 --> 00:02:57.950
so the nucleophile and
the leaving group--

00:02:57.950 --> 00:03:01.210
are involved in the
rate-determining step.

00:03:01.210 --> 00:03:03.830
And this will actually
be quite fast.

00:03:08.010 --> 00:03:09.720
Right over here.

00:03:09.720 --> 00:03:11.120
Now, let's look over here.

00:03:11.120 --> 00:03:13.090
The only difference--
we already said--

00:03:13.090 --> 00:03:15.759
is the existence of this CH3.

00:03:15.759 --> 00:03:17.800
Now, even though we've
drawn it, written it down,

00:03:17.800 --> 00:03:20.050
CH3, when it's written down,
doesn't look a lot bigger

00:03:20.050 --> 00:03:21.920
than an H. Remember,
CH3-- I mean,

00:03:21.920 --> 00:03:26.100
this is roughly the size of
this thing right over here.

00:03:26.100 --> 00:03:28.820
So if we were to actually draw
it out with the hydrogen bonds,

00:03:28.820 --> 00:03:31.040
this is quite large.

00:03:33.710 --> 00:03:36.670
And remember-- in chemistry
and in organic chemistry--

00:03:36.670 --> 00:03:38.652
we always draw these
neat and clean reactions,

00:03:38.652 --> 00:03:40.610
but that's not the way
it happens in real life.

00:03:40.610 --> 00:03:42.972
These things are all
bumping in a solution

00:03:42.972 --> 00:03:45.065
and bumping from
every direction.

00:03:45.065 --> 00:03:48.040
And depending on how they bump
and their relative energies,

00:03:48.040 --> 00:03:51.840
the bonds, the reactions
may or may not happen.

00:03:51.840 --> 00:03:55.360
And so in this scenario,
this fairly big part

00:03:55.360 --> 00:04:01.960
of this molecule might
block the approach

00:04:01.960 --> 00:04:06.354
of a hydroxide anion
coming from that direction.

00:04:06.354 --> 00:04:07.770
The good thing
is-- well, I guess,

00:04:07.770 --> 00:04:09.764
if you view Sn2
reactions as good--

00:04:09.764 --> 00:04:12.180
is that there's still other
directions that it can come in

00:04:12.180 --> 00:04:12.890
from.

00:04:12.890 --> 00:04:20.769
So you could still
see the Sn2 reaction,

00:04:20.769 --> 00:04:25.190
but it's going to
be a bit slower.

00:04:25.190 --> 00:04:30.590
So maybe here, we could write
a bit slower, but still fast.

00:04:41.109 --> 00:04:42.650
Now, let's think
about this scenario.

00:04:42.650 --> 00:04:44.200
And once again, I think you
can see where it's going.

00:04:44.200 --> 00:04:46.300
The only difference now is
it's the secondary carbon.

00:04:46.300 --> 00:04:47.650
We have these two groups here.

00:04:47.650 --> 00:04:49.822
And remember, these are
much larger than the way

00:04:49.822 --> 00:04:50.613
that I've drawn it.

00:04:50.613 --> 00:04:53.030
You have a carbon
attached to 3 hydrogens.

00:04:53.030 --> 00:04:55.630
So now we're blocking
from two directions.

00:04:55.630 --> 00:05:01.780
So once again, you can still
have your Sn2 reaction,

00:05:01.780 --> 00:05:03.805
but it's going to
happen a lot slower.

00:05:06.540 --> 00:05:11.800
So now, you're going to
have a slow Sn2 reaction.

00:05:11.800 --> 00:05:14.160
And now, finally,
what's happening here?

00:05:14.160 --> 00:05:16.430
Well, here, this
entire side is blocked.

00:05:16.430 --> 00:05:19.980
You have a CH3, CH3, CH3.

00:05:19.980 --> 00:05:23.670
This hydroxide anion isn't
going to be able to get through.

00:05:23.670 --> 00:05:25.780
It is being hindered.

00:05:25.780 --> 00:05:29.005
So here we have no Sn2.

00:05:29.005 --> 00:05:30.880
Now, you might be saying,
wait, why didn't it

00:05:30.880 --> 00:05:31.960
come from the other end?

00:05:31.960 --> 00:05:37.240
Well, remember Sn2 reactions--
the nucleophile is attacking

00:05:37.240 --> 00:05:38.950
from one side, and
the leaving group

00:05:38.950 --> 00:05:40.450
is leaving from the other.

00:05:40.450 --> 00:05:42.740
If the nucleophile were
coming in from here,

00:05:42.740 --> 00:05:46.240
the leaving group would have
trouble leaving from there.

00:05:46.240 --> 00:05:47.820
It wouldn't work out as well.

00:05:47.820 --> 00:05:52.490
Sn2, nucleophile from one side,
leaving group from the other.

00:05:52.490 --> 00:05:55.137
Now, the only thing that could
reasonably happen here-- and it

00:05:55.137 --> 00:05:55.720
is reasonable.

00:05:55.720 --> 00:05:57.261
And actually, we've
even seen it when

00:05:57.261 --> 00:05:59.790
we looked at Sn1 reactions--
is maybe the leaving

00:05:59.790 --> 00:06:01.200
group leaves first.

00:06:01.200 --> 00:06:05.410
And actually, the tertiary
carbon is bad for the Sn2,

00:06:05.410 --> 00:06:10.780
but it's good for having
a stable carbocation.

00:06:10.780 --> 00:06:18.750
If this bromine takes that
pair and becomes bromide,

00:06:18.750 --> 00:06:20.950
then this becomes a
carbocation, which

00:06:20.950 --> 00:06:23.790
will be reasonably stable
because that positive charge

00:06:23.790 --> 00:06:25.920
can kind of be spread
amongst its brothers.

00:06:25.920 --> 00:06:30.040
And then, at that point, it's
not crazy for a nucleophile.

00:06:30.040 --> 00:06:32.610
And you could even have
a weaker nucleophile

00:06:32.610 --> 00:06:38.250
than the hydroxide anion to come
in and actually form a bond.

00:06:38.250 --> 00:06:40.170
Now, this general
idea that we're

00:06:40.170 --> 00:06:42.940
talking about in this video--
this idea that a reaction

00:06:42.940 --> 00:06:47.750
happening or not happening
because of three dimensions

00:06:47.750 --> 00:06:50.070
of something hindering
the reaction-- this

00:06:50.070 --> 00:06:51.605
is called steric hindrance.

00:06:57.510 --> 00:07:00.290
Steric refers to what's
happening in three dimensions--

00:07:00.290 --> 00:07:02.300
the size and shape of molecules.

00:07:02.300 --> 00:07:04.260
Hindrance is that that
three-dimensional size

00:07:04.260 --> 00:07:06.280
and shape is hindering
the reaction.

00:07:06.280 --> 00:07:08.950
So the steric hindrance is
greatest right over here,

00:07:08.950 --> 00:07:12.876
and it's the least right over
here for the Sn2 reaction.


Sn2 reactions

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.410
We're now going to
explore one of the most

00:00:02.410 --> 00:00:05.710
fundamental reactions
in organic chemistry.

00:00:05.710 --> 00:00:07.490
And first, let's
look at what are

00:00:07.490 --> 00:00:09.450
the things that
are going to react.

00:00:09.450 --> 00:00:12.700
So over here, I have
a hydroxide anion.

00:00:12.700 --> 00:00:14.820
It has a negative 1 charge.

00:00:14.820 --> 00:00:17.160
And that negative 1
charge comes from the fact

00:00:17.160 --> 00:00:19.990
that oxygen is neutral if
has six valence electrons.

00:00:19.990 --> 00:00:22.200
But this has one, two,
three, four, five,

00:00:22.200 --> 00:00:26.910
six, plus half of this
bond with hydrogen.

00:00:26.910 --> 00:00:29.530
So you could think of
that as another electron.

00:00:29.530 --> 00:00:32.650
Or it's got half of
these two, so that's

00:00:32.650 --> 00:00:37.490
another electron, which
gives it a negative 1 charge.

00:00:37.490 --> 00:00:39.970
Now, this right over
here, it has one carbon--

00:00:39.970 --> 00:00:43.010
I guess that you could
say-- in its backbone.

00:00:43.010 --> 00:00:44.400
It's a methane.

00:00:44.400 --> 00:00:47.641
And it has a group right over
here consisting of bromine.

00:00:47.641 --> 00:00:48.640
So this is bromomethane.

00:00:52.077 --> 00:00:53.660
So let's think a
little bit about what

00:00:53.660 --> 00:00:56.500
might happen under
the right conditions.

00:00:56.500 --> 00:00:59.714
So this hydroxide anion-- we've
already gone over the fact--

00:00:59.714 --> 00:01:01.380
that you could kind
of view it as having

00:01:01.380 --> 00:01:04.420
an extra electron, which
gives us this negative charge.

00:01:04.420 --> 00:01:07.770
Well, if you have a negative
charged anion-- like this--

00:01:07.770 --> 00:01:10.580
it will be attracted
to positive things.

00:01:10.580 --> 00:01:15.840
And one example of a
positive thing is a nucleus.

00:01:15.840 --> 00:01:17.880
And so in this
reaction, we are going

00:01:17.880 --> 00:01:23.045
to refer to this hydroxide
anion as the nucleophile.

00:01:26.741 --> 00:01:29.770
Phile literally refers to
things that something likes.

00:01:29.770 --> 00:01:32.550
This thing, it's
a nucleus-lover,

00:01:32.550 --> 00:01:33.810
I guess you could say.

00:01:33.810 --> 00:01:39.450
And so it might want to share
maybe a pair of electrons

00:01:39.450 --> 00:01:41.820
with a nucleus.

00:01:41.820 --> 00:01:44.870
Now, over here, I
have this bromo group

00:01:44.870 --> 00:01:46.980
in the bromomethane.

00:01:46.980 --> 00:01:50.600
And bromine is quite
electronegative.

00:01:50.600 --> 00:01:52.550
It is more electronegative
than carbon.

00:01:52.550 --> 00:01:54.640
So even in this bond,
we have two electrons

00:01:54.640 --> 00:01:56.240
in this bond right over here.

00:01:56.240 --> 00:01:59.030
Bromine is going to hog the
electrons a little bit more

00:01:59.030 --> 00:02:00.270
than carbon.

00:02:00.270 --> 00:02:04.175
And also, electronegativity
says that, if this bond were

00:02:04.175 --> 00:02:06.940
to break, it's more likely that
this pair-- the pair that's

00:02:06.940 --> 00:02:09.070
in this bond-- is going
to go to the bromine

00:02:09.070 --> 00:02:11.090
than it's going to
go to the carbon.

00:02:11.090 --> 00:02:13.800
So the way that this reaction
works-- and it's really

00:02:13.800 --> 00:02:15.310
kind of happening in one step.

00:02:15.310 --> 00:02:17.240
And this is key to the
name of this reaction,

00:02:17.240 --> 00:02:20.810
as we'll see in a second-- is
that this nucleophile right

00:02:20.810 --> 00:02:23.860
over here might say, hey,
I got this extra charge.

00:02:23.860 --> 00:02:28.240
Let me share some of that
charge with a nucleus--

00:02:28.240 --> 00:02:29.530
in this case, the carbon.

00:02:29.530 --> 00:02:30.630
Let me draw it like this.

00:02:34.330 --> 00:02:37.220
So it would share this pair,
which would form a bond.

00:02:37.220 --> 00:02:39.050
Carbon would have
half of the pair.

00:02:39.050 --> 00:02:40.740
Oxygen would have
half of the pair.

00:02:40.740 --> 00:02:45.240
So it would share
this pair with carbon.

00:02:45.240 --> 00:02:47.520
Now, if carbon is now
sharing half of this,

00:02:47.520 --> 00:02:51.830
it's essentially kind
of getting an electron.

00:02:51.830 --> 00:02:54.370
It doesn't need this electron,
which bromine is already

00:02:54.370 --> 00:02:55.620
hogging a little bit.

00:02:55.620 --> 00:02:58.960
And so bromine would take
this entire pair back.

00:02:58.960 --> 00:03:02.245
Whenever we draw full arrows
like this-- just normal arrows

00:03:02.245 --> 00:03:03.620
that you're used
to seeing-- it's

00:03:03.620 --> 00:03:05.710
referring to the
action of pairs.

00:03:05.710 --> 00:03:07.580
So this whole pair
is going to bromine.

00:03:07.580 --> 00:03:09.510
This pair right over
here is now going

00:03:09.510 --> 00:03:11.120
to be shared with the carbon.

00:03:11.120 --> 00:03:13.670
It's going to form a
bond with the carbon.

00:03:13.670 --> 00:03:15.970
So when we do both
of these steps,

00:03:15.970 --> 00:03:19.960
after both of these steps,
it would look like this.

00:03:19.960 --> 00:03:22.260
Let me draw my
bromine first, which

00:03:22.260 --> 00:03:25.190
is now going to become
a bromide anion.

00:03:25.190 --> 00:03:29.030
So bromine one, two,
three, four, five, six.

00:03:29.030 --> 00:03:33.350
One, two, three,
four, five, six.

00:03:33.350 --> 00:03:39.670
And now, it gets both of these
electrons from this bond.

00:03:39.670 --> 00:03:42.940
So it has both of those
electrons right over there.

00:03:42.940 --> 00:03:45.987
Now, this will have
a negative 1 charge.

00:03:45.987 --> 00:03:46.820
How do we know that?

00:03:46.820 --> 00:03:50.995
Well, bromine is neutral, but
it has seven valence electrons.

00:03:50.995 --> 00:03:52.700
And this thing now has eight.

00:03:52.700 --> 00:03:57.340
So this now has a
negative charge.

00:03:57.340 --> 00:03:58.970
Now, let's think
about what happens

00:03:58.970 --> 00:04:00.660
to all of this business.

00:04:00.660 --> 00:04:04.270
So let me draw my carbon first.

00:04:04.270 --> 00:04:07.130
And let me draw this
hydrogen that's popping out.

00:04:07.130 --> 00:04:10.615
So this hydrogen that's popping
out, let me draw it like that.

00:04:10.615 --> 00:04:11.660
So that's the bond.

00:04:11.660 --> 00:04:14.260
I'm trying to show that's
popping out of the page.

00:04:14.260 --> 00:04:15.545
So that's that hydrogen.

00:04:15.545 --> 00:04:16.920
I just made them
different colors

00:04:16.920 --> 00:04:18.980
so we can keep track of
the different hydrogens.

00:04:18.980 --> 00:04:21.620
Then the hydrogen that's
going into the page, that's

00:04:21.620 --> 00:04:23.190
that one right over there.

00:04:23.190 --> 00:04:25.470
And you have this
hydrogen that's going up.

00:04:25.470 --> 00:04:27.700
So that hydrogen
right over there.

00:04:27.700 --> 00:04:31.510
And now, the hydroxide anion
has bonded to this carbon.

00:04:31.510 --> 00:04:34.560
Now it'll be a hydroxyl group.

00:04:34.560 --> 00:04:37.840
So this thing has now bonded.

00:04:37.840 --> 00:04:40.570
So let me draw the bond and the
color of these two electrons.

00:04:40.570 --> 00:04:42.250
It has now bonded.

00:04:42.250 --> 00:04:44.930
So those two electrons
now make up this blur,

00:04:44.930 --> 00:04:46.930
this bond right over here.

00:04:46.930 --> 00:04:52.380
And now, we have our
oxygen but with those two,

00:04:52.380 --> 00:04:55.440
these two right over here,
these two right over here,

00:04:55.440 --> 00:04:58.152
and then, the bond
to the hydrogen.

00:04:58.152 --> 00:04:59.360
Now, what just happened here?

00:04:59.360 --> 00:05:01.670
The oxygen is now neutral.

00:05:01.670 --> 00:05:02.420
Why is it neutral?

00:05:02.420 --> 00:05:04.350
Well, it has one,
two, three, four.

00:05:04.350 --> 00:05:05.960
And then, it has
half of this bond.

00:05:05.960 --> 00:05:08.150
So you could view
that as an electron

00:05:08.150 --> 00:05:10.080
or the equivalent
of electron charge.

00:05:10.080 --> 00:05:11.590
And then, it has
half of this bond.

00:05:11.590 --> 00:05:13.880
So it has the electron
charge equivalent

00:05:13.880 --> 00:05:15.480
of six valence
electrons-- which makes

00:05:15.480 --> 00:05:17.507
oxygen neutral-- and
now bromine is negative.

00:05:17.507 --> 00:05:19.840
So one way to think about it
is that the negative charge

00:05:19.840 --> 00:05:23.630
has been transferred to the
bromine in this reaction.

00:05:23.630 --> 00:05:26.430
Now, in this reaction,
bromine has left.

00:05:26.430 --> 00:05:36.466
And that's why we call this the
leaving group of the reaction.

00:05:36.466 --> 00:05:38.090
This thing right over
here is the thing

00:05:38.090 --> 00:05:39.910
that is going to leave.

00:05:39.910 --> 00:05:41.636
That is the leaving group.

00:05:41.636 --> 00:05:44.010
Now, the other thing that you
might find interesting-- it

00:05:44.010 --> 00:05:46.640
looks like kind of
this hydroxide anion.

00:05:46.640 --> 00:05:49.470
Some people would even say
attacked this bromomethane,

00:05:49.470 --> 00:05:50.765
the bromomethane left.

00:05:55.010 --> 00:05:57.060
The bromine left-- not
the bromomethane left--

00:05:57.060 --> 00:06:01.487
that attacked the
bromomethane, the bromine left.

00:06:01.487 --> 00:06:03.570
So in some ways, you would
say that there's really

00:06:03.570 --> 00:06:05.280
two reactants over here.

00:06:05.280 --> 00:06:09.310
You have the hydroxide anion,
and you have the bromide anion.

00:06:09.310 --> 00:06:11.670
And this carbon right over
here with the hydrogens,

00:06:11.670 --> 00:06:13.880
this was kind of the
thing that to some degree

00:06:13.880 --> 00:06:15.450
facilitated the reaction.

00:06:15.450 --> 00:06:17.200
This thing bumped
into it and attached.

00:06:17.200 --> 00:06:18.890
And then, this thing the left.

00:06:18.890 --> 00:06:21.915
And so in this context,
we would call the carbon--

00:06:21.915 --> 00:06:24.415
with the three hydrogens-- we
would call this the substrate.

00:06:28.930 --> 00:06:32.600
Now, what should we
call this reaction?

00:06:32.600 --> 00:06:36.000
Well, one way to think about
it is we substituted this bromo

00:06:36.000 --> 00:06:38.860
group with a hydroxyl group.

00:06:38.860 --> 00:06:42.540
So it might make sense to call
this a substitution reaction.

00:06:42.540 --> 00:06:44.100
Substitution has occurred.

00:06:46.820 --> 00:06:48.544
Now, what did we
substitute with?

00:06:48.544 --> 00:06:50.210
Well, we substituted
with a nucleophile.

00:06:56.390 --> 00:06:58.440
And how many reactants
were involved

00:06:58.440 --> 00:07:00.120
in the rate-determining step?

00:07:00.120 --> 00:07:02.328
And I know that might not
make a lot of sense to you.

00:07:02.328 --> 00:07:05.750
But this had, essentially,
one determining step.

00:07:05.750 --> 00:07:07.820
This had to want to
bump in the right way.

00:07:07.820 --> 00:07:11.380
And this would want to have to
leave right at the same time,

00:07:11.380 --> 00:07:13.320
or roughly the
same time, in order

00:07:13.320 --> 00:07:15.160
for this reaction to occur.

00:07:15.160 --> 00:07:17.210
So this reactant
and this reactant

00:07:17.210 --> 00:07:19.350
were involved in
figuring out, well,

00:07:19.350 --> 00:07:21.790
how fast is this
reaction going to happen?

00:07:21.790 --> 00:07:24.080
And we're going to see
future reactions where

00:07:24.080 --> 00:07:26.342
you don't have both
reactants involved

00:07:26.342 --> 00:07:27.550
in the rate-determining step.

00:07:27.550 --> 00:07:29.140
We'll see only one
reactant involved

00:07:29.140 --> 00:07:30.590
in the rate-determining step.

00:07:30.590 --> 00:07:33.470
But I'll just write
this right over here,

00:07:33.470 --> 00:07:38.750
2 reactants in
rate-determining step.

00:07:38.750 --> 00:07:41.350
There's a garbage truck outside.

00:07:41.350 --> 00:07:41.850
OK.

00:07:41.850 --> 00:07:43.474
I think the garbage
truck has left now.

00:07:43.474 --> 00:07:44.600
Safe to resume.

00:07:44.600 --> 00:07:47.290
So as we were saying, there
are two reactants involved

00:07:47.290 --> 00:07:56.290
in the rate-determining step.

00:07:56.290 --> 00:07:58.470
And so the shorthand
for this reaction

00:07:58.470 --> 00:08:05.855
is to call it an Sn2 reaction.

00:08:08.700 --> 00:08:11.120
It's substitution with
a nucleophile where

00:08:11.120 --> 00:08:15.942
both reactants are involved
in the rate-determining step.


Rent control and deadweight loss

WEBVTT
Kind: captions
Language: en

00:00:00.719 --> 00:00:03.150
Voiceover: Let's think about
the market for real estate

00:00:03.150 --> 00:00:04.428
in a given city.

00:00:04.428 --> 00:00:05.957
Here on the vertical axis

00:00:05.957 --> 00:00:07.471
I have plotted rent in terms of dollars

00:00:07.471 --> 00:00:09.042
per square foot per month.

00:00:09.042 --> 00:00:10.615
Here on the horizontal axis,

00:00:10.615 --> 00:00:13.693
is essentially the quantity of square foot

00:00:13.693 --> 00:00:16.157
square foot per month
available in millions.

00:00:16.157 --> 00:00:18.256
This is 1 million, 2 million, 3 million,

00:00:18.256 --> 00:00:20.224
4 million, 5 million.

00:00:20.224 --> 00:00:23.199
Here in blue we have the demand curve.

00:00:23.199 --> 00:00:28.536
You see as the price is high,

00:00:28.536 --> 00:00:30.694
one way to view it is that the demand

00:00:30.694 --> 00:00:33.199
for square footage is low and as the price

00:00:33.199 --> 00:00:35.741
is low the demand for
square footage is higher.

00:00:35.741 --> 00:00:37.678
But what I really want to
focus on in this video,

00:00:37.678 --> 00:00:39.364
is viewing the demand curve as the

00:00:39.364 --> 00:00:41.203
marginal benefit curve.

00:00:41.203 --> 00:00:43.401
Marginal benefit curve.

00:00:43.401 --> 00:00:47.203
When that first incremental square foot

00:00:47.203 --> 00:00:49.437
that is added to the market,

00:00:49.437 --> 00:00:51.701
that has a huge marginal benefit

00:00:51.701 --> 00:00:54.647
where people are desperate
to get an apartment.

00:00:54.647 --> 00:00:57.750
To get someplace where they
could rent and they could live.

00:00:57.750 --> 00:01:00.004
So it has a huge marginal benefit.

00:01:00.004 --> 00:01:01.672
Then the marginal benefit for every

00:01:01.672 --> 00:01:06.168
incremental square foot starts to go down.

00:01:06.168 --> 00:01:08.310
Likewise, we can look at the supply curve.

00:01:08.310 --> 00:01:09.801
We're going to look at this as the

00:01:09.801 --> 00:01:11.557
marginal cost curve.

00:01:11.557 --> 00:01:13.860
The marginal cost curve.

00:01:13.860 --> 00:01:16.082
The marginal cost of that very first

00:01:16.082 --> 00:01:19.287
incremental square foot for the suppliers

00:01:19.287 --> 00:01:21.472
for the landlords in the city is $1

00:01:21.472 --> 00:01:22.643
per square foot.

00:01:22.643 --> 00:01:24.358
One way to think about it,

00:01:24.358 --> 00:01:25.520
that very first square foot,

00:01:25.520 --> 00:01:27.540
we don't know what its
price would have been,

00:01:27.540 --> 00:01:31.673
but let's say its price
was at $3 if it was.

00:01:31.673 --> 00:01:33.126
I'm just making that up.

00:01:33.126 --> 00:01:34.475
In that reality,

00:01:34.475 --> 00:01:36.884
if that very first square
foot's price was at $3,

00:01:36.884 --> 00:01:39.827
there is definitely an
incentive for someone

00:01:39.827 --> 00:01:42.014
to make that first square foot

00:01:42.014 --> 00:01:43.783
because their marginal cost is only

00:01:43.783 --> 00:01:47.358
a dollar and they could
rent it out at $3 per month.

00:01:47.358 --> 00:01:51.283
There is definitely an incentive
for someone to rent it.

00:01:51.283 --> 00:01:54.933
The marginal benefit is
$4 and they just have

00:01:54.933 --> 00:01:56.782
to pay $3 for it.

00:01:56.782 --> 00:01:58.470
It could be rented out for anywhere

00:01:58.470 --> 00:01:59.690
or it would exist,

00:01:59.690 --> 00:02:01.744
or this kind of transaction would happen

00:02:01.744 --> 00:02:06.267
as long as its price
was between $1 and $4.

00:02:06.267 --> 00:02:08.630
You could imagine, based
on how this is drawn,

00:02:08.630 --> 00:02:10.802
where the actual equilibrium price is.

00:02:10.802 --> 00:02:14.023
The suppliers will keep
adding more and more

00:02:14.023 --> 00:02:17.155
square foot as long as they
can actually rent it out,

00:02:17.155 --> 00:02:20.384
all the way until the
point that the marginal

00:02:20.384 --> 00:02:23.517
benefit is equal to the marginal cost.

00:02:23.517 --> 00:02:25.927
Right over here marginal benefit is equal

00:02:25.927 --> 00:02:27.767
to marginal cost.

00:02:27.767 --> 00:02:29.462
It wouldn't make sense for suppliers

00:02:29.462 --> 00:02:32.204
to produce an incremental
square foot right over here.

00:02:32.204 --> 00:02:33.922
If they produce an
incremental square foot,

00:02:33.922 --> 00:02:37.167
their marginal cost has gone beyond $3,

00:02:37.167 --> 00:02:39.384
while the marginal benefit is below $3,

00:02:39.384 --> 00:02:42.754
no one is going to rent that thing out.

00:02:42.754 --> 00:02:49.279
We reach an equilibrium point at 2 million

00:02:49.279 --> 00:02:51.267
square feet per month on the market.

00:02:51.267 --> 00:02:53.153
Let me make that line a
little bit straighter.

00:02:53.153 --> 00:02:55.887
2 million square foot
per month on the market

00:02:55.887 --> 00:03:03.089
and at a price of $3 per
square foot per month.

00:03:03.089 --> 00:03:05.161
You can look at the total surplus here.

00:03:05.161 --> 00:03:07.204
In this equilibrium scenario,

00:03:07.204 --> 00:03:10.403
we can calculate the consumer surplus.

00:03:10.403 --> 00:03:12.141
So all of these folks,

00:03:12.141 --> 00:03:15.293
for this first incremental foot,

00:03:15.293 --> 00:03:17.203
someone was willing to
pay $4 per square foot.

00:03:17.203 --> 00:03:18.433
They only have to pay $3.

00:03:18.433 --> 00:03:20.453
The next one a little less than $4.

00:03:20.453 --> 00:03:21.960
Benefit they only have to pay $3.

00:03:21.960 --> 00:03:23.816
All the way to this point right over here.

00:03:23.816 --> 00:03:27.140
So the area of this
triangle right over here,

00:03:27.140 --> 00:03:30.538
this right over here is
the consumer surplus.

00:03:30.538 --> 00:03:33.618
So that right over here
is consumer surplus.

00:03:33.618 --> 00:03:37.998
Consumer surplus.

00:03:37.998 --> 00:03:38.892
We can calculate it.

00:03:38.892 --> 00:03:40.191
This is a triangle.

00:03:40.191 --> 00:03:42.567
I'm assuming actually both the
supply and demand curves are lines.

00:03:42.567 --> 00:03:44.855
So let's see this has a height of 1

00:03:44.855 --> 00:03:48.183
and it has a base of 2 so
its area is going to be

00:03:48.183 --> 00:03:51.133
1 times 2 times one-half.

00:03:51.133 --> 00:03:58.320
That is going to be equal
to 1 million dollars.

00:03:58.320 --> 00:04:02.073
We are multiplying $1 times 2 million

00:04:02.073 --> 00:04:04.024
times one-half.

00:04:04.024 --> 00:04:07.555
That's going to be 1 million dollars

00:04:07.555 --> 00:04:09.736
of consumer surplus per month.

00:04:09.736 --> 00:04:11.738
Let's think about the producer surplus.

00:04:11.738 --> 00:04:14.275
The producer surplus is
going to be this area.

00:04:14.275 --> 00:04:17.565
It's going to be this
area right over here.

00:04:17.565 --> 00:04:19.012
That first incremental foot it only cost

00:04:19.012 --> 00:04:21.888
those producers $1 but they are able

00:04:21.888 --> 00:04:25.323
to rent it out for $3.

00:04:25.323 --> 00:04:27.401
Then they will keep producing,

00:04:27.401 --> 00:04:28.841
keep producing all the way until they

00:04:28.841 --> 00:04:30.345
do 2 million square foot.

00:04:30.345 --> 00:04:32.181
For all of their square feet,

00:04:32.181 --> 00:04:34.359
they are able to rent it out for more

00:04:34.359 --> 00:04:35.671
than it was their cost to produce it.

00:04:35.671 --> 00:04:40.545
So the area of this triangle
is the producer surplus.

00:04:40.545 --> 00:04:44.623
This is the producer surplus.

00:04:44.623 --> 00:04:49.593
Producer surplus.

00:04:49.593 --> 00:04:51.824
We can calculate that as well.

00:04:51.824 --> 00:04:54.665
The height right over here is $2.

00:04:54.665 --> 00:04:59.683
$2 times this width is 2
million square feet per month.

00:04:59.683 --> 00:05:02.767
2 times 2 times one-half is 2.

00:05:02.767 --> 00:05:06.710
This is equal to 2 million dollars.

00:05:06.710 --> 00:05:08.681
If we were to talk about
what the total surplus is,

00:05:08.681 --> 00:05:14.550
it is 3 million dollars.

00:05:14.550 --> 00:05:16.621
Now, this equilibrium rent,

00:05:16.621 --> 00:05:18.802
$3 per square foot per month
is actually quite a lot

00:05:18.802 --> 00:05:21.081
for 1,000 square foot apartment.

00:05:21.081 --> 00:05:23.200
My last apartment was a two bedroom,

00:05:23.200 --> 00:05:24.965
two bath apartment.

00:05:24.965 --> 00:05:26.218
It was about 1,000 square foot.

00:05:26.218 --> 00:05:27.949
So that means you're
going to be paying $3,000

00:05:27.949 --> 00:05:29.106
per month for that.

00:05:29.106 --> 00:05:30.326
That's pretty high rent.

00:05:30.326 --> 00:05:31.991
That's the type of rent you might pay

00:05:31.991 --> 00:05:34.126
in a city like San Francisco.

00:05:34.126 --> 00:05:36.083
Let's say people start
complaining about it.

00:05:36.083 --> 00:05:38.617
So the government says,
"Okay that rent is not fair.

00:05:38.617 --> 00:05:39.951
"It's too high.

00:05:39.951 --> 00:05:43.163
"We want to introduce some
type of price control.

00:05:43.163 --> 00:05:47.948
"We want to introduce rent control."

00:05:47.948 --> 00:05:50.820
I'm oversimplifying how this works,

00:05:50.820 --> 00:05:53.767
but just so that we can deal
with this model right over here,

00:05:53.767 --> 00:05:56.758
Let's say that the city
just sets a ceiling

00:05:56.758 --> 00:06:00.411
on the price per square foot per month.

00:06:00.411 --> 00:06:02.847
Let's just say they set a price ceiling,

00:06:02.847 --> 00:06:09.745
a price ceiling of $2 per
square foot per month.

00:06:09.745 --> 00:06:12.927
$2 per square foot.

00:06:12.927 --> 00:06:14.231
Let me write it this way.

00:06:14.231 --> 00:06:21.211
$2 per square foot per month.

00:06:21.211 --> 00:06:26.728
So they set a price
ceiling right over here.

00:06:26.728 --> 00:06:29.889
Given that, what is going to happen?

00:06:29.889 --> 00:06:32.168
What is going to happen?

00:06:32.168 --> 00:06:33.718
What I really want to think about is what

00:06:33.718 --> 00:06:35.646
is going to be the new consumer surplus

00:06:35.646 --> 00:06:37.047
or the new producer surplus?

00:06:37.047 --> 00:06:38.370
I encourage you to pause the video

00:06:38.370 --> 00:06:40.916
and try to think about that on your own.

00:06:40.916 --> 00:06:42.879
Well let's think about
what's going to happen.

00:06:42.879 --> 00:06:44.386
From the producers point of view,

00:06:44.386 --> 00:06:46.403
it doesn't make sense for them to produce

00:06:46.403 --> 00:06:48.684
more than 1 million units.

00:06:48.684 --> 00:06:51.273
1 million square feet per month.

00:06:51.273 --> 00:06:53.294
I have to rent out more than a million

00:06:53.294 --> 00:06:54.678
square feet per month

00:06:54.678 --> 00:06:57.345
because that extra square foot above that,

00:06:57.345 --> 00:06:58.957
its marginal cost is going to be more than

00:06:58.957 --> 00:07:00.388
what they're going to get.

00:07:00.388 --> 00:07:01.871
The producers are just
going to stop there.

00:07:01.871 --> 00:07:05.036
The producer surplus
is going to be the area

00:07:05.036 --> 00:07:10.168
of this triangle right over here.

00:07:10.168 --> 00:07:13.833
Let's see, this is $1 times 1 million

00:07:13.833 --> 00:07:16.069
times one-half.

00:07:16.069 --> 00:07:17.889
This is now a producer surplus.

00:07:17.889 --> 00:07:21.227
Producer surplus.

00:07:21.227 --> 00:07:22.940
A new producer surplus
under the rent control

00:07:22.940 --> 00:07:30.857
of $500,000, half a million, of $500,000.

00:07:30.857 --> 00:07:31.692
So we see that the rent control

00:07:31.692 --> 00:07:34.316
immediately hit the producers pretty hard.

00:07:34.316 --> 00:07:36.227
The producer surplus has
gone down dramatically.

00:07:36.227 --> 00:07:38.224
Now what about the consumer surplus?

00:07:38.224 --> 00:07:40.743
We're talking about a million units,

00:07:40.743 --> 00:07:43.659
or a million square foot
per month I should say.

00:07:43.659 --> 00:07:48.737
So now the new consumer
surplus is the area,

00:07:48.737 --> 00:07:52.871
is this entire area.

00:07:52.871 --> 00:07:55.071
So you see that at least for this first

00:07:55.071 --> 00:07:57.198
incremental million square feet,

00:07:57.198 --> 00:08:00.298
the consumers have started
to win out a little bit.

00:08:00.298 --> 00:08:03.002
To figure this out,
what the total area is,

00:08:03.002 --> 00:08:04.872
we just need to figure
out what we could break

00:08:04.872 --> 00:08:06.854
this up into two sections.

00:08:06.854 --> 00:08:09.305
We could break this up into two sections.

00:08:09.305 --> 00:08:10.838
This is the point,

00:08:10.838 --> 00:08:14.801
1 million square feet at $3.50
dollars per square foot per month.

00:08:14.801 --> 00:08:16.903
This is right over here,
the point, the 3.5.

00:08:16.903 --> 00:08:19.608
We could use that to
figure out this new area.

00:08:19.608 --> 00:08:21.675
Actually let me do it
in a different color,

00:08:21.675 --> 00:08:25.212
in this green area right over here.

00:08:25.212 --> 00:08:26.816
What is it going to be?

00:08:26.816 --> 00:08:30.545
Well the area of this
thing right over here

00:08:30.545 --> 00:08:33.124
is one-half high.

00:08:33.124 --> 00:08:38.169
One-half times 1 times one-half.

00:08:38.169 --> 00:08:41.458
This right over here is 250K.

00:08:41.458 --> 00:08:45.032
We add that to this area right over here,

00:08:45.032 --> 00:08:51.164
which is one and a half times 1.

00:08:51.164 --> 00:08:54.553
$1.5 dollars per square foot per month

00:08:54.553 --> 00:08:55.888
times 1 million.

00:08:55.888 --> 00:08:57.485
Did I do that right? Yep.

00:08:57.485 --> 00:09:00.618
So that's going to be plus ... plus this,

00:09:00.618 --> 00:09:03.383
which is 1.5 million.

00:09:03.383 --> 00:09:05.474
You add these two together.

00:09:05.474 --> 00:09:07.277
The total consumer surplus ...

00:09:07.277 --> 00:09:18.462
So the total consumer
surplus is now 1.75 million.

00:09:18.462 --> 00:09:20.546
So the consumer surplus definitely

00:09:20.546 --> 00:09:23.949
did go up in total because it gained

00:09:23.949 --> 00:09:26.205
all of this from the producer,

00:09:26.205 --> 00:09:28.093
but let's think about what has now

00:09:28.093 --> 00:09:30.886
happened in our society or in our city.

00:09:30.886 --> 00:09:33.284
The producers definitely
don't want to put out

00:09:33.284 --> 00:09:35.576
as many square foot per rent anymore.

00:09:35.576 --> 00:09:37.916
It does look like we have lost some

00:09:37.916 --> 00:09:40.945
total surplus for our little city here.

00:09:40.945 --> 00:09:44.746
We have lost out on this entire area.

00:09:44.746 --> 00:09:46.719
We can calculate it by looking at what

00:09:46.719 --> 00:09:49.404
the total surplus was
before the rent control

00:09:49.404 --> 00:09:52.337
and what the total surplus
was after the rent control.

00:09:52.337 --> 00:09:54.162
The total surplus before,

00:09:54.162 --> 00:09:59.685
so before the total
surplus was the 2 million

00:09:59.685 --> 00:10:02.669
producer surplus plus the
1 million consumer surplus,

00:10:02.669 --> 00:10:05.984
so it was 3 million.

00:10:05.984 --> 00:10:12.011
After, it is the 1.75
million consumer surplus

00:10:12.011 --> 00:10:14.734
plus the $500,000 producer surplus,

00:10:14.734 --> 00:10:19.786
which is 2.25 million.

00:10:19.786 --> 00:10:23.359
What we've lost is the difference here,

00:10:23.359 --> 00:10:28.601
which is $750,000.

00:10:28.601 --> 00:10:30.050
So this area right over here.

00:10:30.050 --> 00:10:31.275
This is per month.

00:10:31.275 --> 00:10:35.467
This right over here represents
the lost total surplus.

00:10:35.467 --> 00:10:40.541
This lost total surplus
of $750,000 per month

00:10:40.541 --> 00:10:46.891
is referred to as the dead weight loss.

00:10:46.891 --> 00:10:49.158
You can debate about rent control.

00:10:49.158 --> 00:10:50.655
Is it good? Is it bad?

00:10:50.655 --> 00:10:52.864
Is it good for this kind of dynamic?

00:10:52.864 --> 00:10:55.605
Who gets what share of the surplus?

00:10:55.605 --> 00:10:57.746
This of course is an oversimplification

00:10:57.746 --> 00:10:59.079
of a market and even the way rent

00:10:59.079 --> 00:11:00.384
control would be instituted.

00:11:00.384 --> 00:11:02.547
This is a model for beginning to think

00:11:02.547 --> 00:11:05.931
about what happens to the total surplus

00:11:05.931 --> 00:11:09.931
when these types of price
controls are instituted.


Multiplying: 4 digits times 1 digit (using grid)

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:06.140
Let's multiply 6 times 7,981.

00:00:06.140 --> 00:00:07.890
And the way we're going
to do it right now

00:00:07.890 --> 00:00:20.780
is just to represent or expand
out 7,981 as 7,000 plus 900

00:00:20.780 --> 00:00:27.640
plus 80 plus 1.

00:00:27.640 --> 00:00:30.850
And so multiplying 6 times
7,981 is the same thing

00:00:30.850 --> 00:00:36.090
as multiplying 6 times 7,000
plus 6 times 900 plus 6 times

00:00:36.090 --> 00:00:38.330
80 plus 6 times 1.

00:00:38.330 --> 00:00:40.830
You'd essentially
distribute the 6.

00:00:40.830 --> 00:00:42.350
And to help us keep
track of things,

00:00:42.350 --> 00:00:44.100
let me draw a little
grid right over here.

00:00:44.100 --> 00:00:45.820
So this is the 6,
and we're going

00:00:45.820 --> 00:00:48.920
to have to think about what
6 times 7,000 is, 6 times

00:00:48.920 --> 00:00:53.210
900, 6 times 80, and 6 times 1.

00:00:53.210 --> 00:00:57.840
So I'll make a little square for
our rectangle for each of them.

00:00:57.840 --> 00:00:59.370
Let me do that.

00:00:59.370 --> 00:01:01.540
So here we go.

00:01:01.540 --> 00:01:05.670
And so we just need to think
about, what is 6 times 7,000?

00:01:05.670 --> 00:01:08.450
Well 6 times 7 is 42.

00:01:08.450 --> 00:01:11.912
So 6 times 7,000 is 42,000.

00:01:11.912 --> 00:01:16.750
6 times 900, well once
again, 6 times 9 is 54.

00:01:16.750 --> 00:01:20.810
So 6 times 900 is 5,400.

00:01:20.810 --> 00:01:24.500
6 times 80, well
80 is eight 10s.

00:01:24.500 --> 00:01:30.350
So 6 times 8 is 48, but since
it's six times 80 or eight 10s,

00:01:30.350 --> 00:01:34.100
this is going to
be 48 10s, or 480.

00:01:34.100 --> 00:01:40.300
And then finally, 6 times
1, of course, is equal to 6.

00:01:40.300 --> 00:01:42.200
So to find what this
product is, we just

00:01:42.200 --> 00:01:45.010
have to take the sum of
each of these numbers.

00:01:45.010 --> 00:01:48.800
What 6 times 7,000 is plus
6 times 900 plus 6 times

00:01:48.800 --> 00:01:50.950
80 plus 6 times 1.

00:01:50.950 --> 00:01:53.210
So let's do that
right over here.

00:01:53.210 --> 00:02:06.140
So it's going to be 42,000
plus 5,400 plus 480 plus 6.

00:02:06.140 --> 00:02:09.590
And we get, let's see, in the
ones place, we just have a 6.

00:02:09.590 --> 00:02:11.830
In the tens place,
we just have an 8.

00:02:11.830 --> 00:02:15.130
In the hundreds
place, 4 plus 4 is 8.

00:02:15.130 --> 00:02:17.787
In the thousands
place, 2 plus 5 is 7.

00:02:17.787 --> 00:02:19.620
And then finally, the
ten thousandths place,

00:02:19.620 --> 00:02:20.980
we still have a 4.

00:02:20.980 --> 00:02:23.300
So we get 47,886.

00:02:23.300 --> 00:02:28.256
So this Is equal to 47,886.

00:02:28.256 --> 00:02:29.630
And what I encourage
you to do is

00:02:29.630 --> 00:02:32.424
to think about
how this is really

00:02:32.424 --> 00:02:33.840
underlining what
we're doing here.

00:02:33.840 --> 00:02:35.160
It's not that
different than what

00:02:35.160 --> 00:02:37.451
you might have done with the
traditional multiplication

00:02:37.451 --> 00:02:38.276
techniques.

00:02:38.276 --> 00:02:40.650
And this is a useful way of
thinking about it because now

00:02:40.650 --> 00:02:43.172
you really understand
what's going on.

00:02:43.172 --> 00:02:44.880
And actually, when
you start doing things

00:02:44.880 --> 00:02:47.440
like this in your head,
at least for myself,

00:02:47.440 --> 00:02:49.760
this is actually how I try
to tackle the multiplication

00:02:49.760 --> 00:02:50.259
problem.

00:02:50.259 --> 00:02:53.595
When someone says 6 times 7,981,
if I was just looking at this

00:02:53.595 --> 00:02:55.470
and I didn't have any
paper, I would say, OK,

00:02:55.470 --> 00:02:56.660
what's 6 times 7,000?

00:02:56.660 --> 00:02:58.030
I'd say, OK, that's 42,000.

00:02:58.030 --> 00:02:59.170
I'll try to remember that.

00:02:59.170 --> 00:03:01.110
What's 6 times 900?

00:03:01.110 --> 00:03:02.630
Oh that's 5,400.

00:03:02.630 --> 00:03:07.080
Well if I add that to
the 42,000, I get 47,400.

00:03:07.080 --> 00:03:08.961
Then, what's 6 times 80?

00:03:08.961 --> 00:03:09.460
480.

00:03:09.460 --> 00:03:15.150
Have to add that to the
47,400 to get to 47,880.

00:03:15.150 --> 00:03:17.490
And then, what's 6 times 1?

00:03:17.490 --> 00:03:18.370
Well that's 6.

00:03:18.370 --> 00:03:20.770
Well add that to
the 47,880, which

00:03:20.770 --> 00:03:25.461
I've been keeping in my brain,
and that's going to be 47,886.

00:03:25.461 --> 00:03:27.210
So this helps you
understand what's really

00:03:27.210 --> 00:03:29.870
going on when you
multiply multiple digits,

00:03:29.870 --> 00:03:33.402
and it's a useful technique for
doing mental multiplication.


Multiplying: 2 digits numbers (using distributive property)

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:04.390
In this video, I'm going
to multiply 87 times 63.

00:00:04.390 --> 00:00:07.890
But I'm not going to do it
just by using some process,

00:00:07.890 --> 00:00:09.550
just showing you some steps.

00:00:09.550 --> 00:00:12.450
Instead, we're just going to
use the distributive property

00:00:12.450 --> 00:00:14.940
to actually try to
calculate this thing.

00:00:14.940 --> 00:00:18.530
So first, what I'm going
to do-- let me rewrite 87.

00:00:18.530 --> 00:00:21.440
So this is the same thing as 87.

00:00:21.440 --> 00:00:24.040
But instead of
writing 63 like that,

00:00:24.040 --> 00:00:29.457
I'm going to write
63 as 60 plus 3.

00:00:33.370 --> 00:00:35.660
Now, what is this
going to be equal to?

00:00:35.660 --> 00:00:38.720
Well, 87 times 60
plus 3, that's going

00:00:38.720 --> 00:00:41.380
to be the same thing
as-- and let me actually

00:00:41.380 --> 00:00:42.360
copy and paste this.

00:00:42.360 --> 00:00:54.670
So this is going to be the
same thing as 87 times 60

00:00:54.670 --> 00:00:59.850
plus 87 times 3.

00:01:02.420 --> 00:01:08.310
You could say that we've
just distributed the 87.

00:01:08.310 --> 00:01:10.610
We're multiplying
87 times 60 plus 3.

00:01:10.610 --> 00:01:16.520
That's 87 times 60
plus 87 times 3.

00:01:16.520 --> 00:01:20.490
I could put parentheses here to
make it a little bit clearer.

00:01:20.490 --> 00:01:21.240
Well, fair enough.

00:01:21.240 --> 00:01:23.580
But then how do you
calculate what this is?

00:01:23.580 --> 00:01:28.070
Well, now we can
rewrite 87 as 80 plus 7.

00:01:28.070 --> 00:01:29.530
So let's rewrite that.

00:01:29.530 --> 00:01:30.722
So this is the same thing.

00:01:30.722 --> 00:01:32.180
Actually, let me
write it this way.

00:01:32.180 --> 00:01:33.420
I can swap them around.

00:01:33.420 --> 00:01:36.260
So this is the same
thing as 60 times 87.

00:01:36.260 --> 00:01:43.400
But I'll write that
as 60 times 80 plus 7.

00:01:43.400 --> 00:01:44.890
We could do it like this.

00:01:44.890 --> 00:01:58.309
80 plus 7 plus 3 times
80 plus 7, or 3 times 87.

00:01:58.309 --> 00:01:59.850
Let me just copy
and paste that, so I

00:01:59.850 --> 00:02:01.349
don't have to keep
switching colors.

00:02:01.349 --> 00:02:04.190
Plus 3 times 80 plus 7.

00:02:04.190 --> 00:02:07.070
So copy and then
let me paste it.

00:02:07.070 --> 00:02:08.800
And then you have
it just like that.

00:02:08.800 --> 00:02:11.980
So all I did, just to
be clear-- all of what

00:02:11.980 --> 00:02:16.560
you see right over here,
87 times 60, well, that's

00:02:16.560 --> 00:02:18.950
the same thing as
60 times 87, which

00:02:18.950 --> 00:02:22.440
is the same thing as
60 times 80 plus 7.

00:02:22.440 --> 00:02:24.560
All that you see
here, 87 times 3,

00:02:24.560 --> 00:02:27.090
that's the same thing
as 3 times 87, which

00:02:27.090 --> 00:02:30.400
is the same thing as
3 times 80 plus 7.

00:02:30.400 --> 00:02:32.180
That's just that over here.

00:02:32.180 --> 00:02:34.100
But look, we can
distribute again.

00:02:34.100 --> 00:02:37.630
We can distribute the
60 times 80 plus 7.

00:02:37.630 --> 00:02:41.900
So this is going to be 60-- I'm
going to do that same color.

00:02:41.900 --> 00:02:43.320
Color changing is hard.

00:02:43.320 --> 00:03:02.410
This is 60 times 80 plus
60 times 7 plus 3 times

00:03:02.410 --> 00:03:21.235
80 plus 3 times 7
right over here.

00:03:21.235 --> 00:03:23.290
So notice what we
really did is we

00:03:23.290 --> 00:03:25.720
thought about what each
of these digits represent.

00:03:25.720 --> 00:03:27.110
8 represents 80.

00:03:27.110 --> 00:03:28.440
7 represents 7.

00:03:28.440 --> 00:03:30.579
6 represents 60, because
it's in the tens place.

00:03:30.579 --> 00:03:32.120
The 8 was in the
tens place, as well.

00:03:32.120 --> 00:03:34.290
This 3 is in the ones
place, so it's just 3.

00:03:34.290 --> 00:03:36.170
And we just multiplied
them all together.

00:03:36.170 --> 00:03:39.380
We multiplied the
80 times the 60.

00:03:39.380 --> 00:03:41.980
We multiplied the
80 times the 3.

00:03:41.980 --> 00:03:45.030
We multiplied the 7 times
the 60 right over here.

00:03:45.030 --> 00:03:47.120
We multiplied the 7 times the 3.

00:03:47.120 --> 00:03:50.170
And then we add them
all up together.

00:03:50.170 --> 00:03:52.500
And this will actually
give us our product.

00:03:52.500 --> 00:03:56.880
So for example, this right
over here, 6 times 8 is 48.

00:03:56.880 --> 00:03:58.340
But this isn't six 8's.

00:03:58.340 --> 00:03:59.970
This is 60 80's.

00:03:59.970 --> 00:04:02.285
So this is going to be 4,800.

00:04:02.285 --> 00:04:03.900
We've got two 0's
right over here,

00:04:03.900 --> 00:04:06.810
so 48 followed by the two 0's.

00:04:06.810 --> 00:04:12.200
This right over here,
60 times 7, is 420.

00:04:12.200 --> 00:04:13.790
6 times 7 is 42.

00:04:13.790 --> 00:04:17.050
But it's going to be 10 times
as much, because this is a 60.

00:04:17.050 --> 00:04:19.940
And then 3 times 80--
well, same logic.

00:04:19.940 --> 00:04:21.170
3 times 8 is 24.

00:04:21.170 --> 00:04:23.860
So this is going to be 240.

00:04:23.860 --> 00:04:27.062
And then, finally,
3 times 7 is 21.

00:04:27.062 --> 00:04:29.520
And then to get the product,
we can add these two together.

00:04:29.520 --> 00:04:30.300
And you might be
saying, hey, Sal,

00:04:30.300 --> 00:04:31.520
I know faster ways
of doing this.

00:04:31.520 --> 00:04:32.978
But the whole reason
I'm doing this

00:04:32.978 --> 00:04:35.440
is to show you that that fast
way you knew how to do it,

00:04:35.440 --> 00:04:39.610
it's not some magical formula
or some magical process you're

00:04:39.610 --> 00:04:40.180
doing.

00:04:40.180 --> 00:04:42.400
It just comes out of really
the distributive property

00:04:42.400 --> 00:04:44.409
and, hopefully, a little
bit of common sense.

00:04:44.409 --> 00:04:45.950
So what is this
going to be equal to?

00:04:45.950 --> 00:04:47.241
Well, we could add them all up.

00:04:47.241 --> 00:04:53.950
4,800 plus 420 plus 240 plus 21.

00:04:53.950 --> 00:04:55.440
Well, you're going
to get a 1 here.

00:04:55.440 --> 00:04:56.130
Let's see.

00:04:56.130 --> 00:04:59.410
20 plus 40 plus 20 is 80.

00:04:59.410 --> 00:04:59.910
Let's see.

00:04:59.910 --> 00:05:10.420
800 plus 400 is 1,200
plus 200 more is 1,400.

00:05:10.420 --> 00:05:12.740
And so you get 5,481.

00:05:12.740 --> 00:05:16.730
It's equal to 5,481.

00:05:16.730 --> 00:05:19.000
And you might say, gee,
this was a bit of a pain

00:05:19.000 --> 00:05:21.530
to have to do the distributive
property over and over again.

00:05:21.530 --> 00:05:24.010
Is there a simpler way
to maybe visualize this?

00:05:24.010 --> 00:05:24.920
And there is.

00:05:24.920 --> 00:05:26.860
You could actually
write this as a grid.

00:05:26.860 --> 00:05:30.430
So we could say we're
multiplying 87 times 63.

00:05:30.430 --> 00:05:31.800
We could write it like this.

00:05:31.800 --> 00:05:44.760
We could say it's 80
plus 7 times 60 plus 3.

00:05:44.760 --> 00:05:46.680
And then you can set
up a grid like this.

00:05:46.680 --> 00:05:48.840
So let me set up
a little box here.

00:05:48.840 --> 00:05:50.730
It's 2 digits by 2 digits.

00:05:50.730 --> 00:05:55.170
So it's going to be a 2 by 2
grid, 2 rows and 2 columns.

00:05:55.170 --> 00:05:57.340
And then you just
have to calculate.

00:05:57.340 --> 00:05:58.636
Well, what's 60 times 80?

00:05:58.636 --> 00:06:00.010
Well, we already
calculated that.

00:06:00.010 --> 00:06:02.510
That's 4,800.

00:06:02.510 --> 00:06:04.760
What is 60 times 7?

00:06:04.760 --> 00:06:08.650
Well, that's going to be 420.

00:06:08.650 --> 00:06:10.520
What is 3 times 80?

00:06:10.520 --> 00:06:11.930
We already calculated that.

00:06:11.930 --> 00:06:14.245
That is 240.

00:06:14.245 --> 00:06:17.210
And I want to do that
same color-- 240.

00:06:17.210 --> 00:06:19.881
And finally, what is 3 times 7?

00:06:19.881 --> 00:06:20.810
21.

00:06:20.810 --> 00:06:22.310
You add them all together.

00:06:22.310 --> 00:06:25.600
You get 5,481.

00:06:25.600 --> 00:06:28.870
And I encourage you to now just
do this same multiplication

00:06:28.870 --> 00:06:31.850
problem, the same
87 times 63, the way

00:06:31.850 --> 00:06:34.600
that you might have
traditionally learned it.

00:06:34.600 --> 00:06:37.300
And look at the different steps
and why they are making sense

00:06:37.300 --> 00:06:39.050
and why, at the end
of the day, you really

00:06:39.050 --> 00:06:41.534
are doing the same thing that
we just did in this video.

00:06:41.534 --> 00:06:43.200
You're just doing it
in a different way.

00:06:43.200 --> 00:06:45.080
And the whole point of
this whole exercise,

00:06:45.080 --> 00:06:49.510
this whole video, is so
you're not just blindly

00:06:49.510 --> 00:06:52.550
doing some type of steps to
find the product of two numbers.

00:06:52.550 --> 00:06:55.150
But you can actually
understand why those steps work

00:06:55.150 --> 00:06:58.200
and how those numbers
relate to each other.


Formula for continuously compounding interest

WEBVTT
Kind: captions
Language: en

00:00:00.412 --> 00:00:03.805
Let's say that we're
looking to borrow $50.

00:00:03.805 --> 00:00:06.604
We can say that our principal is $50.

00:00:06.604 --> 00:00:09.413
We're going to borrow it for 3 years.

00:00:09.413 --> 00:00:13.746
Our time, let's say T in years is 3.

00:00:13.746 --> 00:00:15.870
Let's say, we're not going
to just compound per year.

00:00:15.870 --> 00:00:19.939
We're going to compound 4 times a year,

00:00:19.939 --> 00:00:22.335
or every 3 months.

00:00:22.335 --> 00:00:26.937
Let's say that our interest rate ...

00:00:26.937 --> 00:00:30.337
if we were to only compound once per year,

00:00:30.337 --> 00:00:31.830
it would be 10%.

00:00:31.830 --> 00:00:33.472
Since we're going to
compound 4 times a year,

00:00:33.472 --> 00:00:34.938
we're going to see ...

00:00:34.938 --> 00:00:36.914
We're going to divide this by 4

00:00:36.914 --> 00:00:38.671
to see how much we compound each period.

00:00:38.671 --> 00:00:42.605
10% is the same thing as 0.10.

00:00:42.605 --> 00:00:43.402
Let's write an expression.

00:00:43.402 --> 00:00:44.937
I encourage you actually
to pause this video

00:00:44.937 --> 00:00:46.470
and try to write an expression

00:00:46.470 --> 00:00:48.920
for the amount that you
would have to pay back

00:00:48.920 --> 00:00:50.521
if you were to do this.

00:00:50.521 --> 00:00:53.788
If you were to borrow $50 over 3 years,

00:00:53.788 --> 00:00:55.789
compounding 4 times a year,

00:00:55.789 --> 00:00:57.188
each period you would be

00:00:57.188 --> 00:01:00.854
compounding 10% divided 4%.

00:01:00.854 --> 00:01:04.253
How much would you have
to pay back in 3 years?

00:01:04.253 --> 00:01:05.836
Let's write it out.

00:01:05.836 --> 00:01:08.670
$50, that's your principal.

00:01:08.670 --> 00:01:11.790
You're going to multiply that,

00:01:11.790 --> 00:01:14.122
so you could compound it.

00:01:14.122 --> 00:01:15.504
Each time, each period,

00:01:15.504 --> 00:01:17.921
each of these 3 x 4 periods.

00:01:17.921 --> 00:01:18.835
You have 3 years,

00:01:18.835 --> 00:01:21.003
each of them divide into 4 sections,

00:01:21.003 --> 00:01:22.855
so you're going to have 12 periods.

00:01:22.855 --> 00:01:28.454
Each of them you're going to
compound by 1 plus this R.

00:01:28.454 --> 00:01:30.190
I'll write that as a decimal.

00:01:30.190 --> 00:01:35.083
0.10 divided by the number of times

00:01:35.083 --> 00:01:39.749
you're compounding per year to the ...

00:01:39.749 --> 00:01:41.331
Well, you would be raising
it to the nth power,

00:01:41.331 --> 00:01:43.187
if this was only over a year.

00:01:43.187 --> 00:01:44.120
There's 4 periods

00:01:44.120 --> 00:01:45.251
and you would raise it to the 4th power

00:01:45.251 --> 00:01:46.250
if it was only a year,

00:01:46.250 --> 00:01:47.749
but this is 3 years.

00:01:47.749 --> 00:01:51.333
You're going to be doing this 3 x 4.

00:01:51.333 --> 00:01:53.921
You're going to have 4 periods, 3 times.

00:01:53.921 --> 00:01:54.986
Let me write this.

00:01:54.986 --> 00:01:57.455
It's going to be 4 ...

00:01:57.455 --> 00:01:59.082
Actually, instead of N right over here

00:01:59.082 --> 00:02:01.917
let me write the 4, so you
can see all the numbers.

00:02:01.917 --> 00:02:10.187
You're going to do this 4
x 3, to the 4 x 3 power.

00:02:10.187 --> 00:02:12.333
I encourage you, if you want,

00:02:12.333 --> 00:02:13.520
you could pause the video

00:02:13.520 --> 00:02:14.664
and you can use your calculator

00:02:14.664 --> 00:02:16.545
to actually calculate what that is.

00:02:16.545 --> 00:02:18.397
The whole point of this is
just to use real numbers

00:02:18.397 --> 00:02:20.378
to see why this actually makes sense.

00:02:20.378 --> 00:02:21.462
This is your principal.

00:02:21.462 --> 00:02:23.731
Each time you're going
to be multiplying that

00:02:23.731 --> 00:02:27.627
times 1.025.

00:02:27.627 --> 00:02:32.462
You're going to be growing it by 2 1/2%

00:02:32.462 --> 00:02:34.294
and you're going to do this 12 times,

00:02:34.294 --> 00:02:36.129
because there's 12 periods.

00:02:36.129 --> 00:02:39.212
4 periods per year times 3 years.

00:02:39.212 --> 00:02:41.795
This is going to be how
much you have to pay back.

00:02:41.795 --> 00:02:42.596
If we wanted to write this

00:02:42.596 --> 00:02:44.546
in a little bit more abstract terms,

00:02:44.546 --> 00:02:50.712
we could write this as P(1 +).

00:02:50.712 --> 00:02:53.930
I'll do this a close parentheses,
since it's the same color.

00:02:53.930 --> 00:03:05.730
R over N to the N x T power.

00:03:05.730 --> 00:03:08.794
You could pick your P,
your Ts, your Ns and your R

00:03:08.794 --> 00:03:09.544
and you could put it here

00:03:09.544 --> 00:03:10.462
and that's essentially how much

00:03:10.462 --> 00:03:12.630
you're going to have to pay back.

00:03:12.630 --> 00:03:13.597
An interesting thing,

00:03:13.597 --> 00:03:14.962
and you saw that we had this up here

00:03:14.962 --> 00:03:16.212
from a previous video,

00:03:16.212 --> 00:03:19.044
where we took a limit as
N approaches infinity.

00:03:19.044 --> 00:03:20.063
Let's do the same thing here.

00:03:20.063 --> 00:03:21.546
Let's think about what that would mean.

00:03:21.546 --> 00:03:24.627
If we took the limit as
N approaches infinity,

00:03:24.627 --> 00:03:29.546
if we took the limit of this
as N approaches infinity,

00:03:29.546 --> 00:03:32.878
what is this conceptually?

00:03:32.878 --> 00:03:36.878
We're dividing our year into more

00:03:36.878 --> 00:03:37.996
and more and more chunks,

00:03:37.996 --> 00:03:39.462
an infinite number of chunks.

00:03:39.462 --> 00:03:40.263
You could really say,

00:03:40.263 --> 00:03:41.663
"This would be the case where we're doing

00:03:41.663 --> 00:03:44.546
continuous compound interest.

00:03:44.546 --> 00:03:45.861
Which is a fascinating concept to me.

00:03:45.861 --> 00:03:48.664
You're dividing your time period

00:03:48.664 --> 00:03:50.794
in an infinite number of chunks

00:03:50.794 --> 00:03:51.996
and then compounding just

00:03:51.996 --> 00:03:54.378
an infinitely small extra amount

00:03:54.378 --> 00:03:55.729
every one of those periods.

00:03:55.729 --> 00:03:58.378
You can actually come up
with an expression for that.

00:03:58.378 --> 00:04:00.212
As we see, that this actually doesn't

00:04:00.212 --> 00:04:02.397
just go unbounded and
give us crazy things,

00:04:02.397 --> 00:04:03.796
that we can actually use this

00:04:03.796 --> 00:04:05.399
to come up with a formula

00:04:05.399 --> 00:04:07.294
for continuously compounding interest.

00:04:07.294 --> 00:04:10.663
Which is used heavily
in finance and banking

00:04:10.663 --> 00:04:12.330
and, as you can imagine,
a bunch of things,

00:04:12.330 --> 00:04:15.733
actually many things outside
of finance and banking,

00:04:15.733 --> 00:04:18.330
exponential growth, etc., etc.

00:04:18.330 --> 00:04:21.878
Let's see if we can
actually try to evaluate

00:04:21.878 --> 00:04:24.128
this thing right over here.

00:04:24.128 --> 00:04:26.331
The one thing I am going to do

00:04:26.331 --> 00:04:27.598
to simplify this,

00:04:27.598 --> 00:04:29.730
is to do a substitution.

00:04:29.730 --> 00:04:32.670
I'm going to define a variable.

00:04:32.670 --> 00:04:34.632
The whole goal is so that
I can get it into a form

00:04:34.632 --> 00:04:36.431
that looks something like this.

00:04:36.431 --> 00:04:39.931
I'm going to define a variable X.

00:04:39.931 --> 00:04:43.263
I'm going to say that X is
the reciprocal of R over N,

00:04:43.263 --> 00:04:45.263
so that I can get a 1
over X right over here.

00:04:45.263 --> 00:04:47.847
I'll write that as N over R.

00:04:47.847 --> 00:04:52.681
X is equal to N over R,

00:04:52.681 --> 00:04:54.126
or we could write this as

00:04:54.126 --> 00:05:04.977
N is equal to X x R.

00:05:04.977 --> 00:05:06.259
If we make that substitution

00:05:06.259 --> 00:05:07.463
the limit is N approaches infinite.

00:05:07.463 --> 00:05:10.194
If we make the limit as
X approaches infinite,

00:05:10.194 --> 00:05:12.333
then N is going to go to infinite as well.

00:05:12.333 --> 00:05:13.261
If N goes to infinite,

00:05:13.261 --> 00:05:15.331
then X is going to go to infinite as well.

00:05:15.331 --> 00:05:17.222
R, right over here, is just a constant.

00:05:17.222 --> 00:05:19.459
We're just assuming that that's a given,

00:05:19.459 --> 00:05:21.789
that N is what we're
really seeing what happens

00:05:21.789 --> 00:05:22.988
as we change it.

00:05:22.988 --> 00:05:25.678
We could rewrite this
thing right over here.

00:05:25.678 --> 00:05:26.519
I'm doing it.

00:05:26.519 --> 00:05:28.010
I'm not being as super rigorous,

00:05:28.010 --> 00:05:29.322
but it's really to give you an intuition

00:05:29.322 --> 00:05:32.054
for where the formula we're
about to see comes from.

00:05:32.054 --> 00:05:38.523
Let's rewrite this as the
limit is X approaches infinite.

00:05:38.523 --> 00:05:42.385
The limit of constant
times some expression.

00:05:42.385 --> 00:05:43.786
We could take the constant out.

00:05:43.786 --> 00:05:44.718
We could say that's going to be

00:05:44.718 --> 00:05:51.354
P times the limit as X
approaches infinite of 1 plus.

00:05:51.354 --> 00:05:54.518
R over N is 1 over X.

00:05:54.518 --> 00:06:01.105
1+1 over X to the ...

00:06:01.105 --> 00:06:03.518
N is X x R.

00:06:03.518 --> 00:06:06.322
N is X x R, so let me write that,

00:06:06.322 --> 00:06:15.460
to the X x R, R x T power.

00:06:15.460 --> 00:06:18.441
All of this business is
the exact same thing.

00:06:18.441 --> 00:06:20.395
Let me rewrite this.

00:06:20.395 --> 00:06:24.794
Let me copy and paste
this part right over here.

00:06:24.794 --> 00:06:25.994
Copy.

00:06:25.994 --> 00:06:28.397
This is the same thing.

00:06:28.397 --> 00:06:31.866
This is equal to P times

00:06:31.866 --> 00:06:33.837
(let me put some parenthesis here)

00:06:33.837 --> 00:06:39.840
times (maybe that's too
big) times the limit.

00:06:39.840 --> 00:06:43.670
This limit right over here.

00:06:43.670 --> 00:06:46.099
If I raise something to
the product of these,

00:06:46.099 --> 00:06:48.016
I'm taking X x R x T,

00:06:48.016 --> 00:06:48.805
that's the same thing as

00:06:48.805 --> 00:06:50.202
doing this whole thing to the X

00:06:50.202 --> 00:06:55.017
and then raising that to the RT power.

00:06:55.017 --> 00:06:58.349
This comes from exponent properties,

00:06:58.349 --> 00:07:00.469
that you might have learned before.

00:07:00.469 --> 00:07:01.766
These 2 things are equivalent.

00:07:01.766 --> 00:07:03.932
I'm doing a couple of
steps in the process here,

00:07:03.932 --> 00:07:08.599
but hopefully this seems
reasonably intuitive for you.

00:07:08.599 --> 00:07:10.599
I'm really just using the property.

00:07:10.599 --> 00:07:19.325
The limit as, let's say,
X approaches C of F of X

00:07:19.325 --> 00:07:25.384
to the, let's call it, to the XRT power.

00:07:25.384 --> 00:07:29.467
This is the same thing as the limit

00:07:29.467 --> 00:07:34.583
as X approaches C of F of X to the X

00:07:34.583 --> 00:07:38.250
and then all of that
raised to the RT power.

00:07:38.250 --> 00:07:40.315
What is this stuff right over here?

00:07:40.315 --> 00:07:42.848
What is all of this business
that's inside the parentheses?

00:07:42.848 --> 00:07:44.714
We've seen that before.

00:07:44.714 --> 00:07:50.800
All of this, all of that is equal to E.

00:07:50.800 --> 00:07:51.634
We can write this.

00:07:51.634 --> 00:07:52.549
This is exciting.

00:07:52.549 --> 00:07:56.982
This is formula for continuous
compounding interest.

00:07:56.982 --> 00:07:58.467
If we continuously compound,

00:07:58.467 --> 00:08:04.921
we're going to have to pay
back our principal times E,

00:08:04.921 --> 00:08:11.491
to the RT power.

00:08:11.491 --> 00:08:13.462
Let's do a concrete example here.

00:08:13.462 --> 00:08:19.698
If you were to borrow $50,
over 3 years, 10% interest,

00:08:19.698 --> 00:08:22.366
but you're not compounding
just 4 times a year,

00:08:22.366 --> 00:08:25.198
you're going to compound
an infinite times per year.

00:08:25.198 --> 00:08:27.948
You're going to be continuous compounding.

00:08:27.948 --> 00:08:30.730
We can see how much you would
actually have to pay back.

00:08:30.730 --> 00:08:38.615
It is going to be 50 x E to the ...

00:08:38.615 --> 00:08:41.329
Our rate is .1.

00:08:41.329 --> 00:08:42.615
Just let me put some parentheses here.

00:08:42.615 --> 00:08:48.329
0.1 x time, so times 3 years.

00:08:48.329 --> 00:08:49.327
T as in years.

00:08:49.327 --> 00:08:50.729
We assumed it was in years.

00:08:50.729 --> 00:08:51.463
We get ...

00:08:51.463 --> 00:08:53.865
You would have to pay back $67.

00:08:53.865 --> 00:08:57.865
If we're to round ...
$67.49 if you were to round.