2015 AP Calculus AB 5b

WEBVTT
Kind: captions
Language: en

00:00:00.248 --> 00:00:02.279
- [Voiceover] Part b,
on what open intervals

00:00:02.279 --> 00:00:04.724
contained in negative three is
less than x is less than four

00:00:04.724 --> 00:00:08.927
is the graph of f both
concave down and decreasing?

00:00:08.927 --> 00:00:11.110
Give a reason for your answer.

00:00:11.110 --> 00:00:13.606
All right, so f and this should be both

00:00:13.606 --> 00:00:16.172
concave down and decreasing.

00:00:16.172 --> 00:00:19.181
So f concave down,

00:00:19.181 --> 00:00:21.596
concave down.

00:00:21.596 --> 00:00:25.173
That's going to be true if and only if

00:00:25.173 --> 00:00:29.230
the second derivative
of f is less than zero

00:00:29.230 --> 00:00:32.380
which is true if and only if

00:00:32.380 --> 00:00:35.796
f prime is decreasing,

00:00:35.796 --> 00:00:38.536
f prime is decreasing.

00:00:38.536 --> 00:00:41.220
So that's how we can tell
when f is concave down.

00:00:41.220 --> 00:00:43.902
I wanna see what do I
need to observe of f prime

00:00:43.902 --> 00:00:45.678
because that's the graph they gave us.

00:00:45.678 --> 00:00:50.055
All right, now for f to be decreasing,

00:00:50.055 --> 00:00:52.566
f is going to be decreasing

00:00:52.566 --> 00:00:53.983
if and only if,

00:00:53.983 --> 00:00:55.805
we are decreasing when
you have a negative slope

00:00:55.805 --> 00:00:57.443
if your relative change is negative.

00:00:57.443 --> 00:01:01.204
So that's true if and only if f prime of x

00:01:01.204 --> 00:01:03.062
is less than zero.

00:01:03.062 --> 00:01:05.186
So both true.

00:01:05.186 --> 00:01:08.119
Both true

00:01:08.119 --> 00:01:09.137
when

00:01:10.292 --> 00:01:13.485
f prime of x is less than zero

00:01:13.485 --> 00:01:15.632
and decreasing,

00:01:15.632 --> 00:01:19.001
and, let me write, and
f prime is decreasing.

00:01:19.001 --> 00:01:22.682
And f prime, decreasing.

00:01:22.682 --> 00:01:24.567
So let's see if we can identify that.

00:01:24.567 --> 00:01:26.343
Well, let's go back to our graph

00:01:26.343 --> 00:01:29.060
and see when is f prime
of x less than zero

00:01:29.060 --> 00:01:31.591
and decreasing.

00:01:31.591 --> 00:01:33.507
So if we look at less than zero,

00:01:33.507 --> 00:01:34.964
let me highlight this,

00:01:34.964 --> 00:01:36.062
so less than zero.

00:01:36.062 --> 00:01:39.707
I'm highlighting it in
magenta right over here.

00:01:39.707 --> 00:01:43.783
This is all the places
where we are less than zero

00:01:43.783 --> 00:01:45.849
in magenta.

00:01:45.849 --> 00:01:48.686
And now of that, where are we decreasing?

00:01:48.686 --> 00:01:53.237
Well, we are decreasing over
this interval right over here.

00:01:53.237 --> 00:01:56.627
So that would be negative
two is less than x

00:01:56.627 --> 00:02:00.226
which is less than negative one.

00:02:00.226 --> 00:02:02.258
So that's we're decreasing

00:02:02.258 --> 00:02:03.762
of the places where we are negative.

00:02:03.762 --> 00:02:06.084
I'm taking the subset
of where we are negative

00:02:06.084 --> 00:02:07.640
to think what we're decreasing.

00:02:07.640 --> 00:02:08.882
We're decreasing right there.

00:02:08.882 --> 00:02:10.984
And we are also decreasing,

00:02:10.984 --> 00:02:13.285
we are also decreasing

00:02:13.285 --> 00:02:14.736
right over there.

00:02:14.736 --> 00:02:19.043
So that is one is less than
x which is less than three.

00:02:19.043 --> 00:02:22.735
Now we are decreasing before x
equals negative two over here

00:02:22.735 --> 00:02:24.616
but we're not less than zero there

00:02:24.616 --> 00:02:26.021
so I'm not gonna include that.

00:02:26.021 --> 00:02:28.018
And so we have our intervals.

00:02:28.018 --> 00:02:29.817
So our open intervals.

00:02:29.817 --> 00:02:34.375
So we have both true for when
these two things are true.

00:02:34.375 --> 00:02:37.591
And so this happens.

00:02:37.591 --> 00:02:39.327
This happens

00:02:40.958 --> 00:02:42.806
when

00:02:42.806 --> 00:02:47.073
negative two is less than
x is less than negative one

00:02:47.073 --> 00:02:50.517
or one is less than x

00:02:50.517 --> 00:02:53.268
is less than three.

00:02:53.268 --> 00:02:55.672
So those are two open intervals.

00:02:55.672 --> 00:02:56.751
This one

00:02:56.751 --> 00:02:58.710
and that one.


2015 AP Calculus AB 5d

WEBVTT
Kind: captions
Language: en

00:00:00.387 --> 00:00:03.405
- [Voiceover] Part d, given
that f of one is equal to three,

00:00:03.405 --> 00:00:08.119
write an expression for f of
x that involves an integral.

00:00:08.119 --> 00:00:09.326
It says it involves an integral.

00:00:09.326 --> 00:00:11.540
We can assume it's going
to involve f prime.

00:00:11.540 --> 00:00:14.408
Thus you can see given us so
much information about f prime

00:00:14.408 --> 00:00:15.952
including its graph and the area

00:00:15.952 --> 00:00:18.146
under or above the curve
of the different integrals.

00:00:18.146 --> 00:00:21.951
And it wanted us to find f of
four and f of negative two.

00:00:21.951 --> 00:00:25.320
So let's think about how we can connect

00:00:25.320 --> 00:00:28.847
f prime, an integral and f of x.

00:00:28.847 --> 00:00:32.368
Well, if we have the integral from a to b

00:00:32.368 --> 00:00:37.368
of f prime of x, dx.

00:00:37.662 --> 00:00:39.670
What is this going to be equal to?

00:00:39.670 --> 00:00:42.248
Well, this is going to be
equal to the antiderivative

00:00:42.248 --> 00:00:45.812
of f prime of x which is f of x.

00:00:45.812 --> 00:00:49.492
And we're going to
evaluate that at b and a

00:00:49.492 --> 00:00:50.770
and subtract the difference.

00:00:50.770 --> 00:00:53.614
So this is going to be f of b

00:00:53.614 --> 00:00:55.959
minus f of a.

00:00:55.959 --> 00:00:59.431
This is straight out of the
fundamental theorem of calculus.

00:00:59.431 --> 00:01:01.416
All right, well this is interesting.

00:01:01.416 --> 00:01:03.103
Well, what if, instead of having a b,

00:01:03.103 --> 00:01:05.308
what if we had an x there?

00:01:05.308 --> 00:01:07.383
And if we're going to put
an x as one of our bounds

00:01:07.383 --> 00:01:09.318
of integration, well, we'll
use a different variable

00:01:09.318 --> 00:01:10.445
for our integration here.

00:01:10.445 --> 00:01:12.011
So let me write it this way.

00:01:12.011 --> 00:01:14.786
So find the integral from a to x

00:01:14.786 --> 00:01:19.786
of f prime of u, du.

00:01:19.813 --> 00:01:21.752
Well, what is this going to be equal to?

00:01:21.752 --> 00:01:24.310
Well, this is going to
be, by the same logic,

00:01:24.310 --> 00:01:26.481
the antiderivative of
this evaluated with x,

00:01:26.481 --> 00:01:31.481
so f of x minus f of a,

00:01:31.503 --> 00:01:34.545
minus f of a.

00:01:34.545 --> 00:01:36.625
Or, if we wanted to solve for f of x,

00:01:36.625 --> 00:01:38.430
we could add f of a to bought sides

00:01:38.430 --> 00:01:42.823
and we would get f of x
is equal to the integral

00:01:42.823 --> 00:01:47.587
from a to x of f prime of u, du.

00:01:47.587 --> 00:01:48.823
Once again, I wanted the u one.

00:01:48.823 --> 00:01:50.213
This is some letter other than x

00:01:50.213 --> 00:01:52.838
since I already used x as one
of my bounds of integration.

00:01:52.838 --> 00:01:54.603
And I'm adding f of a to both sides.

00:01:54.603 --> 00:01:56.867
I swapped the sides too.

00:01:56.867 --> 00:02:00.501
So f of x is going to be
equal to this plus f of a.

00:02:00.501 --> 00:02:03.020
Plus f of a.

00:02:03.020 --> 00:02:06.759
So this is a general form
if a is a lower bound

00:02:06.759 --> 00:02:08.628
but they gave us some information.

00:02:08.628 --> 00:02:10.802
They said that f of one,

00:02:10.802 --> 00:02:13.582
f of one is equal to three.

00:02:13.582 --> 00:02:16.094
So if we choose a to be equal to one,

00:02:16.094 --> 00:02:18.056
if we say this is one right over here.

00:02:18.056 --> 00:02:19.217
This is one.

00:02:19.217 --> 00:02:21.704
Then we know that this is
going to be equal to three.

00:02:21.704 --> 00:02:24.735
So we can write f of x,

00:02:24.735 --> 00:02:28.368
f of x is equal to the integral.

00:02:28.368 --> 00:02:31.712
I'm using one as my a since
I know what f of one is.

00:02:31.712 --> 00:02:36.712
From one to x of f prime of u, du,

00:02:37.076 --> 00:02:40.226
plus f of one, they told us what that is.

00:02:40.226 --> 00:02:42.196
That is going to be equal to three.

00:02:42.196 --> 00:02:44.001
So this is the first part right here.

00:02:44.001 --> 00:02:44.814
This is it.

00:02:44.814 --> 00:02:47.960
That is the first part of the problem.

00:02:47.960 --> 00:02:52.163
Now, let's try to find f of
four and f of negative two.

00:02:52.163 --> 00:02:55.333
All right, f of four.

00:02:55.333 --> 00:02:57.216
Well, whenever see an x,
we substitute to four.

00:02:57.216 --> 00:03:00.093
There's gonna be integral from one to four

00:03:00.093 --> 00:03:04.604
of f prime of u, du plus three.

00:03:04.604 --> 00:03:06.732
So, what is gonna be the
integral from one to four

00:03:06.732 --> 00:03:08.532
of f prime of u, du?

00:03:08.532 --> 00:03:10.482
So let's look up here.

00:03:10.482 --> 00:03:13.574
So the integral from

00:03:13.574 --> 00:03:16.260
one to four of f prime,

00:03:16.260 --> 00:03:18.373
that's the curve right over here.

00:03:18.373 --> 00:03:22.204
Well, that's going to give this area

00:03:22.204 --> 00:03:24.239
but it's going to be
the negative of the area

00:03:24.239 --> 00:03:26.804
because it's the, if you
just take the integrals,

00:03:26.804 --> 00:03:31.124
it's the area that you
could view above the x-axis

00:03:31.124 --> 00:03:32.803
and below where for the intervals of u,

00:03:32.803 --> 00:03:36.216
above the u axis and below the function.

00:03:36.216 --> 00:03:37.316
Well, this is the other way around.

00:03:37.316 --> 00:03:40.689
The function is below
the horizontal axis here.

00:03:40.689 --> 00:03:42.971
So this area, which they
told us in the problem,

00:03:42.971 --> 00:03:46.907
the area bound by the x-axis
in the graph of f prime

00:03:46.907 --> 00:03:49.498
over the interval one, four is 12.

00:03:49.498 --> 00:03:51.855
So this area over here is 12

00:03:51.855 --> 00:03:53.434
but the integral is going to be a negative

00:03:53.434 --> 00:03:57.159
because, once again, our
function is below the x-axis.

00:03:57.159 --> 00:04:00.646
So this integral, this right over here is

00:04:00.646 --> 00:04:03.058
going to be a negative 12.

00:04:03.058 --> 00:04:05.244
This is negative 12.

00:04:05.244 --> 00:04:09.934
So negative 12 plus
three is negative nine.

00:04:09.934 --> 00:04:13.046
All right, now let's
evaluate f of negative two.

00:04:13.046 --> 00:04:16.390
F of negative two is equal to the integral

00:04:16.390 --> 00:04:20.244
from one to negative two of f prime

00:04:20.244 --> 00:04:23.564
of u, du plus three.

00:04:23.564 --> 00:04:25.294
Well, it's kind of feels a little strange

00:04:25.294 --> 00:04:28.475
to have the upper bound to being
lower than the lower bound.

00:04:28.475 --> 00:04:29.729
So we could swap the bounds

00:04:29.729 --> 00:04:31.610
and then add a negative about here.

00:04:31.610 --> 00:04:33.925
So this is going to be
equal to the negative of,

00:04:33.925 --> 00:04:38.089
if we swap the bounds here.

00:04:38.089 --> 00:04:40.934
So from negative two to one,

00:04:40.934 --> 00:04:43.685
f prime of u,

00:04:43.685 --> 00:04:45.990
du plus three.

00:04:45.990 --> 00:04:48.545
So that's the integral
from negative to the one.

00:04:48.545 --> 00:04:50.646
Well, they gave us that.

00:04:50.646 --> 00:04:53.574
From negative two to one.

00:04:53.574 --> 00:04:56.325
They told us that this
area over here is nine.

00:04:56.325 --> 00:04:58.752
But once again, because it's
below the horizontal axis

00:04:58.752 --> 00:05:02.067
and above the curve, we
would say that the integral

00:05:02.067 --> 00:05:04.529
would evaluate to be negative nine.

00:05:04.529 --> 00:05:05.643
So the area is nine.

00:05:05.643 --> 00:05:09.741
But once again, the curve
is below the x-axis.

00:05:09.741 --> 00:05:13.093
So the integral would
give us negative nine.

00:05:15.433 --> 00:05:17.926
This would evaluate to negative nine.

00:05:17.926 --> 00:05:21.150
So take the negative of negative
nine which is positive nine

00:05:21.150 --> 00:05:24.703
plus three which is equal to,

00:05:24.703 --> 00:05:27.512
so nine plus three is equal to 12,

00:05:27.512 --> 00:05:29.023
and we're done.


2015 AP Calculus AB/BC 4ab

WEBVTT
Kind: captions
Language: en

00:00:00.060 --> 00:00:02.080
- [Voiceover] Consider
the differential equation,

00:00:02.080 --> 00:00:03.780
the derivative of y with respect to x

00:00:03.780 --> 00:00:05.662
is equal to 2x minus y.

00:00:05.662 --> 00:00:07.420
"On the axes provided,

00:00:07.420 --> 00:00:10.570
"sketch a slope field for the
given differential equation

00:00:10.570 --> 00:00:12.400
"at the six points indicated."

00:00:12.400 --> 00:00:16.140
And we see one, two, three,
four, five, six points.

00:00:16.140 --> 00:00:18.080
And so what I can do is,

00:00:18.080 --> 00:00:19.922
let me set up a table, actually.

00:00:19.922 --> 00:00:22.320
Let me set up a table with three columns.

00:00:22.320 --> 00:00:25.133
One for x,

00:00:25.133 --> 00:00:27.880
one for y,

00:00:28.880 --> 00:00:31.040
and then one for the derivative of y

00:00:31.040 --> 00:00:32.640
with respect to x,

00:00:32.640 --> 00:00:34.570
which this differential equation tells us

00:00:34.570 --> 00:00:36.182
in terms of x and y.

00:00:36.182 --> 00:00:37.920
And so we can look at
the different points,

00:00:37.920 --> 00:00:39.360
this point right over here

00:00:39.360 --> 00:00:42.440
when x is one

00:00:42.440 --> 00:00:44.750
and y is two.

00:00:44.750 --> 00:00:47.430
x is one, y is two.

00:00:47.430 --> 00:00:51.160
Well, the derivative, they
tell us is two times x minus y.

00:00:51.160 --> 00:00:54.350
So this is 2x minus y.

00:00:54.350 --> 00:00:58.560
So it's gonna be two times one minus two,

00:00:58.560 --> 00:01:02.001
which is equal to zero.

00:01:02.001 --> 00:01:04.990
And so, if the derivative
there is equal to zero,

00:01:04.990 --> 00:01:07.840
then if I were to draw a line

00:01:07.840 --> 00:01:11.100
that indicates the slope at that point,

00:01:11.100 --> 00:01:13.440
well I would draw a
line with a zero slope.

00:01:15.320 --> 00:01:17.881
It would look something like,

00:01:17.881 --> 00:01:19.649
let me draw it a little bit,

00:01:19.649 --> 00:01:22.911
it would look something like that.

00:01:22.911 --> 00:01:24.430
All right, let's keep going.

00:01:24.430 --> 00:01:25.540
Let's look at this point.

00:01:25.540 --> 00:01:28.040
Where x is equal to
zero, y is equal to two.

00:01:28.040 --> 00:01:30.350
x is equal to zero, y is equal to two.

00:01:30.350 --> 00:01:35.350
The derivative is going
to be two times x minus y.

00:01:35.541 --> 00:01:38.880
So it's going to be zero minus two,

00:01:38.880 --> 00:01:41.376
which is equal to negative two.

00:01:41.376 --> 00:01:43.870
So how do we draw a slope of negative two?

00:01:43.870 --> 00:01:45.420
Well, it's going to go from the top left

00:01:45.420 --> 00:01:48.020
to the bottom right, it's
going to be pretty steep.

00:01:49.960 --> 00:01:53.440
It might look something like that.

00:01:53.440 --> 00:01:56.540
That looks like a slope of negative,

00:01:56.540 --> 00:01:59.889
let me try to draw it a little bit better.

00:01:59.889 --> 00:02:02.920
So I could draw it...

00:02:04.150 --> 00:02:05.864
Well, that's pretty good.

00:02:05.864 --> 00:02:08.140
Maybe I shouldn't use my line tool,

00:02:08.140 --> 00:02:10.712
let me just try to draw a reasonable,

00:02:10.712 --> 00:02:12.610
so if I have a slope of negative two

00:02:12.610 --> 00:02:13.740
as I move one to the right,

00:02:13.740 --> 00:02:15.070
I move two down.

00:02:15.070 --> 00:02:17.810
So it should look something like,

00:02:17.810 --> 00:02:20.829
something like that.

00:02:20.829 --> 00:02:22.430
All right, let's keep going.

00:02:22.430 --> 00:02:23.860
x one, y one.

00:02:23.860 --> 00:02:25.570
x is one, y is one.

00:02:25.570 --> 00:02:28.991
It's gonna be two times one minus one.

00:02:28.991 --> 00:02:30.850
So this is two minus one.

00:02:30.850 --> 00:02:33.292
It's gonna have a slope of one.

00:02:33.292 --> 00:02:37.940
And so that would look something like,

00:02:37.940 --> 00:02:40.400
a slope of one looks something like that.

00:02:40.400 --> 00:02:42.780
Let me just draw it by hand.

00:02:42.780 --> 00:02:44.142
That's actually easier.

00:02:44.142 --> 00:02:46.920
And then we have x is zero, y is one.

00:02:46.920 --> 00:02:48.750
x is zero, y is one.

00:02:48.750 --> 00:02:53.230
Slope is gonna be two
times zero minus one,

00:02:53.230 --> 00:02:55.750
which is equal to negative one.

00:02:55.750 --> 00:02:59.330
So now the slope is negative one.

00:02:59.330 --> 00:03:02.200
And just like that,
notice it's less negative

00:03:02.200 --> 00:03:04.910
than up here, it's less steep.

00:03:04.910 --> 00:03:08.450
And then let's go to x is one,

00:03:08.450 --> 00:03:10.080
y is negative one.

00:03:10.080 --> 00:03:12.420
x is one, y is negative one.

00:03:12.420 --> 00:03:16.851
So it's gonna be two times
one minus negative one.

00:03:16.851 --> 00:03:19.820
So this is two plus one.

00:03:19.820 --> 00:03:22.080
This is equal to positive three.

00:03:22.080 --> 00:03:25.940
So here, the slope, it's
going to be even steeper,

00:03:25.940 --> 00:03:27.935
but now in the positive sense.

00:03:27.935 --> 00:03:31.835
So a slope of three would
look something like that.

00:03:31.835 --> 00:03:36.658
And then finally, we have x
is zero, y is negative one.

00:03:36.658 --> 00:03:40.550
It's gonna be two times
zero minus negative one,

00:03:40.550 --> 00:03:42.870
which is going to be
equal to a slope of one,

00:03:42.870 --> 00:03:45.330
which is going to look like this.

00:03:45.330 --> 00:03:46.920
It should be the same,
it should be parallel

00:03:46.920 --> 00:03:49.053
to what we have right over there.

00:03:49.053 --> 00:03:50.330
And we're done.

00:03:50.330 --> 00:03:52.067
All right, let's do part B now.

00:03:52.067 --> 00:03:54.960
Find the second derivative
of y with respect to x

00:03:54.960 --> 00:03:56.340
in terms of x and y.

00:03:56.340 --> 00:03:59.160
Determine the concavity of all solutions

00:03:59.160 --> 00:04:02.190
for the given differential
equation in quadrant two.

00:04:02.190 --> 00:04:04.921
Give a reason for your answer.

00:04:04.921 --> 00:04:07.150
All right, so first let's just
find the second derivative.

00:04:07.150 --> 00:04:10.514
So we already know that dy,

00:04:10.514 --> 00:04:13.350
let me give myself a little
bit more space, actually.

00:04:13.350 --> 00:04:17.180
We already know that the
derivative of y with respect to x

00:04:17.180 --> 00:04:21.331
is equal to 2x minus y.

00:04:21.331 --> 00:04:22.700
Now, to find the second derivative,

00:04:22.700 --> 00:04:25.010
I just wanna take the
derivative of both sides of this

00:04:25.010 --> 00:04:26.577
with respect to x.

00:04:26.577 --> 00:04:27.738
So let's do that.

00:04:27.738 --> 00:04:31.302
So I could take the derivative
of the left-hand side

00:04:31.302 --> 00:04:35.736
with respect to x, and the
derivative of the right-hand side

00:04:35.736 --> 00:04:38.463
with respect to x.

00:04:38.463 --> 00:04:39.960
And so what is this going to be?

00:04:39.960 --> 00:04:41.090
Well, on the left-hand side,

00:04:41.090 --> 00:04:44.350
the notation would just
be the second derivative

00:04:44.350 --> 00:04:47.899
of y with respect to x.

00:04:47.899 --> 00:04:49.920
And then over here,
let's take the derivative

00:04:49.920 --> 00:04:51.450
of each of these with respect to x.

00:04:51.450 --> 00:04:53.910
The derivative of 2x with respect to x

00:04:53.910 --> 00:04:55.944
is going to be equal to two,

00:04:55.944 --> 00:04:58.760
and then minus the derivative
of y with respect to x,

00:04:58.760 --> 00:05:03.580
well, it's just gonna be
minus the derivative of y

00:05:03.580 --> 00:05:06.910
with respect to x.

00:05:08.660 --> 00:05:10.410
So we could say, okay, we
found the second derivative,

00:05:10.410 --> 00:05:13.480
but remember, they're
saying in terms of x and y.

00:05:13.480 --> 00:05:16.830
Right now I've found the
second derivative in terms of,

00:05:16.830 --> 00:05:18.560
well, in terms of a constant,

00:05:18.560 --> 00:05:20.950
and in terms of the first derivative.

00:05:20.950 --> 00:05:22.930
And so we can substitute our expression

00:05:22.930 --> 00:05:24.580
for the first derivative back here

00:05:24.580 --> 00:05:27.133
to have this expression
in terms of x and y.

00:05:27.133 --> 00:05:29.540
And so, this is all going to be equal to,

00:05:29.540 --> 00:05:30.280
let me write it over here.

00:05:30.280 --> 00:05:34.196
The second derivative
of y with respect to x

00:05:34.196 --> 00:05:38.074
is going to be equal to two minus,

00:05:38.074 --> 00:05:39.980
well, the derivative
of y with respect to x,

00:05:39.980 --> 00:05:42.160
we already know is 2x minus y.

00:05:42.160 --> 00:05:44.940
2x minus y.

00:05:44.940 --> 00:05:47.480
And so this is going to be equal to

00:05:47.480 --> 00:05:52.056
two minus 2x plus y.

00:05:52.056 --> 00:05:53.633
Plus y.

00:05:53.633 --> 00:05:56.260
And then they say, determine the concavity

00:05:56.260 --> 00:05:58.840
of all solutions for the
given differential equation

00:05:58.840 --> 00:06:00.180
in quadrant two.

00:06:00.180 --> 00:06:02.856
Remember, if you're thinking
about our coordinate axes,

00:06:02.856 --> 00:06:04.610
if you think about our coordinate axes,

00:06:04.610 --> 00:06:07.270
so if that's our y-axis,
that is our x-axis,

00:06:07.270 --> 00:06:08.680
this is quadrant one,

00:06:08.680 --> 00:06:10.280
this is quadrant two,

00:06:10.280 --> 00:06:11.910
this is quadrant three,

00:06:11.910 --> 00:06:13.760
and this is quadrant four.

00:06:13.760 --> 00:06:15.460
So they're talking about quadrant two.

00:06:15.460 --> 00:06:18.190
What do we know about x
and y in quadrant two?

00:06:18.190 --> 00:06:20.710
We know x is less than zero,

00:06:20.710 --> 00:06:23.734
and we know y is greater than zero.

00:06:23.734 --> 00:06:26.350
So quadrant two, let's write like this.

00:06:26.350 --> 00:06:29.720
Quadrant two.

00:06:29.720 --> 00:06:32.390
We know x is less than zero,

00:06:32.390 --> 00:06:36.173
and y is greater than zero.

00:06:36.173 --> 00:06:40.370
And so, if that is the case, so if we get,

00:06:40.370 --> 00:06:43.943
so we have two minus 2x plus y.

00:06:43.943 --> 00:06:46.340
So this is going to be,
this right over here

00:06:46.340 --> 00:06:47.983
is gonna be greater than zero.

00:06:47.983 --> 00:06:52.790
And negative two times a negative value,

00:06:52.790 --> 00:06:55.394
well, this is also going
to be greater than zero.

00:06:55.394 --> 00:06:58.000
So quadrant two, because of this,

00:06:58.000 --> 00:07:00.320
that means if this is,

00:07:00.320 --> 00:07:02.040
this whole expression
is going to be positive,

00:07:02.040 --> 00:07:04.500
negative two times a negative
number is gonna be positive,

00:07:04.500 --> 00:07:05.980
plus a positive, plus a positive

00:07:05.980 --> 00:07:07.235
is going to be positive.

00:07:07.235 --> 00:07:09.302
So the second derivative

00:07:09.302 --> 00:07:13.300
second derivative with
respect to x is positive.

00:07:13.300 --> 00:07:14.480
Positive.

00:07:16.230 --> 00:07:18.100
Which means that our slope

00:07:18.100 --> 00:07:20.820
is increasing over that interval,

00:07:20.820 --> 00:07:23.220
which means that we have positive,

00:07:23.220 --> 00:07:26.660
we could say that our
con-kuh, our concavity,

00:07:26.660 --> 00:07:28.240
I always have trouble saying that.

00:07:30.270 --> 00:07:33.913
Concavity is upwards.

00:07:33.913 --> 00:07:35.080
Upwards.

00:07:35.080 --> 00:07:37.480
And if you ever forget whether, okay,

00:07:37.480 --> 00:07:39.010
a second derivative being positive,

00:07:39.010 --> 00:07:40.900
is that concavity upwards or downwards?

00:07:40.900 --> 00:07:44.260
I always just like to draw kind of our,

00:07:44.260 --> 00:07:46.490
you know, the canonical concave upwards

00:07:46.490 --> 00:07:48.150
or concave downwards,

00:07:48.150 --> 00:07:50.510
and you can see here that your slope,

00:07:50.510 --> 00:07:52.890
your slope here is turning less negative.

00:07:52.890 --> 00:07:55.580
Or you could say it's
becoming more positive.

00:07:55.580 --> 00:07:56.630
It's increasing.

00:07:56.630 --> 00:08:00.320
And it continues to increase
when you are concave upwards.

00:08:00.320 --> 00:08:02.060
So second derivative positive?

00:08:02.060 --> 00:08:04.100
You are concave upwards.

00:08:04.100 --> 00:08:05.600
Second derivative negative?

00:08:05.600 --> 00:08:08.964
You are going to be concave downwards.

00:08:08.964 --> 00:08:11.100
I'll get rid of this 'cause
you wouldn't put that

00:08:11.100 --> 00:08:12.960
on the actual AP test.


AP US history short answer example 2

WEBVTT
Kind: captions
Language: en

00:00:00.445 --> 00:00:03.022
- [Voiceover] All right, in
this video we're talking about

00:00:03.022 --> 00:00:07.683
the short answer section of
the AP U.S. history exam.

00:00:07.683 --> 00:00:10.771
In the first part of this
video we talked about

00:00:10.771 --> 00:00:14.277
the first two sections of this question

00:00:14.277 --> 00:00:17.481
which asked for examples of how contact

00:00:17.481 --> 00:00:21.080
with Europeans changed
Native American society.

00:00:21.080 --> 00:00:23.821
And the three most important
things we talked about

00:00:23.821 --> 00:00:26.909
were first that you really
have to pay attention

00:00:26.909 --> 00:00:30.461
to the timeline given
to you in the question

00:00:30.461 --> 00:00:32.064
because you don't want to give an answer

00:00:32.064 --> 00:00:34.827
that's outside of that time frame,

00:00:34.827 --> 00:00:39.094
so only talk about the
period from 1492 to 1700.

00:00:39.094 --> 00:00:40.859
Nothing after that.

00:00:40.859 --> 00:00:42.322
The second thing we
mentioned is that there

00:00:42.322 --> 00:00:45.712
could be all sorts of right
answers to this question.

00:00:45.712 --> 00:00:48.428
So choose the things that you know best

00:00:48.428 --> 00:00:50.541
and feel most qualified to talk about.

00:00:50.541 --> 00:00:52.584
And third, if you're stuck,

00:00:52.584 --> 00:00:55.209
consider how a certain issue might affect

00:00:55.209 --> 00:00:57.833
people differently of different races,

00:00:57.833 --> 00:01:00.317
different classes, and different genders.

00:01:00.317 --> 00:01:02.709
You might find a new example
when considering those

00:01:02.709 --> 00:01:05.843
that will help you answer
a more difficult question.

00:01:05.843 --> 00:01:09.141
All right, let's move on to
the last part of this question.

00:01:09.141 --> 00:01:12.670
For part C, we're now looking for kind of

00:01:12.670 --> 00:01:14.203
a counter example,

00:01:14.203 --> 00:01:16.965
a way that Native
Americans resisted changes

00:01:16.965 --> 00:01:20.216
brought on by Europeans
in this time period.

00:01:20.216 --> 00:01:22.120
Now you might be tempted to talk about

00:01:22.120 --> 00:01:23.932
the Ghost Dance,

00:01:23.932 --> 00:01:26.300
but remember that the
Ghost Dance wasn't until

00:01:26.300 --> 00:01:28.344
the end of the 19th century.

00:01:28.344 --> 00:01:30.712
That's the 1880s, 1890s,

00:01:30.712 --> 00:01:33.638
so it's way outside the
scope of our time period

00:01:33.638 --> 00:01:36.028
here ending at 1700.

00:01:36.028 --> 00:01:38.746
So we're thinking back to the early period

00:01:38.746 --> 00:01:41.974
of contact, the early colonial period.

00:01:41.974 --> 00:01:45.433
So what are some examples of resistance

00:01:45.433 --> 00:01:47.361
to change in this time period?

00:01:47.361 --> 00:01:50.704
Well, I think an obvious
example would be war.

00:01:50.704 --> 00:01:53.815
Many Native Americans went to war against

00:01:53.815 --> 00:01:55.743
European invaders.

00:01:55.743 --> 00:02:00.272
Some good examples of this
might be King Philip's War

00:02:00.272 --> 00:02:03.430
in New England,

00:02:03.430 --> 00:02:07.917
or maybe Popé's Rebellion in the West.

00:02:10.847 --> 00:02:15.220
Let me give us a little bit
more space to write here.

00:02:15.220 --> 00:02:17.659
These are two really specific examples,

00:02:17.659 --> 00:02:19.214
and I would recommend that you give

00:02:19.214 --> 00:02:21.768
specific examples whenever possible,

00:02:21.768 --> 00:02:23.835
even though it's not necessary at times,

00:02:23.835 --> 00:02:25.136
but it will show that you really know

00:02:25.136 --> 00:02:27.411
what you're talking about.

00:02:27.411 --> 00:02:30.174
Another example of resistance might be

00:02:30.174 --> 00:02:31.790
going against some of the things we've

00:02:31.790 --> 00:02:33.206
talked about earlier.

00:02:33.206 --> 00:02:34.647
So labor, for example.

00:02:34.647 --> 00:02:36.666
Well, not all Native
Americans were part of

00:02:36.666 --> 00:02:38.501
an encomienda system.

00:02:38.501 --> 00:02:41.540
Some of them chose to flee.

00:02:41.540 --> 00:02:44.025
It's one of the reasons why Europeans

00:02:44.025 --> 00:02:47.461
resorted to African
slavery in this time period

00:02:47.461 --> 00:02:49.528
because Native Americans
knew the landscape

00:02:49.528 --> 00:02:51.711
very well and could get away when someone

00:02:51.711 --> 00:02:53.405
tried to enslave them.

00:02:53.405 --> 00:02:55.866
You could also talk about
some cultural resistance

00:02:55.866 --> 00:02:57.445
to European ways.

00:02:57.445 --> 00:03:01.829
For example, refusal to
convert to Christianity.

00:03:03.158 --> 00:03:07.779
Many Native Americans were targeted

00:03:07.779 --> 00:03:11.493
by missionaries who were
coming to the New World

00:03:11.493 --> 00:03:14.884
specifically to convert Native Americans

00:03:14.884 --> 00:03:18.924
to Christianity along with
getting very wealthy, of course.

00:03:18.924 --> 00:03:21.222
But in many cases Native Americans

00:03:21.222 --> 00:03:24.079
outright refused to
convert to Christianity

00:03:24.079 --> 00:03:26.889
or they may have incorporated Christianity

00:03:26.889 --> 00:03:29.629
into their own previous beliefs.

00:03:29.629 --> 00:03:32.461
So any of these are good options.

00:03:32.461 --> 00:03:34.667
And I should think two
out of the three of these

00:03:34.667 --> 00:03:36.316
are fairly simple.

00:03:36.316 --> 00:03:40.472
I think disease and also warfare

00:03:41.632 --> 00:03:43.479
are things that are going to come to mind

00:03:43.479 --> 00:03:45.476
right away when we think about this early

00:03:45.476 --> 00:03:47.403
period of contact.

00:03:47.403 --> 00:03:50.538
It's the second example that's
really gonna make you reach,

00:03:50.538 --> 00:03:53.834
and for that I recommend this strategy

00:03:53.834 --> 00:03:56.506
of considering race, class and gender,

00:03:56.506 --> 00:03:58.619
because among one of
them there's definitely

00:03:58.619 --> 00:04:01.080
going to be something that is relevant.

00:04:01.080 --> 00:04:05.585
So remember, any of
these answers is correct.

00:04:05.585 --> 00:04:09.058
You only need to choose
the ones that you know,

00:04:09.058 --> 00:04:11.961
and all you need to do
is write one sentence

00:04:11.961 --> 00:04:14.817
about each of these options.

00:04:14.817 --> 00:04:17.882
Keep yourself short,
simple and to the point.

00:04:17.882 --> 00:04:20.598
Answer the question and move on.


AP US history short answer example 1

WEBVTT
Kind: captions
Language: en

00:00:00.584 --> 00:00:03.532
- [Voiceover] So this video
is about the short answer

00:00:03.532 --> 00:00:07.433
section on the AP U.S. history exam.

00:00:07.433 --> 00:00:11.080
So this is a real practice
problem from the AP exam

00:00:11.080 --> 00:00:14.331
and I'd like to go through
it step-by-step with you

00:00:14.331 --> 00:00:16.768
to give you an idea of how to approach

00:00:16.768 --> 00:00:19.207
these problems really well.

00:00:19.207 --> 00:00:21.482
So each of the short answer problems is

00:00:21.482 --> 00:00:24.941
three questions long and they all discuss

00:00:24.941 --> 00:00:28.564
one facet of American history.

00:00:28.564 --> 00:00:30.747
And so your job in these questions is not

00:00:30.747 --> 00:00:32.860
to try to write an essay,

00:00:32.860 --> 00:00:36.181
but merely to just answer A, B, and C,

00:00:36.181 --> 00:00:38.432
which will be related to each other,

00:00:38.432 --> 00:00:40.058
and you have about the front of one

00:00:40.058 --> 00:00:41.823
blue book page to do this,

00:00:41.823 --> 00:00:44.447
so really no more than 200 words.

00:00:44.447 --> 00:00:45.794
And what's great about these

00:00:45.794 --> 00:00:48.835
is that there are many right answers.

00:00:48.835 --> 00:00:52.040
They're asking you to
display your knowledge

00:00:52.040 --> 00:00:54.245
about a topic in the past,

00:00:54.245 --> 00:00:56.543
but what parts of that
knowledge you display

00:00:56.543 --> 00:00:57.868
is up to you.

00:00:57.868 --> 00:01:00.235
So that means you can really
play to your strengths.

00:01:00.235 --> 00:01:01.746
Bring up the thing that you know.

00:01:01.746 --> 00:01:04.508
You don't have to worry
about knowing everything.

00:01:04.508 --> 00:01:07.644
Just enough to answer a few questions.

00:01:07.644 --> 00:01:09.803
So let's read this
question and we're gonna

00:01:09.803 --> 00:01:11.312
read it really carefully,

00:01:11.312 --> 00:01:14.029
which you'll see is important
for a number of reasons.

00:01:14.029 --> 00:01:17.117
So answer A. B, and C.

00:01:17.117 --> 00:01:21.342
A, "Briefly explain one
example of how contact

00:01:21.342 --> 00:01:24.082
"between Native Americans and Europeans

00:01:24.082 --> 00:01:26.953
"brought changes to
Native American societies

00:01:26.953 --> 00:01:30.994
"in the period 1492 to 1700."

00:01:30.994 --> 00:01:33.315
Now this is really important.

00:01:33.315 --> 00:01:36.172
Making sure that you know the date range

00:01:36.172 --> 00:01:38.076
that you're being asked to write about

00:01:38.076 --> 00:01:40.444
is really key to giving a good answer

00:01:40.444 --> 00:01:42.789
because we could talk
about Native American

00:01:42.789 --> 00:01:46.528
history in American society
really right up to the present,

00:01:46.528 --> 00:01:51.036
so 1700 is our end date.

00:01:51.036 --> 00:01:55.145
B, "Briefly explain a second
example of how contact

00:01:55.145 --> 00:01:57.919
"between Native Americans and Europeans

00:01:57.919 --> 00:02:00.723
"brought changes to
Native American societies

00:02:00.723 --> 00:02:02.257
"in the same period."

00:02:02.257 --> 00:02:05.183
And C, "Briefly explain one example of how

00:02:05.183 --> 00:02:08.364
"Native American societies resisted change

00:02:08.364 --> 00:02:12.172
"brought by contact with
Europeans in the same period."

00:02:12.172 --> 00:02:14.819
All right, so basically
what we're being asked here

00:02:14.819 --> 00:02:18.551
is to give two examples of change

00:02:19.996 --> 00:02:24.426
and one example of
resistance to that change.

00:02:27.474 --> 00:02:28.727
Now the way I see it,

00:02:28.727 --> 00:02:31.444
we have one really important stricture

00:02:31.444 --> 00:02:32.954
like I mentioned already,

00:02:32.954 --> 00:02:35.438
which is that we can only
talk about the period

00:02:35.438 --> 00:02:39.849
in-between European
contact and the year 1700.

00:02:39.849 --> 00:02:42.335
So that means a lot of
things that you might

00:02:42.335 --> 00:02:45.956
want to be talking about
from the 1800s and 1900s

00:02:45.956 --> 00:02:48.859
you can't here, because you
won't get credit for it.

00:02:48.859 --> 00:02:51.553
So we're talking about
the period long before

00:02:51.553 --> 00:02:53.202
the American Revolution,

00:02:53.202 --> 00:02:55.778
long before even the
French and Indian War,

00:02:55.778 --> 00:02:57.242
Seven Years' War.

00:02:57.242 --> 00:03:00.120
So this is really early contact.

00:03:00.120 --> 00:03:03.046
But the question actually
doesn't give any strictures

00:03:03.046 --> 00:03:05.229
about societies,

00:03:05.229 --> 00:03:07.993
so we could talk about Native Americans

00:03:07.993 --> 00:03:10.617
in North America at large,

00:03:10.617 --> 00:03:14.006
we could talk about Native
Americans in Jamestown,

00:03:14.006 --> 00:03:17.767
or we could talk about
Native Americans in Mexico.

00:03:17.767 --> 00:03:20.391
We could talk about Native Americans

00:03:20.391 --> 00:03:24.084
and the French in the
Northwest Territories,

00:03:24.084 --> 00:03:26.987
so our geography is really broad here.

00:03:26.987 --> 00:03:29.936
It's our timeline that's relatively short.

00:03:29.936 --> 00:03:32.118
So let's brainstorm
what we could talk about

00:03:32.118 --> 00:03:34.579
for answers A or B.

00:03:34.579 --> 00:03:36.576
Well, we could mention,

00:03:36.576 --> 00:03:40.036
I think an excellent
example would be disease.

00:03:42.416 --> 00:03:45.446
Native Americans experienced an epidemic

00:03:45.446 --> 00:03:49.185
of disease related to pathogens
brought over by Europeans,

00:03:49.185 --> 00:03:52.853
so you could talk about disease
really reshaped society,

00:03:52.853 --> 00:03:56.453
particularly there were morning wars

00:03:57.983 --> 00:03:59.863
in the Northeast,

00:03:59.863 --> 00:04:02.836
where Native Americans
would raid New England

00:04:02.836 --> 00:04:06.250
settlements trying to find
replacements for the people

00:04:06.250 --> 00:04:08.363
who had died in disease.

00:04:08.363 --> 00:04:11.125
You could also talk about trade

00:04:11.125 --> 00:04:15.375
and for example horses

00:04:15.375 --> 00:04:17.720
might be a good example of this.

00:04:17.720 --> 00:04:19.717
The Spanish brought
horses to the New World,

00:04:19.717 --> 00:04:22.202
which completely changed the form

00:04:22.202 --> 00:04:25.174
of transportation and wealth for a lot

00:04:25.174 --> 00:04:28.772
of Native American societies
on the Great Plains.

00:04:28.772 --> 00:04:30.467
What else could we talk about?

00:04:30.467 --> 00:04:33.022
Well, we could also mention labor.

00:04:33.022 --> 00:04:34.903
The Spanish tried to implement a system

00:04:34.903 --> 00:04:37.829
of encomienda labor,

00:04:37.829 --> 00:04:41.080
which would grant land to the Spanish

00:04:41.080 --> 00:04:44.191
for Native Americans to labor upon

00:04:44.191 --> 00:04:46.954
in a system of slavery.

00:04:46.954 --> 00:04:49.973
You might also talk about gender roles

00:04:49.973 --> 00:04:53.363
and intermarriage between

00:04:53.363 --> 00:04:57.194
Native Americans and Europeans

00:04:57.194 --> 00:05:00.676
which changed racial patterns

00:05:00.676 --> 00:05:04.903
and created new racial caste systems

00:05:04.903 --> 00:05:06.784
and also changed the way that women

00:05:06.784 --> 00:05:09.269
were often viewed in
Native American societies,

00:05:09.269 --> 00:05:11.544
often resulting in them having less power

00:05:11.544 --> 00:05:14.331
than they did before
the Europeans arrived.

00:05:14.331 --> 00:05:15.793
When you're thinking about how to answer

00:05:15.793 --> 00:05:16.978
these types of questions,

00:05:16.978 --> 00:05:18.975
one of the most useful things you can do

00:05:18.975 --> 00:05:22.590
is think about race,

00:05:23.595 --> 00:05:27.421
class and gender.

00:05:28.308 --> 00:05:30.352
These are things that
historians think about

00:05:30.352 --> 00:05:32.024
all the time,

00:05:32.024 --> 00:05:34.509
and what they mean by
these things is to say

00:05:34.509 --> 00:05:36.970
how does a certain issue affect

00:05:36.970 --> 00:05:39.687
say white people differently than it might

00:05:39.687 --> 00:05:41.961
affect black people?

00:05:41.961 --> 00:05:43.772
Or Native American people

00:05:43.772 --> 00:05:46.234
differently than it
might affect Europeans?

00:05:46.234 --> 00:05:49.276
Or does this affect rich people

00:05:49.276 --> 00:05:52.179
more than it affects poor people?

00:05:52.179 --> 00:05:54.640
And that might also include people who

00:05:54.640 --> 00:05:59.089
own a factory, for example,

00:05:59.089 --> 00:06:02.480
versus the people who work in a factory

00:06:02.480 --> 00:06:04.988
or labor in a factory.

00:06:04.988 --> 00:06:06.381
And lastly, you might think about whether

00:06:06.381 --> 00:06:10.073
something affects men or women differently

00:06:10.073 --> 00:06:14.363
or people of different gender
expressions or sexualities.

00:06:14.363 --> 00:06:17.034
So if you're stuck on what to write,

00:06:17.034 --> 00:06:19.867
take some time to think
about how a certain event

00:06:19.867 --> 00:06:23.209
may have affected anyone
in these three groups

00:06:23.209 --> 00:06:25.880
differently than another group.

00:06:25.880 --> 00:06:28.876
All right, so we answered
the first two questions,

00:06:28.876 --> 00:06:32.033
and you'll see that any one of these

00:06:32.033 --> 00:06:34.542
is a perfectly appropriate answer

00:06:34.542 --> 00:06:38.048
and you only have to
talk about one for each.

00:06:38.048 --> 00:06:40.509
And for each of these I would really only

00:06:40.509 --> 00:06:42.808
just write one sentence.

00:06:42.808 --> 00:06:45.988
Make sure that that
sentence gets at the heart

00:06:45.988 --> 00:06:48.079
of what it is that you're trying to say.

00:06:48.079 --> 00:06:52.072
For example, don't just write, "Disease,"

00:06:53.082 --> 00:06:55.950
but rather write a sentence that says,

00:06:55.950 --> 00:06:58.829
"The introduction of pathogens brought

00:06:58.829 --> 00:07:02.150
"from the Old World to the New affected

00:07:02.150 --> 00:07:04.658
"Native American societies by drastically

00:07:04.658 --> 00:07:07.908
"reducing their populations
and their settlement patterns."

00:07:07.908 --> 00:07:09.952
All right, let's pause here for now

00:07:09.952 --> 00:07:13.401
and in the next video we'll tackle part C.


2015 AP Chemistry free response 7

WEBVTT
Kind: captions
Language: en

00:00:00.588 --> 00:00:01.989
- [Voiceover] Aluminum
metal can be recycled

00:00:01.989 --> 00:00:03.309
from scrap metal by melting the

00:00:03.309 --> 00:00:06.288
metal to evaporate impurities.

00:00:06.288 --> 00:00:08.988
Calculate the amount of
head needed to purify 1.00

00:00:08.988 --> 00:00:13.068
mole of aluminum originally
at 298 kelvin by melting it.

00:00:13.068 --> 00:00:17.108
The melting point of
aluminum is 933 kelvin.

00:00:17.108 --> 00:00:19.408
The molar heat capacity of aluminum is

00:00:19.408 --> 00:00:21.808
24 joules per mole kelvin.

00:00:21.808 --> 00:00:24.268
And the heat of fusion of aluminum

00:00:24.268 --> 00:00:27.468
is 10.7 kilojoules per mole.

00:00:27.468 --> 00:00:30.426
Alright, so we didn't think
about the heat they'd need to

00:00:30.426 --> 00:00:33.425
raise its temperature from 298 kelvin

00:00:33.425 --> 00:00:36.505
we raise the temperature of
this one mole of aluminum

00:00:36.505 --> 00:00:40.535
from 298 kelvin to 933 kelvin,

00:00:40.535 --> 00:00:41.485
and we're going to do that by looking

00:00:41.485 --> 00:00:43.425
at the molar heat capacity.

00:00:43.425 --> 00:00:45.565
And then we're going to add to that

00:00:45.565 --> 00:00:47.385
the heat of fusion, the heat necessary

00:00:47.385 --> 00:00:49.585
to actually melt it.

00:00:49.585 --> 00:00:50.973
So let's do that in steps,

00:00:50.973 --> 00:00:54.723
so heat needed

00:00:54.723 --> 00:00:59.723
heat needed to raise

00:00:59.863 --> 00:01:03.213
temperature

00:01:03.213 --> 00:01:05.893
by how much are we raising the

00:01:05.893 --> 00:01:06.943
temperature by?

00:01:06.943 --> 00:01:09.386
We went from 298 to 933,

00:01:09.386 --> 00:01:11.323
so if we raise it by two we get 300

00:01:11.323 --> 00:01:14.624
and then you have to
raise it by another 633.

00:01:14.624 --> 00:01:16.263
So that's going to be,
so we're raising our

00:01:16.263 --> 00:01:20.683
temperature by 635 kelvin.

00:01:20.683 --> 00:01:23.024
That's to get us to the melting point.

00:01:23.024 --> 00:01:26.103
So the heat needed to raise
temperature by 635 kelvin.

00:01:26.103 --> 00:01:30.939
Well, we have one mole

00:01:30.939 --> 00:01:32.959
one mole of aluminum

00:01:32.959 --> 00:01:36.349
I can write 1.00, so we can
have two significant digits,

00:01:36.349 --> 00:01:38.101
or three significant digits there.

00:01:38.101 --> 00:01:41.021
One mole of aluminum times the specific

00:01:41.021 --> 00:01:44.301
or the molar heat capacity I should say.

00:01:44.301 --> 00:01:48.201
So 24 joules per mole

00:01:48.201 --> 00:01:51.451
per mole kelvin

00:01:51.451 --> 00:01:53.631
times, so this right
over here would be the

00:01:53.631 --> 00:01:55.961
amount of heat to raise it by one kelvin

00:01:55.961 --> 00:01:59.417
but now lets multiply by, we
have to raise it by 635 kelvin

00:01:59.417 --> 00:02:03.044
635 kelvin

00:02:03.044 --> 00:02:05.457
And the units work out,
moles cancel with moles.

00:02:05.457 --> 00:02:10.457
Kelvin cancels with
kelvin, and then we get to

00:02:11.847 --> 00:02:15.555
let's see let me get
my calculator out here.

00:02:15.555 --> 00:02:19.143
So 24 times 635

00:02:19.143 --> 00:02:23.413
is equal to 15,240.

00:02:23.413 --> 00:02:26.393
and we're taking the
product of a bunch of stuff

00:02:26.393 --> 00:02:28.493
three significant figures,
two significant figures

00:02:28.493 --> 00:02:32.133
three significant figures,
so we should write only

00:02:32.133 --> 00:02:36.033
two significant figures
we'll round to 15,000 joules.

00:02:37.533 --> 00:02:42.533
So this is going to be 15,000 joules

00:02:43.163 --> 00:02:45.033
temperature to 933 kelvin.

00:02:45.033 --> 00:02:50.033
And then we have the heat of fusion

00:02:50.773 --> 00:02:55.773
heat to melt once at melting point.

00:02:56.833 --> 00:03:00.053
Melting point.

00:03:01.053 --> 00:03:04.952
Well that is going to be 1.00 mole

00:03:04.952 --> 00:03:09.702
1.00 mole times the heat of fusion.

00:03:09.702 --> 00:03:13.590
times 10.7 kilojoules

00:03:13.590 --> 00:03:17.770
kilojoules per mole.

00:03:17.770 --> 00:03:21.130
So this is going to
give us 10.7 kilojoules

00:03:21.130 --> 00:03:22.830
or if we want to write it in joules,

00:03:22.830 --> 00:03:26.720
so this is 10.7 kilojoules

00:03:26.720 --> 00:03:31.530
which is equal to 10,700 joules.

00:03:31.530 --> 00:03:35.911
And so the total is going
to be the sum of these

00:03:35.911 --> 00:03:40.110
so we could say it is
25,700, but we only have

00:03:40.110 --> 00:03:42.790
two significant figures
here so we could round that

00:03:42.790 --> 00:03:47.790
so we could say 26,000 joules or

00:03:48.410 --> 00:03:52.310
I guess we could write 26 kilojoules.

00:03:52.310 --> 00:03:55.330
Alright, now part B.

00:03:55.330 --> 00:03:57.430
Part B.

00:03:57.430 --> 00:04:00.070
The equation for the overall
process of extracting

00:04:00.070 --> 00:04:05.070
aluminum from aluminum oxide
is shown below, alright?

00:04:06.619 --> 00:04:10.623
Which requires less energy,
recycling existing aluminum

00:04:10.623 --> 00:04:13.983
or extracting aluminum
from aluminum oxide?

00:04:13.983 --> 00:04:17.733
Justify your answer with a calculation.

00:04:17.733 --> 00:04:22.162
Alright, so this reaction right over here

00:04:22.162 --> 00:04:26.902
and this gives us the
heat necessary for a mole

00:04:26.902 --> 00:04:30.202
of the reaction, so if you
get a mole of aluminum oxide

00:04:30.202 --> 00:04:32.782
you put in this much heat, you're going

00:04:32.782 --> 00:04:36.142
to get 2.00 moles of aluminum and

00:04:36.142 --> 00:04:39.442
1.5 moles of molecular oxygen.

00:04:39.442 --> 00:04:44.442
So the equation for the overall

00:04:44.912 --> 00:04:49.912
so to extract 1.00 mole

00:04:51.270 --> 00:04:54.199
1.00 mole of aluminum

00:04:54.199 --> 00:04:59.199
from aluminum oxide requires

00:05:02.018 --> 00:05:05.119
so this much heat will produce 2.00 moles.

00:05:05.119 --> 00:05:07.218
So to do 1.00 mole you just
have to have half of that.

00:05:07.218 --> 00:05:12.218
Requires 1,675 kilojoules divided by two.

00:05:12.919 --> 00:05:17.919
So this is going to be
1,675 divided by two

00:05:19.258 --> 00:05:24.258
is equal to, let's see
it's going to be 837.5.

00:05:25.258 --> 00:05:30.258
837.5

00:05:30.538 --> 00:05:34.598
837.5 kilojoules, yep that's half here.

00:05:34.598 --> 00:05:35.718
and I had four significant digits here

00:05:35.718 --> 00:05:38.397
and I still have four significant digits.

00:05:38.397 --> 00:05:40.417
Extracting, doing it from,

00:05:40.417 --> 00:05:42.598
which requires less recycling existing

00:05:42.598 --> 00:05:44.218
aluminum or extracting it?

00:05:44.218 --> 00:05:48.632
Well to recycle

00:05:49.662 --> 00:05:53.762
1.00 mole requires

00:05:54.853 --> 00:05:57.033
we just figured out, 26 kilojoules.

00:05:57.033 --> 00:05:58.033
26 kilojoules.

00:05:58.033 --> 00:06:01.343
Everything here is in kilojoules,

00:06:01.343 --> 00:06:05.793
which is much less energy.

00:06:05.793 --> 00:06:09.483
Which is much less energy.

00:06:12.553 --> 00:06:14.283
And we are done.


2015 AP Chemistry free response 4

WEBVTT
Kind: captions
Language: en

00:00:00.335 --> 00:00:01.577
- [Voiceover] Answer
the following questions

00:00:01.577 --> 00:00:03.899
about the solubility of calcium hydroxide,

00:00:03.899 --> 00:00:05.955
and they give us the solubility product.

00:00:05.955 --> 00:00:08.056
Write a balanced chemical equation

00:00:08.056 --> 00:00:10.796
for the dissolution of
solid calcium hydroxide

00:00:10.796 --> 00:00:12.363
in pure water.

00:00:12.363 --> 00:00:16.550
So we're going to start
off with calcium hydroxide.

00:00:17.820 --> 00:00:19.480
And it is solid,

00:00:19.480 --> 00:00:21.825
and it is going to be in equilibrium.

00:00:21.825 --> 00:00:23.915
When it's dissolved, you have calcium ions

00:00:23.915 --> 00:00:26.296
that have a positive two charge.

00:00:26.296 --> 00:00:28.478
Writing aqueous here since it's dissolved

00:00:28.478 --> 00:00:30.718
in our solution, a solution of water,

00:00:30.718 --> 00:00:33.749
and then plus you have
your hydroxide ions,

00:00:33.749 --> 00:00:36.733
also dissolved in the water.

00:00:36.733 --> 00:00:39.670
For every calcium, you
have two hydroxides.

00:00:39.670 --> 00:00:40.948
We see that we are balanced,

00:00:40.948 --> 00:00:42.295
one calcium, one calcium,

00:00:42.295 --> 00:00:45.708
two hydroxides, two hydroxides.

00:00:45.708 --> 00:00:47.634
All right, let's do the next part.

00:00:47.634 --> 00:00:51.408
Calculate the molar solubility
of calcium hydroxide,

00:00:51.408 --> 00:00:53.375
not in pure water, but in a solution

00:00:53.375 --> 00:00:54.752
that already has a concentration

00:00:54.752 --> 00:00:57.098
of another salt, calcium nitrate.

00:00:57.098 --> 00:00:59.896
It's a .1 molar concentration,

00:00:59.896 --> 00:01:02.973
.10 molar concentration
of calcium nitrate,

00:01:02.973 --> 00:01:03.936
which you might recognize

00:01:03.936 --> 00:01:05.794
is a very soluble salt.

00:01:05.794 --> 00:01:08.394
It's very soluble, it's a nitrate salt,

00:01:08.394 --> 00:01:12.597
which are very soluble.

00:01:12.597 --> 00:01:13.492
So how do we think about this?

00:01:13.492 --> 00:01:15.837
How do we think about the molar solubility

00:01:15.837 --> 00:01:17.276
of calcium hydroxide?

00:01:17.276 --> 00:01:19.412
How much incremental calcium

00:01:19.412 --> 00:01:21.316
can be dissolved when you already

00:01:21.316 --> 00:01:22.558
have some calcium?

00:01:22.558 --> 00:01:24.555
And for whatever the
concentration of calcium

00:01:24.555 --> 00:01:25.438
that gets dissolved, you're gonna have

00:01:25.438 --> 00:01:28.712
twice the concentration of
hydroxide that gets dissolved.

00:01:28.712 --> 00:01:31.637
You might have guessed,
or intuited, or have known

00:01:31.637 --> 00:01:34.970
that this will involve
the solubility product.

00:01:34.970 --> 00:01:37.616
So the solubility product

00:01:37.616 --> 00:01:39.799
is equal to the concentration

00:01:39.799 --> 00:01:42.018
of the product of the concentrations

00:01:42.018 --> 00:01:44.096
of the two things that
actually get dissolved.

00:01:44.096 --> 00:01:46.395
So you have your calcium ions

00:01:46.395 --> 00:01:49.075
and you have your hydroxide ions,

00:01:49.075 --> 00:01:51.014
hydroxide ions,

00:01:51.014 --> 00:01:52.141
and since you two have moles of this

00:01:52.141 --> 00:01:53.905
for every mole of the calcium,

00:01:53.905 --> 00:01:56.448
we square it just like that.

00:01:56.448 --> 00:01:57.238
Now some of you might be saying,

00:01:57.238 --> 00:01:58.584
hey, this looks kind of like

00:01:58.584 --> 00:01:59.826
an equilibrium constant,

00:01:59.826 --> 00:02:01.464
but if we were dealing with
an equilibrium constant,

00:02:01.464 --> 00:02:04.041
we would divide by the concentration

00:02:04.041 --> 00:02:05.422
of the reactants.

00:02:05.422 --> 00:02:09.939
And we could have done an
equilibrium constant here,

00:02:09.939 --> 00:02:11.448
used an equilibrium constant,

00:02:11.448 --> 00:02:13.608
but the concentration of the reactant,

00:02:13.608 --> 00:02:15.511
this is in its solid state.

00:02:15.511 --> 00:02:17.405
If we imagine what's
happening right over here,

00:02:17.405 --> 00:02:18.565
we have a block,

00:02:18.565 --> 00:02:19.505
you can imagine having a block

00:02:19.505 --> 00:02:21.934
of solid calcium hydroxide here

00:02:21.934 --> 00:02:23.015
inside the water,

00:02:23.015 --> 00:02:24.676
and we've been saturated.

00:02:24.676 --> 00:02:27.659
So we've saturated as
much of the ions as we can

00:02:27.659 --> 00:02:29.076
into our solution.

00:02:29.076 --> 00:02:31.955
And so every time you have
some calcium hydroxide

00:02:31.955 --> 00:02:33.116
that gets dissolved,

00:02:33.116 --> 00:02:35.195
you have an equal amount that forms

00:02:35.195 --> 00:02:37.076
from the solution.

00:02:37.076 --> 00:02:40.233
And so if you were doing
a equilibrium constant,

00:02:40.233 --> 00:02:43.077
you would divide by the concentration

00:02:43.077 --> 00:02:45.179
of the solid calcium hydroxide.

00:02:45.179 --> 00:02:47.557
We're not used to thinking
about concentrations

00:02:47.557 --> 00:02:49.187
of solids in terms of molarity

00:02:49.187 --> 00:02:50.290
and things like that,

00:02:50.290 --> 00:02:51.149
but that's going to be constant

00:02:51.149 --> 00:02:53.786
because if there's any
of the calcium hydroxide

00:02:53.786 --> 00:02:55.944
that gets dissolved, well
the volume goes down,

00:02:55.944 --> 00:02:58.951
and if any gets formed,
the volume goes up.

00:02:58.951 --> 00:03:00.971
So the concentration stays constant.

00:03:00.971 --> 00:03:03.235
The convention is you multiply both sides

00:03:03.235 --> 00:03:05.488
by that to get a solubility product,

00:03:05.488 --> 00:03:07.311
and that's where the
solubility product comes from.

00:03:07.311 --> 00:03:09.435
I'll do another video that goes into much

00:03:09.435 --> 00:03:10.573
more depth than that.

00:03:10.573 --> 00:03:12.722
But anyway, how do we use this?

00:03:12.722 --> 00:03:14.779
Well, we already have,

00:03:14.779 --> 00:03:16.056
let me write this down,

00:03:16.056 --> 00:03:18.343
we already have some calcium ions.

00:03:18.343 --> 00:03:22.523
So we already have

00:03:22.523 --> 00:03:27.523
0.10 molar calcium ions dissolved.

00:03:31.266 --> 00:03:32.624
I don't even have to
write dissolved there.

00:03:32.624 --> 00:03:35.422
We already have that in the solution.

00:03:35.422 --> 00:03:37.912
We are going to add

00:03:40.077 --> 00:03:43.316
x molar, I guess I could say,

00:03:43.316 --> 00:03:48.316
calcium ions through dissolving

00:03:51.491 --> 00:03:54.528
the calcium hydroxide.

00:03:57.238 --> 00:04:00.954
And that means that we
would have two x molar

00:04:00.954 --> 00:04:04.158
concentration, I could write it this way,

00:04:04.158 --> 00:04:07.090
concentration of

00:04:08.790 --> 00:04:11.158
hydroxide

00:04:11.158 --> 00:04:13.632
because for every,
whatever the concentration

00:04:13.632 --> 00:04:15.117
of calcium ions, you're gonna have twice

00:04:15.117 --> 00:04:16.371
the concentration of hydroxide

00:04:16.371 --> 00:04:17.857
because for every calcium molecule,

00:04:17.857 --> 00:04:20.470
you're gonna have two hydroxide molecules.

00:04:20.470 --> 00:04:21.738
So how do we use that

00:04:21.738 --> 00:04:24.453
with this solubility product?

00:04:24.453 --> 00:04:27.588
Well, you already have
this right over here,

00:04:27.588 --> 00:04:32.000
you already have 0.10 already,

00:04:32.000 --> 00:04:34.101
and then you're going to add x.

00:04:34.101 --> 00:04:36.061
So that's that right over there.

00:04:36.061 --> 00:04:37.663
I think this notation that I wrote here

00:04:37.663 --> 00:04:39.196
might be a little bit confusing.

00:04:39.196 --> 00:04:41.356
So bear with me.

00:04:41.356 --> 00:04:44.537
So maybe not putting the brackets there

00:04:44.537 --> 00:04:47.512
will make it a little
bit easier to understand

00:04:48.429 --> 00:04:49.388
because there's clearly already

00:04:49.388 --> 00:04:51.699
a concentration.

00:04:51.699 --> 00:04:53.939
All right, so this is going
to be the concentration

00:04:53.939 --> 00:04:55.019
of calcium ions,

00:04:55.019 --> 00:04:55.832
the ones that you already had

00:04:55.832 --> 00:04:57.132
plus the ones that you add.

00:04:57.132 --> 00:05:00.070
And then your concentration of hydroxide,

00:05:00.070 --> 00:05:01.719
well, that's going to be two x.

00:05:01.719 --> 00:05:02.531
For every calcium you add,

00:05:02.531 --> 00:05:04.551
you're gonna add twice as many hydroxides.

00:05:04.551 --> 00:05:06.920
And of course, you square that.

00:05:06.920 --> 00:05:09.253
That�s going to be equal
to the solubility product,

00:05:09.253 --> 00:05:13.351
which they tell us is
1.3 times 10 to the -6th.

00:05:13.351 --> 00:05:18.351
So we have 1.3 times 10 to the -6th.

00:05:18.576 --> 00:05:20.992
Now if you try to solve this outright,

00:05:20.992 --> 00:05:22.292
it's going to get quite complicated.

00:05:22.292 --> 00:05:23.761
You're gonna end up,

00:05:23.761 --> 00:05:25.103
this is going to be four x squared,

00:05:25.103 --> 00:05:26.159
and you multiply it times this,

00:05:26.159 --> 00:05:27.379
you're gonna get an x squared term,

00:05:27.379 --> 00:05:29.284
and then you�re getting
an extra third term.

00:05:29.284 --> 00:05:30.967
Not that easy to solve.

00:05:30.967 --> 00:05:33.137
A simplifying assumption that we make

00:05:33.137 --> 00:05:35.205
when we solve these, especially because

00:05:35.205 --> 00:05:37.318
this nitrate salt is so soluble,

00:05:37.318 --> 00:05:40.616
much more soluble than this salt

00:05:40.616 --> 00:05:43.240
right over here, this hydroxide salt,

00:05:43.240 --> 00:05:45.632
and so you can assume this is,

00:05:45.632 --> 00:05:46.572
this part right over here

00:05:46.572 --> 00:05:49.335
is approximately 0.1

00:05:49.335 --> 00:05:54.335
since x is much smaller than 0.10.

00:05:55.140 --> 00:05:56.254
That's the assumption you make.

00:05:56.254 --> 00:05:58.054
You have to know that okay,

00:05:58.054 --> 00:05:58.994
well first of all, this is going to be

00:05:58.994 --> 00:06:00.562
hard to solve the way it is,

00:06:00.562 --> 00:06:02.860
but then the fact that we're adding

00:06:02.860 --> 00:06:04.417
a much less soluble thing

00:06:04.417 --> 00:06:08.596
to something that is nowhere close

00:06:08.596 --> 00:06:13.415
to saturation, this is a
very high concentration.

00:06:13.415 --> 00:06:14.517
We can go back fill to see

00:06:14.517 --> 00:06:16.154
if we make this assumption,

00:06:16.154 --> 00:06:17.617
whatever we solve for x,

00:06:17.617 --> 00:06:19.004
if it gets us still pretty close

00:06:19.004 --> 00:06:20.751
to this solubility product.

00:06:20.751 --> 00:06:22.435
So if we make that assumption,

00:06:22.435 --> 00:06:25.779
we get 1.3 times 10 to the -6th

00:06:25.779 --> 00:06:29.099
is equal to 0.10

00:06:29.099 --> 00:06:31.854
times four x squared,

00:06:31.854 --> 00:06:35.081
or we could get x squared

00:06:35.081 --> 00:06:37.020
is equal to 1.3

00:06:37.020 --> 00:06:39.319
times 10 to the -6th

00:06:39.319 --> 00:06:42.662
divided by 0.10,

00:06:42.662 --> 00:06:45.716
and then times four,

00:06:45.716 --> 00:06:48.185
or we could say divided by

00:06:51.441 --> 00:06:56.121
0.40, or we could say that x

00:06:56.121 --> 00:06:58.094
is going to be the square root of this.

00:06:58.094 --> 00:07:00.532
So x is going to be equal to,

00:07:00.532 --> 00:07:02.715
let's get a calculator out,

00:07:02.715 --> 00:07:06.384
so we have 1.3 EE

00:07:06.384 --> 00:07:09.124
times 10 to the -6th

00:07:09.124 --> 00:07:12.387
divided by .4

00:07:12.387 --> 00:07:13.582
is equal to that,

00:07:13.582 --> 00:07:15.706
and now we want to take
the square root of that.

00:07:15.706 --> 00:07:17.262
And let's see, we're dealing

00:07:17.262 --> 00:07:19.108
with how many significant figures

00:07:19.108 --> 00:07:20.141
are over here?

00:07:20.141 --> 00:07:22.069
Well looks like it's about two.

00:07:22.069 --> 00:07:25.077
So 0018.

00:07:25.077 --> 00:07:27.275
So approximately,

00:07:29.132 --> 00:07:30.472
let me write that a little bit neater,

00:07:30.472 --> 00:07:35.472
approximately 0.0018 molar.

00:07:37.998 --> 00:07:38.916
Let me see if that's,

00:07:38.916 --> 00:07:39.384
is that what I got?

00:07:39.384 --> 00:07:44.384
Yeah, .0018 molar.

00:07:45.038 --> 00:07:45.863
And so that's our answer.

00:07:45.863 --> 00:07:47.465
If you want to feel better about it,

00:07:47.465 --> 00:07:49.264
you can substitute this back

00:07:49.264 --> 00:07:53.119
into our original equation here

00:07:53.119 --> 00:07:54.779
and say, okay, this does still get me

00:07:54.779 --> 00:07:56.289
something close to 1.3

00:07:56.289 --> 00:07:59.086
times 10 to the -6th.

00:07:59.086 --> 00:07:59.864
And if you try that,

00:07:59.864 --> 00:08:03.464
you will see that that
is actually the case.

00:08:03.464 --> 00:08:05.727
Now let's do this last part.

00:08:05.727 --> 00:08:07.143
In the box below,

00:08:07.143 --> 00:08:09.349
complete the particle
representation diagram

00:08:09.349 --> 00:08:11.648
that includes four water molecules

00:08:11.648 --> 00:08:13.328
with proper orientation

00:08:13.328 --> 00:08:15.285
around the calcium ions.

00:08:15.285 --> 00:08:16.376
Well, the water molecules we know

00:08:16.376 --> 00:08:17.571
are polar.

00:08:17.571 --> 00:08:18.896
If this is the oxygen here

00:08:18.896 --> 00:08:19.790
and this is a hydrogen,

00:08:19.790 --> 00:08:21.135
this is a hydrogen,

00:08:21.135 --> 00:08:23.191
we have our lone pairs of electrons here.

00:08:23.191 --> 00:08:25.385
So you have a partially
negative charge there,

00:08:25.385 --> 00:08:26.975
and you have partially positive charges

00:08:26.975 --> 00:08:29.564
at the hydrogens that get
their electrons hogged,

00:08:29.564 --> 00:08:32.292
and so you're gonna orient it.

00:08:32.292 --> 00:08:34.127
So they just want us to draw four.

00:08:34.127 --> 00:08:35.787
So the oxygen end is going

00:08:35.787 --> 00:08:39.294
to be oriented towards
the positive calcium

00:08:39.294 --> 00:08:42.033
because you have a
partially negative charge

00:08:42.033 --> 00:08:44.262
being attracted to the positive charge.

00:08:44.262 --> 00:08:45.748
So that's one,

00:08:45.748 --> 00:08:48.565
that is two,

00:08:48.565 --> 00:08:51.128
that is three,

00:08:51.128 --> 00:08:52.569
and that is four.

00:08:52.569 --> 00:08:55.725
They told us to draw only
four right over there.

00:08:55.725 --> 00:08:56.573
If you want to be clear,

00:08:56.573 --> 00:08:58.246
you can say those are oxygens

00:08:58.246 --> 00:08:59.430
right over there,

00:08:59.430 --> 00:09:03.783
and these are the
hydrogens right over here.

00:09:03.783 --> 00:09:05.431
The hydrogens have a partially,

00:09:05.431 --> 00:09:07.823
that end of the molecule has
a partially positive charge

00:09:07.823 --> 00:09:08.613
so it's going to be repelled

00:09:08.613 --> 00:09:09.913
from the calcium ion,

00:09:09.913 --> 00:09:12.386
and then the other side
that has the lone pairs,

00:09:12.386 --> 00:09:14.023
well, that's gonna have a
partially negative charge

00:09:14.023 --> 00:09:14.870
so it's going to be attracted

00:09:14.870 --> 00:09:16.763
to the positive charge.

00:09:16.763 --> 00:09:18.162
And there you go.


2015 AP Chemistry free response 6

WEBVTT
Kind: captions
Language: en

00:00:00.422 --> 00:00:02.547
- [Voiceover] A student
learns that ionic compounds

00:00:02.547 --> 00:00:04.730
have significant covalent character

00:00:04.730 --> 00:00:09.304
when a cation has a polarizing
effect on a large anion.

00:00:09.304 --> 00:00:10.217
So what are they talking about?

00:00:10.217 --> 00:00:12.214
So if I have a cation,

00:00:12.214 --> 00:00:14.861
so this is my cation,

00:00:14.861 --> 00:00:17.903
and then this is my large anion,

00:00:17.903 --> 00:00:19.923
my large anion right over here.

00:00:19.923 --> 00:00:21.293
And actually, let me do it this way,

00:00:21.293 --> 00:00:23.453
let me make it a little
bit clearer 'cause a cation

00:00:23.453 --> 00:00:27.493
still has electrons, so
you have your nucleus here,

00:00:27.493 --> 00:00:31.429
of your cation, you have
your electron cloud,

00:00:31.429 --> 00:00:36.429
you have your electron cloud
of your cation right over here,

00:00:36.941 --> 00:00:40.239
but you have fewer electrons
than you have protons,

00:00:40.239 --> 00:00:42.723
so you're going to have a positive charge.

00:00:42.723 --> 00:00:45.904
And that's why we call it a cation.

00:00:45.904 --> 00:00:50.904
And the anion, it still
has a positive nucleus,

00:00:51.326 --> 00:00:52.329
and it has an electron cloud.

00:00:52.329 --> 00:00:57.329
I'm drawing a large anion here, a large,

00:00:58.124 --> 00:01:02.381
a large, a large anion right over here.

00:01:02.381 --> 00:01:05.291
That's an electron cloud,
but it has more electrons

00:01:05.291 --> 00:01:07.311
in the cloud than it has
protons in the nucleus,

00:01:07.311 --> 00:01:09.273
so it has a negative charge.

00:01:09.273 --> 00:01:11.003
That is the anion.

00:01:11.003 --> 00:01:12.857
Now what they're saying is you start

00:01:12.857 --> 00:01:14.761
to have significant covalent character,

00:01:14.761 --> 00:01:16.235
remember covelent bonds or electrons

00:01:16.235 --> 00:01:18.542
are essentially shared between two atoms

00:01:18.542 --> 00:01:21.804
when a cation has a polarizing
effect on a large anion.

00:01:21.804 --> 00:01:26.111
So one way to think
about is these electrons

00:01:26.111 --> 00:01:28.561
that are out here, well
they're gonna want to get away

00:01:28.561 --> 00:01:30.407
from all their negative friends here.

00:01:30.407 --> 00:01:31.823
Negative charges repel each other

00:01:31.823 --> 00:01:33.224
and they might want to spend more time

00:01:33.224 --> 00:01:35.558
closer to this cation.

00:01:35.558 --> 00:01:37.137
And so their gonna start
having covalent character,

00:01:37.137 --> 00:01:39.215
they're gonna be more and more shared

00:01:39.215 --> 00:01:41.839
between the two, and
there really is a spectrum

00:01:41.839 --> 00:01:45.775
between ionic bonds and covalent bonds.

00:01:45.775 --> 00:01:47.795
And over here, you might
have a polarizing effect.

00:01:47.795 --> 00:01:49.699
These things spend more
time on the left hand side,

00:01:49.699 --> 00:01:51.727
the way I've drawn it, than
on the right hand side.

00:01:51.727 --> 00:01:53.515
So you're gonna have more of the electrons

00:01:53.515 --> 00:01:56.034
and more of the negative
charge hanging out over here

00:01:56.034 --> 00:01:58.449
than on the right, that's
what they're talking about.

00:01:58.449 --> 00:02:01.092
And as a result, the student hypothesizes

00:02:01.092 --> 00:02:05.707
that salts composed of small
cations and large anions

00:02:05.707 --> 00:02:09.006
should have relatively low melting points.

00:02:09.006 --> 00:02:12.021
Alright, so it should be
easy to take less energy

00:02:12.021 --> 00:02:13.432
to break them out of their,

00:02:13.432 --> 00:02:14.861
kind of the salt lattice structure

00:02:14.861 --> 00:02:16.544
when they're in solid form.

00:02:16.544 --> 00:02:18.495
Select two compounds from the table,

00:02:18.495 --> 00:02:20.225
and explain how the data supports

00:02:20.225 --> 00:02:22.140
the student's hypothesis.

00:02:22.140 --> 00:02:23.719
Let's see, these are all salts here,

00:02:23.719 --> 00:02:27.729
and if these were to disassociate,

00:02:27.729 --> 00:02:30.828
or if you think about
what the cations would be,

00:02:30.828 --> 00:02:35.828
the cations, the cations that make up,

00:02:36.557 --> 00:02:40.000
that make up these compounds,

00:02:40.000 --> 00:02:43.298
you have lithium, lithium,

00:02:43.298 --> 00:02:48.298
you have sodium, and you have potassium.

00:02:49.346 --> 00:02:51.517
And then the anions,

00:02:51.517 --> 00:02:54.722
the anions that make up these compounds,

00:02:54.722 --> 00:02:59.722
you have iodide, you have
iodide and you have fluoride.

00:03:02.628 --> 00:03:04.156
Let me write it this way,

00:03:04.156 --> 00:03:07.500
flouride and iodide.

00:03:07.500 --> 00:03:10.437
So which of these are small
and which of these are big?

00:03:10.437 --> 00:03:15.093
Well, let's compare lithium,
sodium, and potassium.

00:03:15.093 --> 00:03:17.752
And we see lithium, sodium, and potassium,

00:03:17.752 --> 00:03:19.934
they're all group one elements.

00:03:19.934 --> 00:03:21.560
And in general, as we go down a group,

00:03:21.560 --> 00:03:23.371
as we go down a column
in the periodic table,

00:03:23.371 --> 00:03:27.132
we're adding shells,
and so size increases.

00:03:27.132 --> 00:03:30.395
So size increases as we go down like this.

00:03:30.395 --> 00:03:35.395
Size, size increases.

00:03:35.770 --> 00:03:39.726
And so of these, lithium is
the smallest and potassium

00:03:39.726 --> 00:03:42.396
is the largest of these three.

00:03:42.396 --> 00:03:44.729
So lithium, let me write that down,

00:03:44.729 --> 00:03:49.176
lithium is the smallest, smallest,

00:03:49.176 --> 00:03:52.752
and potassium is the largest of these,

00:03:52.752 --> 00:03:54.075
of these three.

00:03:54.075 --> 00:03:56.130
Now if you look at the anions,

00:03:56.130 --> 00:03:59.637
well you have once again,
fluorine and iodine

00:03:59.637 --> 00:04:02.330
are in the same, are in the same group,

00:04:02.330 --> 00:04:05.430
and iodine is below it,
it has way more electrons

00:04:05.430 --> 00:04:07.055
and you've added way more shells here

00:04:07.055 --> 00:04:08.677
is actually the important part.

00:04:08.677 --> 00:04:11.068
And so you can see that since iodine

00:04:11.068 --> 00:04:12.856
is larger than fluorine,

00:04:12.856 --> 00:04:16.548
well iodide, which is when
they gain an electron,

00:04:16.548 --> 00:04:19.393
it's gonna be larger than fluoride.

00:04:19.393 --> 00:04:24.393
So iodide, this one is larger,

00:04:25.430 --> 00:04:28.588
and the fluorine is smaller.

00:04:28.588 --> 00:04:33.588
So let's see, lithium-iodide
is a case of small cation,

00:04:34.776 --> 00:04:39.776
small cation and large, and large anion.

00:04:39.908 --> 00:04:43.216
And so that's actually the direction,

00:04:43.216 --> 00:04:46.177
that's kind of the best example of small,

00:04:46.177 --> 00:04:48.069
of what the student is talking about,

00:04:48.069 --> 00:04:50.914
small cation and large anion.

00:04:50.914 --> 00:04:55.914
Lithium iodide, lithium iodide
is a great example of that.

00:04:56.997 --> 00:04:59.737
And as we can see, it has
the lowest melting point

00:04:59.737 --> 00:05:01.456
of everyone on the table.

00:05:01.456 --> 00:05:02.733
Now let's take the other extreme.

00:05:02.733 --> 00:05:06.274
What happens if we take a large cation

00:05:06.274 --> 00:05:09.478
and a small anion.

00:05:09.478 --> 00:05:13.948
Well, let's see, they don't
have that combination here,

00:05:13.948 --> 00:05:17.036
but let's see, if let's see, sodium,

00:05:17.036 --> 00:05:22.036
well sodium fluoride is interesting

00:05:22.156 --> 00:05:24.490
because we have a larger,

00:05:24.490 --> 00:05:26.974
we have a larger cation,

00:05:26.974 --> 00:05:30.214
so this is a larger cation,

00:05:30.214 --> 00:05:33.070
and then we have a smaller anion,

00:05:33.070 --> 00:05:35.949
smaller anion.

00:05:35.949 --> 00:05:38.663
And notice, that has the highest,

00:05:38.663 --> 00:05:41.115
that has the highest melting point.

00:05:41.115 --> 00:05:43.495
And so these are two
that are good to compare.

00:05:43.495 --> 00:05:46.270
So part a, we'll do that in blue,

00:05:46.270 --> 00:05:51.270
compare lithium iodide, which
has small, small cation,

00:06:03.418 --> 00:06:08.418
plus large anion, to sodium fluoride

00:06:13.751 --> 00:06:18.751
which has larger cation
plus smaller anion,

00:06:23.770 --> 00:06:28.770
smaller, smaller anion.

00:06:28.972 --> 00:06:33.972
And we see, and we see,

00:06:34.753 --> 00:06:39.753
that their melting point's,
their melting point's,

00:06:40.396 --> 00:06:45.396
melting point's consistent
with hypothesis,

00:06:46.317 --> 00:06:51.317
consistent with hypothesis.

00:06:55.257 --> 00:07:00.257
The small cation, or lithium iodide,

00:07:03.107 --> 00:07:05.922
I'll just write melting point,

00:07:05.922 --> 00:07:10.922
melting point lower,

00:07:12.227 --> 00:07:14.409
lower, and actually much lower

00:07:14.409 --> 00:07:19.409
than sodium fluoride's melting point.

00:07:22.629 --> 00:07:23.987
So at least if you compare those two

00:07:23.987 --> 00:07:26.008
and those seem to be good extremes,

00:07:26.008 --> 00:07:27.228
if we took the lowest melting point,

00:07:27.228 --> 00:07:29.480
which completely typified
what the student was saying

00:07:29.480 --> 00:07:30.966
as having a low melting point,

00:07:30.966 --> 00:07:32.611
as we looked at the highest melting point

00:07:32.611 --> 00:07:35.873
which had a larger cation
and a smaller anion.

00:07:35.873 --> 00:07:36.989
And so just looking at that,

00:07:36.989 --> 00:07:40.481
it seems to be consistent
with the student's hypothesis.

00:07:40.481 --> 00:07:42.385
Alright, part B.

00:07:42.385 --> 00:07:44.382
Identify a compound from the table

00:07:44.382 --> 00:07:49.382
that can be dissolved in water
to produce a basic solution.

00:07:49.444 --> 00:07:53.020
Write the net ionic
equation for the reaction

00:07:53.020 --> 00:07:56.677
that occurs to cause the
solution to be basic.

00:07:56.677 --> 00:07:59.103
Alright, so let's think
about how we can form

00:07:59.103 --> 00:08:00.543
a basic solution.

00:08:00.543 --> 00:08:02.447
So essentially, we would need to

00:08:02.447 --> 00:08:06.046
nab some hydrogens from
the water molecules

00:08:06.046 --> 00:08:08.983
in order to have some
hydroxide laying around.

00:08:08.983 --> 00:08:10.945
And so there's a couple of candidates here

00:08:10.945 --> 00:08:12.083
that could do that.

00:08:12.083 --> 00:08:15.427
You have all of these halides,

00:08:15.427 --> 00:08:17.958
you have the iodide,
you have the fluoride,

00:08:17.958 --> 00:08:20.721
and the important thing to recognize

00:08:20.721 --> 00:08:24.785
is that hydrogen iodide
or hydrochloric acid,

00:08:24.785 --> 00:08:26.340
since we're dissolving it in water now,

00:08:26.340 --> 00:08:31.340
hydroiotic acid is a strong acid.

00:08:32.923 --> 00:08:37.923
So let's see, HI, hydroiotic
acid, strong acid,

00:08:41.213 --> 00:08:46.213
strong acid, while hydrofluoric acid

00:08:47.157 --> 00:08:52.157
is a weaker acid, is a weaker acid.

00:08:53.055 --> 00:08:57.002
So a strong acid is not going to be good

00:08:57.002 --> 00:08:59.022
at nabbing hydrogens.

00:08:59.022 --> 00:09:02.761
In fact, it wants to
give away its hyrdorogens

00:09:02.761 --> 00:09:05.404
really, really, really badly.

00:09:05.404 --> 00:09:10.161
So the iodine does not
seem like a good candidate.

00:09:10.161 --> 00:09:15.161
The fluoride does seem like
an interesting candidate.

00:09:15.347 --> 00:09:17.217
so let's take one of these,

00:09:17.217 --> 00:09:19.074
let's take one of these
candidates out here,

00:09:19.074 --> 00:09:20.858
you can use either the lithium fluoride

00:09:20.858 --> 00:09:22.088
or the sodium fluoride,

00:09:22.088 --> 00:09:23.465
let's just take, I know
we've already dealt

00:09:23.465 --> 00:09:26.403
with the sodium fluoride,
so let's keep using that.

00:09:26.403 --> 00:09:29.108
So if you take sodium fluoride,

00:09:29.108 --> 00:09:31.999
sodium fluoride, and so let me,

00:09:31.999 --> 00:09:35.958
since they say write
the net ionic equation

00:09:35.958 --> 00:09:37.130
for the reaction that occurs

00:09:37.130 --> 00:09:39.151
to cause the solution to be basic.

00:09:39.151 --> 00:09:42.076
So we're gonna focus on sodium fluoride,

00:09:42.076 --> 00:09:47.076
sodium, sodium fluoride is what we select.

00:09:47.916 --> 00:09:49.251
We select sodium fluoride,

00:09:49.251 --> 00:09:50.993
that's the first part.

00:09:50.993 --> 00:09:53.009
We identified a compound that
can be dissolved in water

00:09:53.009 --> 00:09:56.318
to produce a basic solution.

00:09:56.318 --> 00:09:58.605
We select that, and now let's draw,

00:09:58.605 --> 00:10:00.161
let's do first the ionic equation,

00:10:00.161 --> 00:10:01.949
then we'll do the net ionic equation.

00:10:01.949 --> 00:10:03.922
So if you dissolve this in water,

00:10:03.922 --> 00:10:08.346
you're going to have sodium cations

00:10:08.346 --> 00:10:10.145
dissolved in our aqueous solution,

00:10:10.145 --> 00:10:13.737
plus fluoride anions dissolved in our,

00:10:13.737 --> 00:10:18.737
dissolved in our aqueous solution,

00:10:19.326 --> 00:10:23.924
and it is going to be in equilibrium,

00:10:23.924 --> 00:10:28.641
in equilibrium with,
actually let me draw it,

00:10:28.641 --> 00:10:30.827
let me draw it this way,

00:10:30.827 --> 00:10:32.000
let me draw it this way,

00:10:32.000 --> 00:10:37.000
plus H2O, because that's what
it's going to react with,

00:10:37.828 --> 00:10:39.848
and obviously, I don't have to
say it's an aqueous solution,

00:10:39.848 --> 00:10:41.485
it is the aqueous solution,

00:10:41.485 --> 00:10:43.203
so it's making the solution aqueous.

00:10:43.203 --> 00:10:45.444
This is going to be in equilibrium with,

00:10:45.444 --> 00:10:46.596
and I'll go to the next row here.

00:10:46.596 --> 00:10:48.406
If I was actually taking the AP test,

00:10:48.406 --> 00:10:50.263
well actually, let me just,

00:10:50.263 --> 00:10:51.865
let me just copy and paste it here,

00:10:51.865 --> 00:10:55.766
just so we don't have to
deal with it on two rows.

00:10:55.766 --> 00:11:00.766
So put it right over there.

00:11:00.805 --> 00:11:04.451
So that is going to be in,

00:11:04.451 --> 00:11:09.164
that is going to be in equilibrium.

00:11:09.164 --> 00:11:10.883
Let me get the right tool out.

00:11:10.883 --> 00:11:13.669
That is going to be in equilibrium

00:11:13.669 --> 00:11:17.384
with this fluoride nabbing a hydrogen.

00:11:17.384 --> 00:11:22.384
So hydrofluoric acid, we could say HF

00:11:26.324 --> 00:11:31.324
in our aqueous solution, plus OH minus

00:11:31.664 --> 00:11:33.905
in our aqueous solution,

00:11:33.905 --> 00:11:38.340
and then you still have the sodium cation

00:11:38.340 --> 00:11:41.103
in our aqueous solution.

00:11:41.103 --> 00:11:43.983
And so this, as you can see
will perform at equilibrium,

00:11:43.983 --> 00:11:45.863
and so you're gonna have
more hydroxide around,

00:11:45.863 --> 00:11:47.090
so you're forming your basic,

00:11:47.090 --> 00:11:49.401
you're forming your basic solution.

00:11:49.401 --> 00:11:52.361
If this was an iodide right over here,

00:11:52.361 --> 00:11:55.589
then this reaction would go
strongly in that direction.

00:11:55.589 --> 00:11:57.864
If this was hydrogen
iodide right over here,

00:11:57.864 --> 00:12:00.244
this wouldn't be some
kind of nice equilibrium.

00:12:00.244 --> 00:12:03.263
This would be a strong
acid, it would definitely

00:12:03.263 --> 00:12:05.341
want to get rid of its hydrogens.

00:12:05.341 --> 00:12:07.501
So that's why we don't want
to use hydrogen iodide.

00:12:07.501 --> 00:12:09.091
We want to use hydrogen fluoride.

00:12:09.091 --> 00:12:10.922
When you put it in water,
it becomes hydrofluoric,

00:12:10.922 --> 00:12:14.184
it's likely to donate,

00:12:14.184 --> 00:12:16.406
when it's in its gassiest
form or it's by itself,

00:12:16.406 --> 00:12:18.205
we would call it a hydrogen fluoride,

00:12:18.205 --> 00:12:19.912
but now when we put it in a solution,

00:12:19.912 --> 00:12:22.548
we can consider it to
be hydrofluoric acid.

00:12:22.548 --> 00:12:25.009
And so if you want your
net ionic equation,

00:12:25.009 --> 00:12:26.623
well we don't have to worry too much

00:12:26.623 --> 00:12:29.247
about these sodiums that
are on both sides of this.

00:12:29.247 --> 00:12:31.081
And so if you just focus
on this right over here,

00:12:31.081 --> 00:12:36.081
this is your net ionic,
net ionic equation.


2015 AP Chemistry free response 5

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:02.162
- [Voiceover] Blue food
coloring can be oxidized

00:00:02.162 --> 00:00:05.807
by household bleach, which
contains hypochlorite,

00:00:05.807 --> 00:00:08.304
household bleach will usually
use sodium hypochlorite,

00:00:08.304 --> 00:00:10.463
to form colorless products,

00:00:10.463 --> 00:00:12.598
as represented by the equation above.

00:00:12.598 --> 00:00:13.923
So this is the food coloring,

00:00:13.923 --> 00:00:15.223
reacts with the hypochlorite,

00:00:15.223 --> 00:00:17.141
produces colorless a product.

00:00:17.141 --> 00:00:19.275
A student uses a spectrophotometer

00:00:19.275 --> 00:00:22.858
set at a wavelength of 635 nanometers

00:00:22.858 --> 00:00:25.523
to study the absorbance
of the food coloring

00:00:25.523 --> 00:00:28.622
over time during the bleaching process.

00:00:28.622 --> 00:00:30.249
So since we're talking
about blue food coloring,

00:00:30.249 --> 00:00:31.549
I'm guessing this is a wavelength

00:00:31.549 --> 00:00:34.519
of light that is blue, since
that's gonna be optimally

00:00:34.519 --> 00:00:36.512
absorbed by blue food coloring.

00:00:37.563 --> 00:00:40.548
In the study, bleach is
present in large excess

00:00:40.548 --> 00:00:42.625
so the concentration of hypochlorite

00:00:42.625 --> 00:00:45.708
is essentially constant
through the reaction.

00:00:45.708 --> 00:00:46.708
Alright.

00:00:46.708 --> 00:00:49.727
The student used data from the study

00:00:49.727 --> 00:00:51.504
to generate the graphs below.

00:00:51.504 --> 00:00:52.594
So we're graphing,

00:00:52.594 --> 00:00:54.730
see the vertical access we have absorbents

00:00:54.730 --> 00:00:56.193
and you can view absorbents,

00:00:56.193 --> 00:00:59.084
if we have a high concentration
of blue food coloring,

00:00:59.084 --> 00:01:01.093
then we're gonna have a high absorbance,

00:01:01.093 --> 00:01:03.705
and if we have a low concentration
of blue food coloring,

00:01:03.705 --> 00:01:05.086
we're gonna have a low absorbance.

00:01:05.086 --> 00:01:06.771
So you can view this as a,

00:01:07.851 --> 00:01:12.397
as a proxy for concentration
of food coloring,

00:01:12.397 --> 00:01:17.397
food coloring, food
coloring concentration.

00:01:17.809 --> 00:01:20.272
And so, here they just plotted absorbance

00:01:20.272 --> 00:01:21.234
relative to time,

00:01:21.234 --> 00:01:22.361
here they have the natural log

00:01:22.361 --> 00:01:23.614
of absorbance relative to time.

00:01:23.614 --> 00:01:28.614
Here, one over absorbance
relative to time.

00:01:28.630 --> 00:01:31.010
And so let's look at the questions here.

00:01:31.010 --> 00:01:32.494
Based on the graphs above,

00:01:32.494 --> 00:01:34.564
what is the order of the reaction

00:01:34.564 --> 00:01:37.337
with respect to the blue food coloring?

00:01:37.337 --> 00:01:39.694
With the respect to the
blue food blue coloring.

00:01:39.694 --> 00:01:40.480
So let's think of,

00:01:40.480 --> 00:01:42.214
I'll do a little super fast primer,

00:01:42.214 --> 00:01:44.453
so if we're talking about a zero,

00:01:44.453 --> 00:01:46.834
if we're talking about
a zero order reaction

00:01:46.834 --> 00:01:50.952
that means that the rate
of reaction is constant.

00:01:50.952 --> 00:01:52.923
Rate, constant.

00:01:54.021 --> 00:01:57.655
And it's independent of, the independent

00:01:57.655 --> 00:02:00.755
of the concentration
of blue food coloring.

00:02:00.755 --> 00:02:02.068
Independent

00:02:06.018 --> 00:02:07.838
of the concentration,

00:02:07.838 --> 00:02:09.769
I'll just say of the coloring.

00:02:09.769 --> 00:02:11.471
Concentration of the coloring.

00:02:11.471 --> 00:02:13.107
Is that the case here?

00:02:13.107 --> 00:02:14.536
Well no, the rate isn't constant,

00:02:14.536 --> 00:02:15.778
if we look at just absorbance,

00:02:15.778 --> 00:02:16.732
which is just once again,

00:02:16.732 --> 00:02:19.521
a proxy for our concentration
of food coloring,

00:02:19.521 --> 00:02:20.898
up here our rate is pretty fast,

00:02:20.898 --> 00:02:22.895
we have a steep slope over here

00:02:22.895 --> 00:02:27.040
and then the slope gets
less and less steep

00:02:27.040 --> 00:02:30.387
as more as our concentration

00:02:30.387 --> 00:02:32.021
of food coloring goes down,

00:02:32.021 --> 00:02:33.681
as the reaction proceeds.

00:02:33.681 --> 00:02:36.490
So this is definitely not
a zero order reaction,

00:02:36.490 --> 00:02:38.282
if this was a zero order reaction

00:02:38.282 --> 00:02:39.857
when we just plot absorbance,

00:02:39.857 --> 00:02:40.972
which is once again the proxy

00:02:40.972 --> 00:02:43.547
for concentration of food
coloring versus time,

00:02:43.547 --> 00:02:46.574
we would expect to see
something more of a line.

00:02:46.574 --> 00:02:49.250
So, if you saw something like that,

00:02:49.250 --> 00:02:50.887
then you would say "okay,

00:02:50.887 --> 00:02:53.354
"that looks like a zero order reaction."

00:02:53.354 --> 00:02:55.240
Now when we took the
natural log of absorbance,

00:02:55.240 --> 00:02:57.771
which is once again a
proxy for the natural log

00:02:57.771 --> 00:02:59.513
of the concentration of food coloring,

00:02:59.513 --> 00:03:01.880
here we get a clear line,

00:03:01.880 --> 00:03:04.772
here we actually do get a clear line.

00:03:04.772 --> 00:03:05.794
And I'm not gonna go into it,

00:03:05.794 --> 00:03:06.839
it takes a little bit of calculus

00:03:06.839 --> 00:03:08.988
and a little bit of basic
differential equations

00:03:08.988 --> 00:03:13.654
to realize it, but this is a
give away for a first order,

00:03:13.654 --> 00:03:15.915
for a first order reaction.

00:03:15.915 --> 00:03:17.500
So in a first order of reaction,

00:03:18.410 --> 00:03:19.875
in first order,

00:03:20.740 --> 00:03:25.348
the rate, the rate is proportional,

00:03:25.348 --> 00:03:26.400
proportional,

00:03:28.147 --> 00:03:31.718
is proportional to the concentration,

00:03:32.768 --> 00:03:35.184
concentration, let me just write it,

00:03:35.184 --> 00:03:37.110
is proportional to the concentration

00:03:37.110 --> 00:03:38.061
since we're saying with respect

00:03:38.061 --> 00:03:39.441
to the blue food coloring,

00:03:39.441 --> 00:03:40.663
is proportional to the concentration

00:03:40.663 --> 00:03:42.345
of blue food coloring.

00:03:42.345 --> 00:03:46.541
I'll just write coloring,
coloring for short.

00:03:46.541 --> 00:03:48.256
And I'll throw a little calculus here,

00:03:48.256 --> 00:03:50.140
you could say the rate of reaction,

00:03:50.140 --> 00:03:51.403
which is the rate in change

00:03:51.403 --> 00:03:53.101
of concentration of our coloring,

00:03:54.976 --> 00:03:56.811
with respect to time,

00:03:56.811 --> 00:03:58.889
and if this looks
completely unfamiliar to you

00:03:58.889 --> 00:04:00.119
and you've never taken a calculus class

00:04:00.119 --> 00:04:03.679
ignore what I'm about to
say for the next 20 seconds.

00:04:03.679 --> 00:04:04.915
This needs to be proportional

00:04:04.915 --> 00:04:08.377
to the concentration of coloring.

00:04:08.377 --> 00:04:12.406
Coloring, alright, COL dot for short.

00:04:12.406 --> 00:04:14.688
And if you saw this, you would see

00:04:14.688 --> 00:04:17.507
that the natural log of the
concentration of coloring

00:04:17.507 --> 00:04:20.518
with respect to, if you
plot that versus time,

00:04:20.518 --> 00:04:21.947
is you're going to get a line.

00:04:21.947 --> 00:04:23.063
So this is a key,

00:04:23.063 --> 00:04:27.256
this is a key signature of a first order,

00:04:27.256 --> 00:04:28.240
first order reaction.

00:04:28.240 --> 00:04:29.401
But you can even see it here,

00:04:29.401 --> 00:04:31.362
up here when the concentration

00:04:31.362 --> 00:04:33.941
of our coloring is high, our rate is high,

00:04:33.941 --> 00:04:35.090
we have a steep slope

00:04:35.090 --> 00:04:37.470
and then when our
concentration becomes lower,

00:04:37.470 --> 00:04:39.722
we also have our slope being lower.

00:04:39.722 --> 00:04:40.988
So you actually don't even need calculus,

00:04:40.988 --> 00:04:42.021
you could look at this one and see

00:04:42.021 --> 00:04:44.205
that something very similar
to that is happening.

00:04:44.205 --> 00:04:46.181
So this is a first order reaction.

00:04:46.181 --> 00:04:47.583
If you're thinking about second order,

00:04:47.583 --> 00:04:48.790
why do they even show us this?

00:04:48.790 --> 00:04:51.590
Well a second order if you
plot one over absorbance

00:04:51.590 --> 00:04:53.248
versus time, or one
over the concentration,

00:04:53.248 --> 00:04:56.098
cause' as we said absorbance is a proxy

00:04:56.098 --> 00:04:58.101
for the concentration
of our food coloring,

00:04:58.101 --> 00:05:00.759
well then this would be a linear plot,

00:05:00.759 --> 00:05:02.090
but as we can see it is not.

00:05:02.090 --> 00:05:03.650
But if this linear
plot, then you could say

00:05:03.650 --> 00:05:05.183
"hey maybe this is a second order,"

00:05:05.183 --> 00:05:06.425
but just to answer the question,

00:05:06.425 --> 00:05:07.913
this is a first order reaction

00:05:08.833 --> 00:05:11.699
with respect to blue food coloring.

00:05:11.699 --> 00:05:14.459
Alright, let's do part B now.

00:05:14.459 --> 00:05:16.529
The reaction is known to be first order

00:05:16.529 --> 00:05:18.404
with respect to bleach.

00:05:18.404 --> 00:05:19.561
Alright, so now we're talking about

00:05:19.561 --> 00:05:20.601
the reaction order with respect

00:05:20.601 --> 00:05:22.366
to bleach, not the food coloring.

00:05:22.366 --> 00:05:24.026
In a second experiment,

00:05:24.026 --> 00:05:25.500
the student prepares solutions

00:05:25.500 --> 00:05:28.009
of food coloring and
bleach with concentrations

00:05:28.009 --> 00:05:30.922
that differ from those used
in the first experiment.

00:05:30.922 --> 00:05:31.688
Alright.

00:05:31.688 --> 00:05:33.697
When the solutions are combined,

00:05:33.697 --> 00:05:35.945
the student observes
that the reaction mixture

00:05:35.945 --> 00:05:39.860
reaches an absorbance
near zero too rapidly.

00:05:39.860 --> 00:05:42.970
So it's getting to no color too fast.

00:05:42.970 --> 00:05:44.529
In order to correct the problem,

00:05:44.529 --> 00:05:45.887
the student proposes the following

00:05:45.887 --> 00:05:48.465
three possible modifications
to the experiment.

00:05:48.465 --> 00:05:50.295
So the solution doesn't want,

00:05:51.355 --> 00:05:53.490
the student does not want the solution

00:05:53.490 --> 00:05:56.023
to become colorless that fast,

00:05:56.023 --> 00:05:57.440
so what should they do?

00:05:57.440 --> 00:05:59.618
Should they increase the temperature?

00:05:59.618 --> 00:06:00.644
Well increasing the temperature

00:06:00.644 --> 00:06:02.827
is just going to make the
reaction happen even faster,

00:06:02.827 --> 00:06:04.801
the molecules are gonna
bump into each other

00:06:04.801 --> 00:06:07.399
with more energy and more frequently,

00:06:07.399 --> 00:06:08.990
and so, that's just gonna make,

00:06:08.990 --> 00:06:11.104
that's gonna get you to
colorless even faster.

00:06:11.104 --> 00:06:12.706
So that's gonna go in
the opposite direction,

00:06:12.706 --> 00:06:14.182
so we can rule that out.

00:06:14.182 --> 00:06:16.811
Increasing the concentration
of blue food coloring.

00:06:17.688 --> 00:06:19.661
Well, that makes sense,

00:06:19.661 --> 00:06:20.729
well they say blue food coloring

00:06:20.729 --> 00:06:23.168
but I'm assuming it's blue, whatever.

00:06:23.168 --> 00:06:25.002
Because, well if it's
getting clear too fast,

00:06:25.002 --> 00:06:27.149
well, if you add more food coloring

00:06:27.149 --> 00:06:30.168
well then it's just gonna,
it's going to have a higher

00:06:30.168 --> 00:06:31.840
absorbance and it's going to take longer

00:06:31.840 --> 00:06:32.943
to get to clear.

00:06:32.943 --> 00:06:34.871
So, this one seems interesting.

00:06:34.871 --> 00:06:35.590
Now what about this,

00:06:35.590 --> 00:06:37.784
increasing the concentration of bleach?

00:06:37.784 --> 00:06:39.839
Well once again the bleach is the thing

00:06:39.839 --> 00:06:41.244
that's getting the food,

00:06:41.244 --> 00:06:43.427
is reacting with the food
coloring to make it clear,

00:06:43.427 --> 00:06:44.707
so if you increase its concentration,

00:06:44.707 --> 00:06:46.481
you're gonna get clear even faster

00:06:46.481 --> 00:06:47.780
which is not what the student wants,

00:06:47.780 --> 00:06:49.359
this is the opposite of
what the student wants.

00:06:49.359 --> 00:06:51.484
So once again we would cross that one out.

00:06:51.484 --> 00:06:53.161
And the one that we like is definitely

00:06:53.161 --> 00:06:57.770
increasing the concentration
of the food coloring.

00:06:57.770 --> 00:07:00.110
They say circle the one
proposed modification,

00:07:00.110 --> 00:07:02.220
so let me make sure I'm circling it,

00:07:02.220 --> 00:07:04.658
I guess I'm more rectangling
it, but you get the idea.

00:07:04.658 --> 00:07:05.897
Correct the problem and explain

00:07:05.897 --> 00:07:07.641
how that modification increases the time

00:07:07.641 --> 00:07:09.262
for the reaction mixture

00:07:09.262 --> 00:07:12.762
to reach an absorbance near zero.

00:07:12.762 --> 00:07:14.344
So I'll write, more coloring,

00:07:15.502 --> 00:07:16.856
more coloring,

00:07:18.476 --> 00:07:21.750
results in

00:07:21.750 --> 00:07:24.850
higher initial absorbance,

00:07:24.850 --> 00:07:26.825
higher initial,

00:07:28.445 --> 00:07:32.067
initial absorbance,

00:07:33.907 --> 00:07:35.132
and thus,

00:07:36.452 --> 00:07:37.532
and thus,

00:07:39.092 --> 00:07:40.242
more time,

00:07:41.582 --> 00:07:43.062
more time,

00:07:44.222 --> 00:07:45.892
for mixture

00:07:48.232 --> 00:07:49.654
to reach,

00:07:49.654 --> 00:07:53.047
to reach near

00:07:53.887 --> 00:07:57.297
zero absorbance.

00:07:57.297 --> 00:08:01.592
Absorbance, the word that I
really have trouble saying.

00:08:01.592 --> 00:08:03.183
Alright, part C.

00:08:03.183 --> 00:08:05.460
In another experiment, a student wishes

00:08:05.460 --> 00:08:09.022
to study the oxidation
of red food coloring,

00:08:09.022 --> 00:08:11.762
just in the spirit of that one
I'll underline it with red,

00:08:11.762 --> 00:08:14.113
of red food coloring with bleach.

00:08:14.113 --> 00:08:15.876
How would the student need to modify

00:08:15.876 --> 00:08:17.766
the original experiment procedure,

00:08:17.766 --> 00:08:19.066
experimental procedure,

00:08:19.066 --> 00:08:20.992
to determine the order of the reaction

00:08:20.992 --> 00:08:24.568
with respect to the red food coloring?

00:08:24.568 --> 00:08:27.537
Well overall, this is a
pretty good experiment,

00:08:27.537 --> 00:08:29.711
they plotted it in three different ways,

00:08:29.711 --> 00:08:33.063
which was, as we saw,
a very good indicator

00:08:33.063 --> 00:08:35.528
of what order of reaction
we were talking about.

00:08:35.528 --> 00:08:37.256
But at the very beginning
of this question,

00:08:38.136 --> 00:08:40.417
I talked a little bit about
this wavelength of light,

00:08:40.417 --> 00:08:41.628
this is blue light,

00:08:41.628 --> 00:08:43.509
and even if you didn't know that off hand,

00:08:43.509 --> 00:08:44.973
you would be able to
say "well we're studying

00:08:44.973 --> 00:08:46.320
"blue food coloring, they probably picked

00:08:46.320 --> 00:08:49.420
"a wavelength of light that
gets absorbed by blue."

00:08:49.420 --> 00:08:51.610
But if we now care about red,

00:08:51.610 --> 00:08:52.810
well we would probably want to use

00:08:52.810 --> 00:08:54.550
a wavelength of light that is optimally

00:08:54.550 --> 00:08:55.874
absorbed by red,

00:08:55.874 --> 00:08:58.033
so a red wavelength of light,

00:08:58.033 --> 00:09:00.448
which will be a lower wavelength of light.

00:09:00.448 --> 00:09:04.541
So, change the wavelength of light,

00:09:05.452 --> 00:09:09.324
change the wavelength

00:09:12.059 --> 00:09:14.340
to be suitable,

00:09:16.970 --> 00:09:18.610
suitable

00:09:18.610 --> 00:09:23.018
for absorbance, or absorption

00:09:24.468 --> 00:09:28.239
by red coloring, red coloring.

00:09:28.239 --> 00:09:30.264
Or you could say you could
lower the wavelength of light

00:09:30.264 --> 00:09:31.524
or the wavelength of light

00:09:31.524 --> 00:09:33.177
should be red part of the spectrum

00:09:33.177 --> 00:09:35.923
to match the red food coloring.

00:09:35.923 --> 00:09:38.089
Everything else seems
completely reasonable.