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PROFESSOR: Hi, I'm Sal.

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00:00:22,890 --> 00:00:28,340
Today we will be doing problem
number 4 of fall 2009, exam 1.

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00:00:28,340 --> 00:00:31,230
Now before you attempt the
problems there's certain

11
00:00:31,230 --> 00:00:33,520
background information that
you should know before

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00:00:33,520 --> 00:00:34,040
attempting it.

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00:00:34,040 --> 00:00:37,320
One is knowledge in
ionic bonding.

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00:00:37,320 --> 00:00:40,700
Because the problem
deals with ions.

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00:00:40,700 --> 00:00:42,815
Two, how to draw an energy-level
diagram.

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00:00:42,815 --> 00:00:44,590
That's very important.

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00:00:44,590 --> 00:00:48,010
And three, what Madelung's
Constant is.

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00:00:48,010 --> 00:00:51,920
Now before attempting the
problem, it's a good thing to

19
00:00:51,920 --> 00:00:54,330
read the whole problem in detail
and make sure you don't

20
00:00:54,330 --> 00:00:55,930
skip anything out of it.

21
00:00:55,930 --> 00:00:57,170
So that you get all
the information

22
00:00:57,170 --> 00:00:58,130
that's given to you.

23
00:00:58,130 --> 00:01:00,100
Because that's very important.

24
00:01:00,100 --> 00:01:02,120
So the problem reads
as follows.

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00:01:02,120 --> 00:01:06,140
For a given cation, c and
anion a, show that the

26
00:01:06,140 --> 00:01:08,130
following four energy
states on the same

27
00:01:08,130 --> 00:01:09,470
energy-level diagram.

28
00:01:09,470 --> 00:01:12,850
1, Ions in infinite
separation.

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00:01:12,850 --> 00:01:18,900
2, 9-pair c and a, c stands for
cation, a stands anion.

30
00:01:18,900 --> 00:01:24,080
3, An ion line, c a c a c a,
just a repetitive line of

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00:01:24,080 --> 00:01:25,720
anions and cations.

32
00:01:25,720 --> 00:01:28,220
And 4, crystalline
solid of c a.

33
00:01:28,220 --> 00:01:32,460
So a three-dimensional crystal
structure with t and a.

34
00:01:32,460 --> 00:01:35,330
And it says also that, assume
that the comparison is based

35
00:01:35,330 --> 00:01:38,450
upon identical numbers of
ions in all four states.

36
00:01:38,450 --> 00:01:41,410
The diagram need not to be drawn
to scale, however you

37
00:01:41,410 --> 00:01:44,740
must convey relative values of
different energy states.

38
00:01:44,740 --> 00:01:46,470
So that's the problem reads.

39
00:01:46,470 --> 00:01:51,550
Now if you recall from lecture
there's an important equation

40
00:01:51,550 --> 00:01:56,280
that actually describes the
energy between two ions,

41
00:01:56,280 --> 00:01:58,470
between a cation and an anion.

42
00:01:58,470 --> 00:02:01,380
And this energy is directly
proportional to the product of

43
00:02:01,380 --> 00:02:02,750
the two charges.

44
00:02:02,750 --> 00:02:04,970
And it's inversely proportional
to the separation

45
00:02:04,970 --> 00:02:06,860
distance between them.

46
00:02:06,860 --> 00:02:10,060
And it's multiplied
by some constants.

47
00:02:10,060 --> 00:02:12,260
So what does this mean?

48
00:02:12,260 --> 00:02:17,380
Well the fact that it's directly
proportional to the

49
00:02:17,380 --> 00:02:21,800
product of your two charges
means that your energy for any

50
00:02:21,800 --> 00:02:25,820
ion or anything that
ionically bonds,

51
00:02:25,820 --> 00:02:27,060
it's going to be negative.

52
00:02:27,060 --> 00:02:30,290
Because your z minus will be
negative and that's the whole

53
00:02:30,290 --> 00:02:33,020
value of your energy
will be negative.

54
00:02:33,020 --> 00:02:34,980
And that's what dictates
stability.

55
00:02:34,980 --> 00:02:37,100
How negative the energy is.

56
00:02:37,100 --> 00:02:38,910
So with that in mind,
we can go ahead

57
00:02:38,910 --> 00:02:40,990
and start the problem.

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00:02:40,990 --> 00:02:44,520
So we have a cation
a and anion c.

59
00:02:44,520 --> 00:02:47,200
Now I like to draw diagrams,
or draw little pictures

60
00:02:47,200 --> 00:02:50,160
because that helps me when
I'm solving the problem.

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00:02:50,160 --> 00:02:51,620
So I'm just going to
draw pictures of a

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00:02:51,620 --> 00:02:52,680
cation and an anion.

63
00:02:52,680 --> 00:03:01,250
So I have a cation, c and
then I have an anion, a.

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00:03:01,250 --> 00:03:08,680
So c and a, and the value of r
naught from our equation is

65
00:03:08,680 --> 00:03:11,350
pretty much the separation
distance between the two.

66
00:03:11,350 --> 00:03:18,010
So project this up,
this is r naught.

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00:03:18,010 --> 00:03:22,170
And if you look from the
picture, r naught is just

68
00:03:22,170 --> 00:03:24,900
simply the radius of your
cation plus the

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00:03:24,900 --> 00:03:28,140
radius of your anion.

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00:03:28,140 --> 00:03:34,530
Now this is already describing
the energy between just a

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00:03:34,530 --> 00:03:37,930
cation and an anion with these
magnitude of charges.

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00:03:37,930 --> 00:03:44,240
So all that tells me is that,
this is already answering

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00:03:44,240 --> 00:03:47,490
number 2, pretty much from the
problem that we're asked for,

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00:03:47,490 --> 00:03:48,980
which is the ion pair.

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00:03:48,980 --> 00:03:52,290
So the problem asked you to draw
an energy-level diagram.

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00:03:52,290 --> 00:03:53,310
That's what I'm going
to do over here.

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00:03:53,310 --> 00:03:54,540
So I'm going to go ahead
and start drawing the

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00:03:54,540 --> 00:03:56,010
energy-level diagram.

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00:03:56,010 --> 00:03:59,730
Now I'm going to go ahead and
label my energy axis to be

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00:03:59,730 --> 00:04:02,740
positive to be up.

81
00:04:02,740 --> 00:04:05,790
And I'm going to draw a
baseline of 0 here.

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00:04:05,790 --> 00:04:12,880
So I know that to be able to
incorporate my ion pair--

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00:04:12,880 --> 00:04:15,880
like I said the energy is
negative, so if the energy

84
00:04:15,880 --> 00:04:17,965
goes up to here and let's say
the value of the energy here

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00:04:17,965 --> 00:04:22,810
is 0, then that ion pair should
lie somewhere below it.

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00:04:22,810 --> 00:04:25,100
So I'm going to go ahead
and draw this line.

87
00:04:25,100 --> 00:04:30,480
And I'll label this, s 2 for
our energy level diagram.

88
00:04:30,480 --> 00:04:31,730
So this is the ion pair.

89
00:04:37,490 --> 00:04:41,270
Now number 1 says ions at
infinite separation.

90
00:04:41,270 --> 00:04:44,190
What is the energy of two
charged species that are

91
00:04:44,190 --> 00:04:45,480
separated by infinity?

92
00:04:45,480 --> 00:04:47,940
Well if you look at your
equation, your equation tells

93
00:04:47,940 --> 00:04:49,890
you that it's inversely
proportional to

94
00:04:49,890 --> 00:04:51,120
the separation distance.

95
00:04:51,120 --> 00:04:54,500
So all that tells you is that
if you divide by infinity,

96
00:04:54,500 --> 00:04:56,260
your energy should
be essentially 0.

97
00:04:56,260 --> 00:04:59,330
Which makes physical sense
because those two charged

98
00:04:59,330 --> 00:05:02,500
species can't feel each other
when they're separated.

99
00:05:02,500 --> 00:05:08,870
So this is actually already
my number 1, which is 0.

100
00:05:08,870 --> 00:05:10,360
Separated away.

101
00:05:10,360 --> 00:05:13,820
So now we need to find what the
energy is of an ion pair.

102
00:05:13,820 --> 00:05:15,440
Which is the line of c a.

103
00:05:15,440 --> 00:05:18,880
So we want to draw
a little picture.

104
00:05:18,880 --> 00:05:21,060
It's pretty much going to
be a repetition of that.

105
00:05:21,060 --> 00:05:24,700
And if I draw a line--

106
00:05:24,700 --> 00:05:27,990
that's not very straight--

107
00:05:27,990 --> 00:05:31,170
I'm going to draw a bunch
of cations and anions.

108
00:05:35,930 --> 00:05:37,830
Essentially I can just imagine
that it extends to

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00:05:37,830 --> 00:05:39,840
infinity in both ways.

110
00:05:39,840 --> 00:05:45,120
So with my energy between these
two ions right here is

111
00:05:45,120 --> 00:05:49,500
given by my equation, then I
should be able to figure out

112
00:05:49,500 --> 00:05:51,090
what the energy is of
the whole system.

113
00:05:51,090 --> 00:05:53,650
Because we're talking about
electrostatic interaction,

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00:05:53,650 --> 00:05:55,900
which you can just
add linearly.

115
00:05:55,900 --> 00:05:58,240
So I can make one assumption.

116
00:05:58,240 --> 00:06:01,920
But look at my cation and anion
pair, I can go ahead and

117
00:06:01,920 --> 00:06:05,200
simplify my life by letting
z plus be 1.

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00:06:05,200 --> 00:06:06,430
So I'll write that down.

119
00:06:06,430 --> 00:06:14,560
I'll let z plus equal to
plus 1, which is the

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00:06:14,560 --> 00:06:17,510
charge on our cation.

121
00:06:17,510 --> 00:06:22,100
And I'll let z minus equal minus
1, which is the charge

122
00:06:22,100 --> 00:06:23,650
on the anion.

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00:06:23,650 --> 00:06:26,350
So that's to help
with the math.

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00:06:26,350 --> 00:06:29,690
So if I come back over here and
if I look at my equation

125
00:06:29,690 --> 00:06:32,230
line then I know that
if I draw--

126
00:06:32,230 --> 00:06:35,130
let's see-- a reference
access here.

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00:06:35,130 --> 00:06:37,270
Then the separation
between the cation

128
00:06:37,270 --> 00:06:39,165
and anion is r naught.

129
00:06:39,165 --> 00:06:41,160
And between the cation and
the other cation is

130
00:06:41,160 --> 00:06:42,700
going to be 2 r naught.

131
00:06:42,700 --> 00:06:44,110
And it's going to
extend for that.

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00:06:44,110 --> 00:06:49,100
So if I go ahead and relabel my
energy equation for a line.

133
00:06:49,100 --> 00:06:57,120
I'll go ahead and call it e line
of function of r naught.

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00:06:57,120 --> 00:07:00,050
It's simply going to be the
first one, which is going to

135
00:07:00,050 --> 00:07:08,420
be negative e squared over 4 pi
epsilon naught, r naught,

136
00:07:08,420 --> 00:07:12,310
multiply the 1 over
minus 1 over n.

137
00:07:12,310 --> 00:07:18,040
So that's the energy between
these two ions.

138
00:07:18,040 --> 00:07:23,030
So now because the energy is
electrostatic, this cation

139
00:07:23,030 --> 00:07:26,310
also feels the repulsion
of this other cation.

140
00:07:26,310 --> 00:07:31,690
So therefore with that you
have got to add what that

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00:07:31,690 --> 00:07:32,490
interaction is.

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00:07:32,490 --> 00:07:35,470
So the product of the two
charges is plus 1, which gives

143
00:07:35,470 --> 00:07:39,870
it a plus in front
of your equation.

144
00:07:39,870 --> 00:07:46,040
And I end up getting e squared
over 4 pi epsilon naught.

145
00:07:46,040 --> 00:07:47,110
Now here's something
that you should

146
00:07:47,110 --> 00:07:49,660
pay particular attention.

147
00:07:49,660 --> 00:07:54,030
The separation between these
two cation is not r naught.

148
00:07:54,030 --> 00:07:55,610
It's actually 2 r naught.

149
00:07:55,610 --> 00:07:58,950
Because of where exactly where
it sits on the line.

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00:07:58,950 --> 00:08:04,620
So my separation distance now
becomes 2 r naught and it's

151
00:08:04,620 --> 00:08:08,860
multiplied by the same 1
minus 1 over n factor.

152
00:08:08,860 --> 00:08:12,390
Then if you repeat it again,
you're analyzing the

153
00:08:12,390 --> 00:08:14,800
interaction energy between the
cation and the other anion

154
00:08:14,800 --> 00:08:17,640
that's next on the line.

155
00:08:17,640 --> 00:08:22,880
So as you can see you end
up getting this nice--

156
00:08:22,880 --> 00:08:29,480
4 pi epsilon naught, 3 r naught,
1 minus 1 over n.

157
00:08:29,480 --> 00:08:34,550
And essentially it repeats
until it ends.

158
00:08:34,550 --> 00:08:39,290
Now I'm just analyzing
on the right side.

159
00:08:39,290 --> 00:08:42,130
Because it extends, it's an
infinite line, because it

160
00:08:42,130 --> 00:08:45,350
extends from negative infinity
to infinity, then you can also

161
00:08:45,350 --> 00:08:47,760
assume that there's an
anion to the left.

162
00:08:47,760 --> 00:08:52,330
So all you really have to do
is, you take your overall

163
00:08:52,330 --> 00:08:55,730
energy for the right side and
you multiply it by 2.

164
00:08:55,730 --> 00:08:57,300
Because it's additive.

165
00:08:57,300 --> 00:09:03,200
So this whole thing gets
multiplied by 2.

166
00:09:03,200 --> 00:09:07,210
And that's essentially what
the energy is of a line.

167
00:09:07,210 --> 00:09:11,550
Now in order to be able to
know where it lies on the

168
00:09:11,550 --> 00:09:14,270
energy-level diagram, it's
important to try to get it

169
00:09:14,270 --> 00:09:17,640
into the same basic form as
the energy of an ion pair,

170
00:09:17,640 --> 00:09:20,530
just something simple
like over there.

171
00:09:20,530 --> 00:09:24,330
So just by looking at the
equation I know there's a lot

172
00:09:24,330 --> 00:09:26,090
of factors here that
are common.

173
00:09:26,090 --> 00:09:30,590
Like that e squared, 4 pi
epsilon naught, even r naught

174
00:09:30,590 --> 00:09:32,920
and this factor can all come
out of the equation.

175
00:09:32,920 --> 00:09:34,290
You can take it out.

176
00:09:34,290 --> 00:09:40,510
So if I do that, my equation of
my line then simply becomes

177
00:09:40,510 --> 00:09:48,160
negative e squared over
4 pi epsilon naught.

178
00:09:48,160 --> 00:09:52,720
Now I'm going to go ahead and
take out the r naught as well.

179
00:09:52,720 --> 00:09:56,830
And I'm going to multiply
it by the factor.

180
00:09:56,830 --> 00:09:59,950
And essentially what
stays from here is

181
00:09:59,950 --> 00:10:02,570
just your 1 over 1.

182
00:10:02,570 --> 00:10:04,970
From this one it's your 1/2
and then your 1/3 and then

183
00:10:04,970 --> 00:10:05,670
we're happy.

184
00:10:05,670 --> 00:10:07,650
And because I also took out the
negative, I want to make

185
00:10:07,650 --> 00:10:09,130
sure that I take out the
negative from all

186
00:10:09,130 --> 00:10:10,470
the other ones too.

187
00:10:10,470 --> 00:10:17,420
So this product then gets
multiplied by 1 minus 1 over

188
00:10:17,420 --> 00:10:23,540
2, plus 1/3 minus 1/4 plus...

189
00:10:23,540 --> 00:10:25,930
and it goes on forever.

190
00:10:25,930 --> 00:10:31,760
Now it so happens that this is
actually a series that we have

191
00:10:31,760 --> 00:10:34,490
an answer or an answer to.

192
00:10:34,490 --> 00:10:38,960
And this product that's being
multiplied by your fundamental

193
00:10:38,960 --> 00:10:42,440
base of your equation
is just simply--

194
00:10:42,440 --> 00:10:45,590
oh sorry I forgot the times 2,
don't forget that, that's very

195
00:10:45,590 --> 00:10:49,510
important because you're summing
from both sides--

196
00:10:49,510 --> 00:10:56,290
this essentially right here it's
simply arised because of

197
00:10:56,290 --> 00:10:57,260
your geometry.

198
00:10:57,260 --> 00:11:00,950
You're analyzing your systems.
And this value happens to be

199
00:11:00,950 --> 00:11:03,470
greater than 1, which
is a good thing.

200
00:11:03,470 --> 00:11:10,280
And the value of this
is actually 1.386.

201
00:11:10,280 --> 00:11:14,720
So the fact that the whole
equation is still negative,

202
00:11:14,720 --> 00:11:18,470
because we're talking about
coulombic interaction between

203
00:11:18,470 --> 00:11:22,860
a negative charged species and
a positive charged species.

204
00:11:22,860 --> 00:11:26,950
And we're multiplying that by a
value that is greater than 1

205
00:11:26,950 --> 00:11:31,570
this means that if this is 1
right here, then our energy

206
00:11:31,570 --> 00:11:34,410
level diagram should
lie somewhere below

207
00:11:34,410 --> 00:11:35,660
it, like right here.

208
00:11:38,740 --> 00:11:40,831
So this is 3.

209
00:11:40,831 --> 00:11:42,350
And that's your ion line.

210
00:11:45,360 --> 00:11:48,280
Now in real life we know
that crystals don't

211
00:11:48,280 --> 00:11:49,660
exist in lines really.

212
00:11:49,660 --> 00:11:52,800
Or they're not just an ion pair.
it's literally something

213
00:11:52,800 --> 00:11:56,850
that is three-dimensional and
has an ordered structure.

214
00:11:56,850 --> 00:12:00,700
Sometimes could be different
geometries, but normally it's

215
00:12:00,700 --> 00:12:02,860
a cubic structure.

216
00:12:02,860 --> 00:12:07,810
So because that forms in nature,
then that tells me

217
00:12:07,810 --> 00:12:11,610
that the energy of a
three-dimensional crystal,

218
00:12:11,610 --> 00:12:13,680
which is the fourth one that
we need to put on the

219
00:12:13,680 --> 00:12:15,980
energy-level diagram,
the value should be

220
00:12:15,980 --> 00:12:19,050
greater than 1.386.

221
00:12:19,050 --> 00:12:22,175
And this value of 1.386 is
actually Madelung's Constant,

222
00:12:22,175 --> 00:12:23,895
which you should know.

223
00:12:23,895 --> 00:12:28,530
It was a required preliminary
to the problem.

224
00:12:28,530 --> 00:12:39,050
So I know that below this
value of 1.386 I have my

225
00:12:39,050 --> 00:12:42,790
three-dimensional crystal.

226
00:12:42,790 --> 00:12:47,810
Now an energy-level diagram is
pretty much quantized by

227
00:12:47,810 --> 00:12:49,090
different steps.

228
00:12:49,090 --> 00:12:55,710
Now what is quantizing
our problem here?

229
00:12:55,710 --> 00:12:59,460
Well this is 0 because
our energy is 0.

230
00:12:59,460 --> 00:13:01,860
So we can almost assume that
we're multiplying--

231
00:13:01,860 --> 00:13:04,560
or that the Madelung's Constant
for ion pair with

232
00:13:04,560 --> 00:13:07,110
infinite separation is 0.

233
00:13:07,110 --> 00:13:11,120
And this is 1, so we can assume
that the Madelung

234
00:13:11,120 --> 00:13:15,130
Constant is 1 or we know that
it's 1 for the ion pair.

235
00:13:15,130 --> 00:13:17,950
So if I start writing this out
on the side here-- give me

236
00:13:17,950 --> 00:13:19,680
some space--

237
00:13:19,680 --> 00:13:20,510
I know that's one.

238
00:13:20,510 --> 00:13:22,530
I'm going to go ahead and
label this by Madelung's

239
00:13:22,530 --> 00:13:25,510
Constant, which is 1.386.

240
00:13:25,510 --> 00:13:27,900
So I come over here.

241
00:13:27,900 --> 00:13:29,150
Relabel this.

242
00:13:31,140 --> 00:13:34,810
This value here's is 1.386.

243
00:13:34,810 --> 00:13:39,320
And this value here has to be
greater than 1.386 for a

244
00:13:39,320 --> 00:13:41,960
three-dimensional crystal.

245
00:13:41,960 --> 00:13:48,470
And this little diagram here is
the answer to your question

246
00:13:48,470 --> 00:13:49,700
from the exam.

247
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So when you take the exam you
don't really have to go

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00:13:51,990 --> 00:13:54,470
through all this math to
answer the problem.

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00:13:54,470 --> 00:13:58,330
If you know the material from
lecture, then the solution

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00:13:58,330 --> 00:14:00,310
should be very straight-forward
very easily.

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00:14:00,310 --> 00:14:02,910
And also if you keep in mind
that there's a lot of material

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00:14:02,910 --> 00:14:05,720
that you need to know before you
take the exam, and knowing

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00:14:05,720 --> 00:14:09,180
how to absorb that material and
apply it will help you in

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00:14:09,180 --> 00:14:12,140
solving these type of
problems, which

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can be pretty difficult.