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SAL: Hi.

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00:00:22,450 --> 00:00:22,880
I'm Sal.

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00:00:22,880 --> 00:00:27,830
Today, we're going to solve
problem nine of fall 2009, the

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00:00:27,830 --> 00:00:30,060
final for 3091.

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00:00:30,060 --> 00:00:33,820
The problem reads, the phase
diagram of the binary system

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00:00:33,820 --> 00:00:38,400
neodymium, praseodymium and
EPR is given below.

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00:00:38,400 --> 00:00:39,800
There are two allotropes--

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00:00:39,800 --> 00:00:41,780
alpha, which is hexagonal
close pack--

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00:00:41,780 --> 00:00:42,870
HCP--

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00:00:42,870 --> 00:00:44,780
and beta, which is body
center cubic--

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00:00:44,780 --> 00:00:46,000
BCC.

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00:00:46,000 --> 00:00:49,170
Now this problem obviously is
about phase diagrams. So

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00:00:49,170 --> 00:00:51,760
before you attempt to solve the
problem, you need to make

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00:00:51,760 --> 00:00:56,130
sure that you understand how
to read phase diagrams and

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00:00:56,130 --> 00:00:59,920
also how to apply the lever
rule and know where in the

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00:00:59,920 --> 00:01:02,580
phase diagram to apply
the lever rule.

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00:01:02,580 --> 00:01:04,880
So if we look at our phase
diagram, which I have drawn

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00:01:04,880 --> 00:01:09,100
over here for part A--

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00:01:09,100 --> 00:01:13,010
the part A asks, explain why a
lenticular phase diagram is to

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00:01:13,010 --> 00:01:15,800
be expected for the
binary system.

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00:01:15,800 --> 00:01:22,310
So this is a lenticular phase
diagram and a lot of students

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00:01:22,310 --> 00:01:25,690
when answering this problem
get confused and say, oh,

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00:01:25,690 --> 00:01:29,140
because the components are
soluble together, that's why

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00:01:29,140 --> 00:01:31,650
you get a lenticular.

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00:01:31,650 --> 00:01:33,570
That's correct, but that
information you can get from

33
00:01:33,570 --> 00:01:34,620
the phase diagram.

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00:01:34,620 --> 00:01:38,130
It's obvious by looking that you
have an alpha phase that

35
00:01:38,130 --> 00:01:42,320
you have from zero to a 100%
that can be missable together,

36
00:01:42,320 --> 00:01:44,870
but it doesn't really
explain the answer.

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00:01:44,870 --> 00:01:48,670
Now the best thing to do is to
look at the periodic table of

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00:01:48,670 --> 00:01:51,070
elements and look at the
physical and chemical

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00:01:51,070 --> 00:01:52,530
properties of the elements.

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00:01:52,530 --> 00:02:01,460
So here we're comparing to--
we're comparing neodymium and

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00:02:01,460 --> 00:02:03,570
praseodymium, which are
two components that

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00:02:03,570 --> 00:02:05,310
are making the alloy.

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00:02:05,310 --> 00:02:10,900
So here this is atomic percent
praseodymium on your x axis

44
00:02:10,900 --> 00:02:12,340
and on your y axis, you have

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00:02:12,340 --> 00:02:14,640
temperature in degrees Celsius.

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00:02:14,640 --> 00:02:18,890
So at zero percent praseodymium,
we have a 100%

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00:02:18,890 --> 00:02:22,940
neodymium and at 100%
praseodymium, we have zero

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00:02:22,940 --> 00:02:24,320
percent neodymium.

49
00:02:24,320 --> 00:02:26,790
That pretty much comes from
the phase diagram.

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00:02:26,790 --> 00:02:30,530
So why would these
two have formed a

51
00:02:30,530 --> 00:02:32,280
lenticular phase diagram?

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00:02:32,280 --> 00:02:35,290
Well, if you look at the
periodic table, you'll notice

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00:02:35,290 --> 00:02:37,990
that these two elements lie
right next to each other.

54
00:02:37,990 --> 00:02:39,100
So what does that mean?

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00:02:39,100 --> 00:02:40,740
That means that chemically,
they're pretty

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00:02:40,740 --> 00:02:43,950
much almost the same.

57
00:02:43,950 --> 00:02:48,240
To answer this question for part
A, I would look at it and

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00:02:48,240 --> 00:02:52,260
I would notice that actually
both these elements have the

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00:02:52,260 --> 00:02:53,510
same crystal structure.

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00:02:53,510 --> 00:02:55,030
So I would write down,
they have the

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00:02:55,030 --> 00:02:56,280
same crystal structure.

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00:03:02,580 --> 00:03:06,760
They also have pretty much the
same electronegativity or very

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00:03:06,760 --> 00:03:08,450
similar electronegativity.

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00:03:08,450 --> 00:03:16,770
So very similar
electronegativity.

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00:03:23,460 --> 00:03:26,020
Also, they have pretty
much the same atomic

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00:03:26,020 --> 00:03:27,270
radii--atomic radius.

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00:03:47,280 --> 00:03:52,640
And also very similar, very
close to each other--

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00:03:52,640 --> 00:03:53,410
the same density.

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00:03:53,410 --> 00:04:02,350
So the density is
almost the same.

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00:04:02,350 --> 00:04:07,070
So because these two components
have the same

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00:04:07,070 --> 00:04:10,310
properties together-- the same
size, same crystal structure,

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00:04:10,310 --> 00:04:15,870
then you should expect that they
should form a missable

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00:04:15,870 --> 00:04:18,720
lenticular phase diagram, which
is what we have here.

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00:04:18,720 --> 00:04:22,050
So if you were to answer that
for part A, that would give

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00:04:22,050 --> 00:04:24,550
you pretty much all the points
and that hints that you

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00:04:24,550 --> 00:04:27,380
understand how to look at the
chemical properties of the

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00:04:27,380 --> 00:04:30,300
elements and the physical
properties of the elements and

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00:04:30,300 --> 00:04:33,510
compare them together and
rationalize why the phase

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00:04:33,510 --> 00:04:36,520
diagram looks the way it does.

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00:04:36,520 --> 00:04:38,750
Part B asks--

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00:04:38,750 --> 00:04:40,000
I'll come over here
to do part B.

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00:04:43,820 --> 00:04:48,590
At each point, I identify all
the phases present in

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00:04:48,590 --> 00:04:50,660
equilibrium.

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00:04:50,660 --> 00:04:52,200
II--

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00:04:52,200 --> 00:04:54,960
state the composition of
each phase and III--

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00:04:54,960 --> 00:04:59,240
calculate the relative amount
of all phase presence.

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00:04:59,240 --> 00:05:01,570
And it asks us to analyze the
three points that we have on

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00:05:01,570 --> 00:05:02,820
our phase diagram.

89
00:05:05,320 --> 00:05:13,190
So we have one for
I, II, III--

90
00:05:13,190 --> 00:05:19,170
So number one-- if we look at
our phase diagram, we lie

91
00:05:19,170 --> 00:05:21,600
exactly where your
alpha phase is.

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00:05:21,600 --> 00:05:25,560
So to answer the first part of
the problem is identify all

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00:05:25,560 --> 00:05:26,810
the phases present--

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00:05:26,810 --> 00:05:29,470
we only have the alpha
phase here.

95
00:05:29,470 --> 00:05:34,480
So if you were to write down
alpha phase, that's correct.

96
00:05:34,480 --> 00:05:39,010
The part two says, state the
composition of each phase.

97
00:05:39,010 --> 00:05:43,390
So what's the composition of
the alpha phase in here?

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00:05:43,390 --> 00:05:48,770
Well, if I trace down my line
down to my x axis, I notice

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00:05:48,770 --> 00:05:52,980
that for the atomic percent
of praseodymium, pretty

100
00:05:52,980 --> 00:05:54,420
close to about 18%.

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00:05:54,420 --> 00:06:02,150
So it's 18 atomic
percent here.

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00:06:02,150 --> 00:06:06,150
So to answer the question for
part I, I would notice that--

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00:06:06,150 --> 00:06:08,220
I would write down that
it's alpha phase.

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00:06:13,760 --> 00:06:18,840
For the second part,
we have 18%--

105
00:06:18,840 --> 00:06:25,700
18 atomic percent praseodymium,
which leaves

106
00:06:25,700 --> 00:06:36,610
behind about 82% atomic
percent of neodymium.

107
00:06:36,610 --> 00:06:38,750
And for part three--

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00:06:38,750 --> 00:06:42,580
part three asks about
the relative amounts

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00:06:42,580 --> 00:06:43,830
of the phases present--

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00:06:43,830 --> 00:06:47,160
we only have the alpha phase
in that part so I would say

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00:06:47,160 --> 00:06:53,790
that we have 100% pure
alpha, which is--

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00:06:53,790 --> 00:06:55,610
if you write this,
that's correct.

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00:06:55,610 --> 00:06:58,090
Now for the second point that
we're going to analyze in our

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00:06:58,090 --> 00:07:01,330
phase diagram number two--

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00:07:01,330 --> 00:07:05,490
if we come over here and look at
it, you'll notice that now

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00:07:05,490 --> 00:07:09,710
we lie in between or between
two phases so where the

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00:07:09,710 --> 00:07:11,910
lenticular region is at.

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00:07:11,910 --> 00:07:16,610
So here, I don't have pure
beta or pure liquid.

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00:07:16,610 --> 00:07:19,780
I have a combination of both
beta and liquid and this is

120
00:07:19,780 --> 00:07:22,590
exactly where you need to
apply the lever rule.

121
00:07:22,590 --> 00:07:25,380
You can't apply the lever rule
here because it's just 100%

122
00:07:25,380 --> 00:07:26,330
pure phase.

123
00:07:26,330 --> 00:07:29,270
You can only apply when you're
in a region where you have two

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00:07:29,270 --> 00:07:32,360
phases that coexist
in equilibrium.

125
00:07:32,360 --> 00:07:37,190
So if I go ahead and
analyze part two--

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00:07:37,190 --> 00:07:38,210
part I--

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00:07:38,210 --> 00:07:44,470
the phases that are present are
beta and the liquid phase.

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00:07:44,470 --> 00:07:49,720
For part II, it asks, what
are the compositions?

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00:07:49,720 --> 00:07:53,205
Well, if I blow up my lenticular
phase diagram--

130
00:07:56,780 --> 00:08:03,240
so I'm sitting right here and
this is sitting at around 62

131
00:08:03,240 --> 00:08:09,870
two atomic percent of Pr--

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00:08:09,870 --> 00:08:14,520
praseodymium and it asks about
the phases or the relative

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00:08:14,520 --> 00:08:16,030
composition of each phase.

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00:08:16,030 --> 00:08:19,390
Well, over here, I have beta,
over here I have the liquid

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00:08:19,390 --> 00:08:21,860
phase and in between I have the
beta plus the liquid that

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00:08:21,860 --> 00:08:25,620
are coexisting in equilibrium.

137
00:08:25,620 --> 00:08:29,510
So if I want to know
how much--

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00:08:29,510 --> 00:08:32,560
what's the composition of my
beta phase that I have here--

139
00:08:32,560 --> 00:08:35,400
well, I'm going to have to
go to the line where both

140
00:08:35,400 --> 00:08:37,730
intersect with the beta and
the beta plus liquid

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00:08:37,730 --> 00:08:38,690
intersects.

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00:08:38,690 --> 00:08:40,880
And if I analyze this point--

143
00:08:40,880 --> 00:08:44,030
if I look down what the
composition of that is for

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00:08:44,030 --> 00:08:52,320
that line, I see that I have
about 56 atomic percent of

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00:08:52,320 --> 00:08:53,600
praseodymium.

146
00:08:53,600 --> 00:08:58,890
So to start answering part two,
I would say that if I

147
00:08:58,890 --> 00:09:00,950
look at beta--

148
00:09:00,950 --> 00:09:10,410
in the beta phase, I have 56% or
atomic percent of Pr and if

149
00:09:10,410 --> 00:09:19,350
I have 56% of Pr, then this
leaves just 44 atomic percent

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00:09:19,350 --> 00:09:24,050
of neodymium and if I
look at my liquid--

151
00:09:24,050 --> 00:09:26,000
so I'm going to just move
this over here.

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00:09:26,000 --> 00:09:30,190
I look at the liquid, which
is going to be where it

153
00:09:30,190 --> 00:09:33,600
intersects in the liquid region
and if I trace it down,

154
00:09:33,600 --> 00:09:37,710
I see that this is
about 70% or so.

155
00:09:37,710 --> 00:09:41,980
So here I have around
roughly 70%.

156
00:09:41,980 --> 00:09:46,290
If you trace it down to your x
axis, this is 70% or 70 atomic

157
00:09:46,290 --> 00:09:55,700
percent Pr, which leaves 30
atomic percent of neodymium.

158
00:09:55,700 --> 00:09:59,260
So if you were to write this
down for the liquid, just

159
00:09:59,260 --> 00:10:02,700
write an arrow here so I won't
have to rewrite that--

160
00:10:02,700 --> 00:10:05,530
that would be the answer for
part two because again, we're

161
00:10:05,530 --> 00:10:08,480
analyzing this point, which is
point number two and we're in

162
00:10:08,480 --> 00:10:10,130
between two phases.

163
00:10:10,130 --> 00:10:14,260
So that covers the second
part of part B.

164
00:10:14,260 --> 00:10:17,210
And for the third part--

165
00:10:17,210 --> 00:10:19,660
I'll go ahead and I'll
put up here.

166
00:10:24,450 --> 00:10:27,290
That one now is where we have
to apply the lever rule

167
00:10:27,290 --> 00:10:30,180
because now we have to figure
out exactly how much of each

168
00:10:30,180 --> 00:10:33,780
phase is present
at equilibrium.

169
00:10:33,780 --> 00:10:37,490
So if I look at my diagram, it's
going to be very simple.

170
00:10:37,490 --> 00:10:40,900
If I want to know how much beta
I have in this mixture,

171
00:10:40,900 --> 00:10:43,540
you're simply going to apply the
lever rule and for beta,

172
00:10:43,540 --> 00:10:47,820
it's going to be whatever this
magnitude is or whatever the

173
00:10:47,820 --> 00:10:50,720
difference between these two
points are over the length of

174
00:10:50,720 --> 00:10:53,960
the whole existence at that
temperature of the length of

175
00:10:53,960 --> 00:10:59,220
the whole composition that you
have. So if I do the simple

176
00:10:59,220 --> 00:11:02,230
calculation for beta--

177
00:11:02,230 --> 00:11:15,470
beta is going to be simply 70
minus 62 over 70 minus 56 and

178
00:11:15,470 --> 00:11:22,370
this essentially just equals
eight over 14, which is about

179
00:11:22,370 --> 00:11:28,620
57 atomic percent
praseodymium.

180
00:11:28,620 --> 00:11:29,860
So--

181
00:11:29,860 --> 00:11:32,270
sorry-- not percent Pr--

182
00:11:32,270 --> 00:11:36,380
percent beta phase.

183
00:11:36,380 --> 00:11:40,760
So in beta, I have 57%.

184
00:11:40,760 --> 00:11:44,030
So that means that
for my liquid--

185
00:11:44,030 --> 00:11:46,520
I can do two things for this
problem, for the liquid.

186
00:11:46,520 --> 00:11:48,780
I know that everything is
normalized to 100%.

187
00:11:48,780 --> 00:11:52,530
So the difference between 157
will be the liquid composition

188
00:11:52,530 --> 00:11:54,150
where I can go ahead and
apply the lever rule

189
00:11:54,150 --> 00:11:56,220
again and do the math.

190
00:11:56,220 --> 00:11:59,520
So if I apply the lever rule
again for my liquid phase,

191
00:11:59,520 --> 00:12:04,820
it's going to simply
be 62 minus 56.

192
00:12:04,820 --> 00:12:06,930
We're coming back on our--

193
00:12:06,930 --> 00:12:09,060
looking at the liquid phase
that is coexisting in this

194
00:12:09,060 --> 00:12:13,160
equilibrium at this temperature
and I want to know

195
00:12:13,160 --> 00:12:14,000
how much liquid--

196
00:12:14,000 --> 00:12:18,220
it's going to be the magnitude
of this over the total length

197
00:12:18,220 --> 00:12:19,920
of my composition there--

198
00:12:19,920 --> 00:12:21,060
that's what I'm doing here.

199
00:12:21,060 --> 00:12:24,900
Again, this is 70 minus 56.

200
00:12:24,900 --> 00:12:31,240
So this would end up being just
six over 14, which is

201
00:12:31,240 --> 00:12:33,140
essentially just--

202
00:12:33,140 --> 00:12:36,050
if you have 57% here, if you do
the math, you're going to

203
00:12:36,050 --> 00:12:45,560
get 43% of liquid.

204
00:12:45,560 --> 00:12:47,270
So part three--

205
00:12:47,270 --> 00:12:53,610
when you apply the lever rule,
you see that at this point you

206
00:12:53,610 --> 00:12:56,710
have more beta than
you have liquid--

207
00:12:56,710 --> 00:12:58,600
and it kind of makes sense
because if you look at your

208
00:12:58,600 --> 00:13:02,730
diagram, your point is sitting
closer to the beta phase.

209
00:13:02,730 --> 00:13:08,030
So I would expect intuitively
that I would have more beta

210
00:13:08,030 --> 00:13:10,910
present then the liquid present
and that's exactly

211
00:13:10,910 --> 00:13:12,780
what the lever rule tells us.

212
00:13:12,780 --> 00:13:17,010
So that's point number
two for part B.

213
00:13:17,010 --> 00:13:20,730
And if I look at point number
three now, which I'll write

214
00:13:20,730 --> 00:13:22,040
right here--

215
00:13:22,040 --> 00:13:25,190
this is essentially the same
thing as point number one.

216
00:13:25,190 --> 00:13:26,610
But now point number three--

217
00:13:26,610 --> 00:13:28,680
we're no longer lying
in alpha.

218
00:13:28,680 --> 00:13:30,440
We're lying in the beta phase.

219
00:13:30,440 --> 00:13:36,260
So if I trace down what my
composition is here, I'm about

220
00:13:36,260 --> 00:13:43,110
78% of Pr--

221
00:13:43,110 --> 00:13:46,810
so to answer the
first part I--

222
00:13:46,810 --> 00:13:48,060
beta--

223
00:13:50,800 --> 00:13:53,600
the second part II, which is
dealing with the compositions

224
00:13:53,600 --> 00:14:04,590
of that beta phase is going to
be 78% or 78 atomic percent Pr

225
00:14:04,590 --> 00:14:12,690
and that leaves just 22 atomic
percent neodymium--

226
00:14:12,690 --> 00:14:13,940
and for the third part III--

227
00:14:20,360 --> 00:14:23,880
what is the relative amounts
of the phases present?

228
00:14:23,880 --> 00:14:26,160
Not the composition,
the phases.

229
00:14:26,160 --> 00:14:30,340
I'm exactly where only pure beta
exists so I can say that

230
00:14:30,340 --> 00:14:37,590
I have 100% beta.

231
00:14:37,590 --> 00:14:40,930
So that pretty much does
it for this problem.

232
00:14:40,930 --> 00:14:44,170
Again, knowing how to read phase
diagrams, knowing how to

233
00:14:44,170 --> 00:14:47,310
apply the lever rule and where
to apply the lever rule will

234
00:14:47,310 --> 00:14:49,120
allow you to very easily
solve the problem.

235
00:14:49,120 --> 00:14:51,950
A lot of students make the
mistake of trying to apply the

236
00:14:51,950 --> 00:14:53,640
lever rule when you
have a pure phase.

237
00:14:53,640 --> 00:14:58,200
I mean, that makes no sense
because said lever rule pretty

238
00:14:58,200 --> 00:15:02,470
much calculates what fraction
of the phases are present in

239
00:15:02,470 --> 00:15:04,080
that equilibrium mixture.

240
00:15:04,080 --> 00:15:07,750
So keep in mind that you can
only do that when you have a

241
00:15:07,750 --> 00:15:10,660
region in your phase diagram,
when you have two phases that

242
00:15:10,660 --> 00:15:13,040
are coexistant in equilibrium.