5.112 Principles Of Chemical Science Worksheet With Answers - Problem Set #4 Solutions - 2011 Page 12
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16Problem set #4 solutions for 5.112, Fall 2011
Page 12 of 16
3
− Mv
2
⎛
⎞
M
2
f (v) = 4 π
2
⎝ ⎜
⎠ ⎟
v
e
2 RT
2 π RT
3
− Mv
2
⎡
−Mv
⎤
⎛
⎞
⎛
⎞
df (v)
M
2
8 π v + 4 π v
=
⎥ = 0
2
⎝ ⎜
⎠ ⎟
e
⎝ ⎜
⎠ ⎟
2 RT
⎢
2 π RT
⎣
⎦
dv
RT
Solving for ν
in terms of T:
mp
1
⎛
⎞
2RT
2
=
v
⎝ ⎜
⎠ ⎟
mp
M
Substitution into f (v) gives:
3
1
⎛
⎞
⎛
⎞
⎛
⎞
M
2RT
M
2
2
) = 4 π
−1
= 4 π
−1
⎝ ⎜
⎠ ⎟
⎝ ⎜
⎠ ⎟
⎝ ⎜
⎠ ⎟
f (v
e
e
2 π RT
2 π RT
mp
M
The problem asks for the temperature that satisfies this relationship:
1
=
f (v
)
f (v
)
mp
T
mp
355
4
1
1
⎛
⎞
⎛
⎞
M
1
M
2
2
4 π
−1
4 π
−1
=
⎝ ⎜
⎠ ⎟
e
e
⎝ ⎜
⎠ ⎟
2 π RT
2 π R(355 K)
4
1
1
⎛
⎞
⎛
⎞
⎛
⎞
1
1
1
2
2
=
⎝ ⎜
⎠ ⎟
⎝ ⎜
⎠ ⎟
⎝ ⎜
⎠ ⎟
T
4
355 K
⎛
⎞
⎛
⎞
⎛
⎞
1
1
1
=
⎝ ⎜
⎠ ⎟
⎝ ⎜
⎠ ⎟
⎝ ⎜
⎠ ⎟
T
16
355 K
T = 16(355 K) = 5680 K
The second temperature is 5680 K
Question 16 (out of 6 points)
Consider methane (CH 4 ) gas at a temperature T.
(a) Calculate the ratio, to 3 significant figures, of the number of methane molecules that
have the average speed v to the number of molecules that have twice the average
speed, 2 v .
Let’s first consider the Maxwell-Boltzmann speed distribution;
3
⎛
⎞
⎛
⎞
ΔN
−mu
2
m
2
⎜ ⎜
⎟ ⎟
⎜ ⎜
⎟ ⎟ ,
= f (u)Δu, where f (u) = 4 π
2
u
exp
2 π k
⎝
⎠
⎝
⎠
N
T
2k
T
B
B
We can compare the fraction of total number of molecules traveling at the average speed v and
compare that to the fraction of total number of molecules traveling at 2v
( )
⎛
⎞
2
−M v
⎜
⎟
exp
( )
( )
⎜
⎟
2
2RT
⎝
⎠
f v
v
=
( )
( )
( )
⎛
⎞
2
2
f 2v
−M 2v
2v
⎜
⎟
exp
⎜
⎟
2RT
⎝
⎠
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