Homework Problem Set 5 Solutions ( )2

5. The Maxwell-Boltzmann distribution for speeds is
12
F(v) = 4π v 2
3/ 2
⎛⎛ m ⎞⎞
⎜⎜ ⎟⎟
⎝⎝ 2π k B T ⎠⎠
⎛⎛
exp − mv2 ⎞⎞
⎜⎜
⎝⎝ 2k B T
⎟⎟ .
⎠⎠
In this equation, m is the mass of a particle, k B is the Boltzmann constant, T is
temperature, and v € is the speed.
velocity vector,
where
v x ,
v y, and
The most probable speed
€ by € the equation €
The speed v is defined to be the magnitude of the
2
v = v x +
2 vy +
2
[ vz ] 1/ 2 ,
v z are the components of the velocity vector.
€
v mp corresponding to the maximum in the distribution is given
€
v mp =
⎛⎛
⎜⎜
⎝⎝
2k B T
m
1/ 2
⎞⎞
⎟⎟ .
⎠⎠
a.) Plot the Maxwell-Boltzmann distribution function F(v) (y-axis) versus v (x-axis) for a
collection of atoms at 300 € K. Assume that one mole of atoms weighs 12.0 g.
b.)
Using the same mass as in part (a), plot € the Maxwell-Boltzmann distribution function
F(v) versus v for a collection of atoms at 1000 K. Discuss the effect of temperature
on the distribution of atomic speeds.
€
Both questions (a) and (b) are answered below.
The Maxwell-Boltzmann distribution function for the atoms at 300 K is given by
⎢⎢ 0.012 kg/mol
( ) = 4π v 2 ⎢⎢
F v
3/ 2
⎡⎡
⎛⎛ 1mol ⎞⎞ ⎤⎤
( ) ⎜⎜
⎝⎝ 6.02217 × 10 23 ⎟⎟
⎠⎠
⎥⎥
⎥⎥
⎢⎢ 2π( 1.38062 × 10 -23 JK −1 )( 300 K)
⎥⎥
⎢⎢
⎥⎥
⎣⎣
⎦⎦
⎧⎧
⎛⎛ 1mol ⎞⎞
( 0.012 kg/mol)
⎜⎜
⎝⎝ 6.02217 × 10
× exp −
23 ⎟⎟ v 2 ⎫⎫
⎪⎪
⎠⎠
⎪⎪
⎪⎪
⎪⎪
⎨⎨
2( 1.38062 × 10 -23 JK −1 ⎬⎬
⎪⎪
)( 300 K)
⎪⎪
⎩⎩ ⎪⎪
⎭⎭ ⎪⎪
€
F( v) = ( 8.41956 × 10 −9 v 2 ) exp{ −2.40549 × 10 −6 v 2 }.
€