402
9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium
This integral is in the form of the error function, defined in Appendix C by 1
erf (x)
2
√
π
x
0
e −t 2 dt
(9.3-38)
Our probability is therefore
(probability)
1
2
erf (1.0055) 0.4225 42.25%
Where the numerical value was obtained from the table of the error function in Appendix C.
We can obtain an approximate probability of a small finite range ∆v x by replacing
the infinitesimal interval dv x in Eq. (9.3-5) by ∆v x :
probability that
v x lies in ∆v x
≈ f (v
x )∆v x (for small ∆v x )
(9.3-39)
where v
x is a value of v x within the range ∆v x .
E X A M P L E 9.6
Find the probability that v x for an argon atom in a system at 273.15 K is in the range
650.00 m s −1 < v x < 651.00 m s −1 .
Solution
From Eq. (9.3-39)
probability ≈
m
2πk B T
1/2
e −mv 2
x /2k B T ∆v x
M
2πRT
1/2
e −Mv 2
x /2RT ∆v x
≈
0.039948 kg mol −1
2π(8.3145 J K −1 mol −1 )(273.15 K)
1/2
× exp
−(0.039948 kg mol −1 )(650 m s −1 ) 2
2(8.3145 J K −1 mol −1 )(273.15 K)
(1.00 m s −1 )
≈ 4.05 × 10 −5
Exercise 9.9
Find the probability that v x for an argon atom in a system at 273.15 K is in the range 650 m s −1
< v x < 652 m s −1 .
Exercise 9.10
a. What fraction of the molecules has x-components of the velocity between −σ v x
and σ v x ?
1 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, U.S. Govt. Printing Office, Washington, DC, 1964, p. 297ff.
9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium
This integral is in the form of the error function, defined in Appendix C by 1
erf (x)
2
√
π
x
0
e −t 2 dt
(9.3-38)
Our probability is therefore
(probability)
1
2
erf (1.0055) 0.4225 42.25%
Where the numerical value was obtained from the table of the error function in Appendix C.
We can obtain an approximate probability of a small finite range ∆v x by replacing
the infinitesimal interval dv x in Eq. (9.3-5) by ∆v x :
probability that
v x lies in ∆v x
≈ f (v
x )∆v x (for small ∆v x )
(9.3-39)
where v
x is a value of v x within the range ∆v x .
E X A M P L E 9.6
Find the probability that v x for an argon atom in a system at 273.15 K is in the range
650.00 m s −1 < v x < 651.00 m s −1 .
Solution
From Eq. (9.3-39)
probability ≈
m
2πk B T
1/2
e −mv 2
x /2k B T ∆v x
M
2πRT
1/2
e −Mv 2
x /2RT ∆v x
≈
0.039948 kg mol −1
2π(8.3145 J K −1 mol −1 )(273.15 K)
1/2
× exp
−(0.039948 kg mol −1 )(650 m s −1 ) 2
2(8.3145 J K −1 mol −1 )(273.15 K)
(1.00 m s −1 )
≈ 4.05 × 10 −5
Exercise 9.9
Find the probability that v x for an argon atom in a system at 273.15 K is in the range 650 m s −1
< v x < 652 m s −1 .
Exercise 9.10
a. What fraction of the molecules has x-components of the velocity between −σ v x
and σ v x ?
1 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, U.S. Govt. Printing Office, Washington, DC, 1964, p. 297ff.