400
9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium
E X A M P L E 9.3
Show that the Gaussian distribution of Eq. (9.3-33) is normalized.
Solution
∞
−∞
f (u)du
1
√
2πσ
∞
−∞
e −(u−µ) 2 /2σ 2 du
1
√
2πσ
∞
−∞
e −w 2 /2σ 2 dw
1
√
2πσ
√
2σ
∞
−∞
e −t 2 dt
1
√
2πσ
√
2σ
√
π 1
where we have looked up the integral in Appendix C.
The Gaussian curve is named for
Johann Carl Friedrich Gauss,
1777–1855, a great German
mathematician who made many
fundamental contributions to
mathematics.
The standard deviation of an arbitrary probability distribution is defined by
σ u
u 2 − u 2 1/2 (definition of σ u )
(9.3-34)
The standard deviation is a convenient measure of the width of a probability distribution.
For a Gaussian distribution the probability that u lies between µ − σ and µ + σ is equal
to 0.683. For most other probability distributions the probability that the independent
variable lies within one standard deviation of the mean is approximately equal to
two-thirds.
Exercise 9.7
a. Show that u µ for the Gaussian distribution of Eq. (9.3-33).
b. Show that the definition in Eq. (9.3-34) when applied to the Gaussian probability distribution
leads to σ u σ.
By comparison of Eq. (9.3-32) and Eq. (9.3-33), the standard deviation σ v x for the
probability distribution of v x is given by
σ v x
k B T
m
RT
M
(9.3-35)
In the second equality M is the molar mass. The value of σ v x for O 2 molecules at 298 K
is equal to 278 m s −1 , which is equal to 623 miles per hour.
The standard deviation of a function h(u) is defined by
σ h
h(u) 2 − h(u) 2 1/2 (definition)
(9.3-36)
E X A M P L E 9.4
Obtain a formula for σ κ x , the standard deviation of the v x contribution to the kinetic energy,
κ x mv 2
x /2.
9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium
E X A M P L E 9.3
Show that the Gaussian distribution of Eq. (9.3-33) is normalized.
Solution
∞
−∞
f (u)du
1
√
2πσ
∞
−∞
e −(u−µ) 2 /2σ 2 du
1
√
2πσ
∞
−∞
e −w 2 /2σ 2 dw
1
√
2πσ
√
2σ
∞
−∞
e −t 2 dt
1
√
2πσ
√
2σ
√
π 1
where we have looked up the integral in Appendix C.
The Gaussian curve is named for
Johann Carl Friedrich Gauss,
1777–1855, a great German
mathematician who made many
fundamental contributions to
mathematics.
The standard deviation of an arbitrary probability distribution is defined by
σ u
u 2 − u 2 1/2 (definition of σ u )
(9.3-34)
The standard deviation is a convenient measure of the width of a probability distribution.
For a Gaussian distribution the probability that u lies between µ − σ and µ + σ is equal
to 0.683. For most other probability distributions the probability that the independent
variable lies within one standard deviation of the mean is approximately equal to
two-thirds.
Exercise 9.7
a. Show that u µ for the Gaussian distribution of Eq. (9.3-33).
b. Show that the definition in Eq. (9.3-34) when applied to the Gaussian probability distribution
leads to σ u σ.
By comparison of Eq. (9.3-32) and Eq. (9.3-33), the standard deviation σ v x for the
probability distribution of v x is given by
σ v x
k B T
m
RT
M
(9.3-35)
In the second equality M is the molar mass. The value of σ v x for O 2 molecules at 298 K
is equal to 278 m s −1 , which is equal to 623 miles per hour.
The standard deviation of a function h(u) is defined by
σ h
h(u) 2 − h(u) 2 1/2 (definition)
(9.3-36)
E X A M P L E 9.4
Obtain a formula for σ κ x , the standard deviation of the v x contribution to the kinetic energy,
κ x mv 2
x /2.