Physical Chemistry Third Edition

9.4 The Distribution of Molecular Speeds 405

say about the mean value of sin^2 (x) for 0<x<π

radians?

9.11 Find the mean value and the sample standard deviation for
all possible throws of two dice, assuming that all outcomes
are equally probable. Thesample standard deviationfor a
sample ofNmembers is different from the standard
deviation of an entire population and is
given by

sx

(

1

N− 1

∑N

i 1

(xi−〈x〉)^2

) 1 / 2

(9.3-43)

where〈x〉is the mean of the set. If the sample is randomly

drawn from a certain population, the sample standard

deviation is said to be an unbiased estimate of the

population standard deviation.

9.12 Find the mean value and the root-mean-square value of
sin(x) for−∞<x<∞, assuming the standard normal
(Gaussian) probability distribution

f(x)

1

√

2 π

e−x

(^2) / 2
9.13 Find the mean value and the root-mean-square value of
cos (x) for−∞<x<∞, assuming the standard normal
probability distribution
f(x)
1
√
2 π
e−x
(^2) / 2
9.14 For nitrogen molecules at 300.0 K, estimate the error in the
normalization integral of Eq. (9.3-20) that is produced by
including velocity components greater than the speed of
light. The following asymptotic formula^2 gives values of
the error function for large values ofz:
erf (z)l−
e−z
2
√
πz
(
1 +
∑∞
m 1
(−1)m
(1)(3)···(2m−1)
(2z^2 )m
)
9.15 For N 2 molecules at 298.15 K, calculate the probability
that|vx|exceeds the speed of light, 2. 997 × 108 ms−^1.
See the asymptotic formula in Problem 9.14.
9.16 a.Use Eq. (9.3-39) to estimate the probability
that an argon atom in a system at 273.15 K has
thexcomponent of its velocity between 0 and
20.0 m s−^1.
b.Find the correct value of the probability and compare it
with your result from part a.
9.17 Find the fraction of the molecules in a gas that have
v^2 x>
kBT
m
Show that this fraction is the same for all gases at all
temperatures.
9.18 The escape velocity is defined as the minimum upward
vertical velocity component required to escape the
earth’s gravity. At the earth’s surface, its value is

1. 12 × 104 ms−^1  2. 5 × 104 miles per hour.
   a. Find the fraction of N 2 molecules at 298 K having
   an upward vertical velocity component exceedingvesc.
   The asymptotic formula in Problem 9.14 can be
   used.
   b. Find the fraction of helium atoms at 298 K having an
   upward vertical velocity component exceedingvesc.
   9.19 The acceleration due to gravity at the surface of the moon
   is 1.67 m s−^2 and its radius is 1. 738 × 106 m. The mean
   daytime temperature at the surface of the moon is roughly
   370 K.
   a. Find the escape velocity at the surface of the moon.
   b. Find the fraction of N 2 molecules at 370 K having an
   upward vertical velocity component exceedingvesc.

9.4 The Distribution of Molecular Speeds

The speed is the magnitude of the velocity. All velocities that have the same magnitude

but different directions correspond to the same speed. In order to obtain a formula for

the probability distribution of speeds, we must add up the probabilities of all velocities

corresponding to the same speed. To do this conveniently, we change tospherical polar

coordinatesin velocity space. The first coordinate is the speedv. The second coordinate

isθ, the angle between the positivevzaxis and the velocity vector. The third coordinate

(^2) M. Abramowitz and I. A. Stegun,op. cit., p. 298 (Note 1).