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. The sample standard deviation for a
sample of N members is different from the standard
deviation of an entire population and is
given by
s x
1
N − 1
N
i1
(x i − 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
√
2π
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
√
2π
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 of z:
erf (z) l −
e −z 2
√
πz
1 +
∞
m1
(−1)
m (1)(3) · · · (2m − 1)
(2z 2 ) m
9.15 For N 2 molecules at 298.15 K, calculate the probability
that |v x | exceeds the speed of light, 2.997 × 10 8 m s −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
the x component 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 >
k B T
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 × 10 4 m s −1 2.5 × 10 4 miles per hour.
a. Find the fraction of N 2 molecules at 298 K having
an upward vertical velocity component exceeding v esc .
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 exceeding v esc .
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 × 10 6 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 exceeding v esc .
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 to spherical polar
coordinates in velocity space. The first coordinate is the speed v. The second coordinate
is θ, the angle between the positive v z axis and the velocity vector. The third coordinate
2 M. Abramowitz and I. A. Stegun, op. cit., p. 298 (Note 1).
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. The sample standard deviation for a
sample of N members is different from the standard
deviation of an entire population and is
given by
s x
1
N − 1
N
i1
(x i − 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
√
2π
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
√
2π
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 of z:
erf (z) l −
e −z 2
√
πz
1 +
∞
m1
(−1)
m (1)(3) · · · (2m − 1)
(2z 2 ) m
9.15 For N 2 molecules at 298.15 K, calculate the probability
that |v x | exceeds the speed of light, 2.997 × 10 8 m s −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
the x component 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 >
k B T
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 × 10 4 m s −1 2.5 × 10 4 miles per hour.
a. Find the fraction of N 2 molecules at 298 K having
an upward vertical velocity component exceeding v esc .
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 exceeding v esc .
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 × 10 6 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 exceeding v esc .
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 to spherical polar
coordinates in velocity space. The first coordinate is the speed v. The second coordinate
is θ, the angle between the positive v z axis and the velocity vector. The third coordinate
2 M. Abramowitz and I. A. Stegun, op. cit., p. 298 (Note 1).