9.3 The Velocity Probability Distribution
403
b. Find a value v
x such that there is a 95% probability that a molecule will have a value of v x
between −v
x and v
x for N 2 molecules at 298.15 K. Find the ratio of this value to the standard
deviation σ v x .
c. Repeat this calculation for N 2 molecules at 1000.0 K.
The three-dimensional probability distribution g(v) can be written as
g(v) f (v x ) f (v y ) f (v z )
m
2πk B T
3/2
e
−m(v 2
x + v 2
y + v 2
z )/2k B T
g(v)
m
2πk B T
3/2
e −mv 2 /2k B T
(9.3-40)
This function is called the Maxwell probability distribution or the Maxwell–Boltzmann
probability distribution. In terms of the molecular kinetic energy,
g(v)
m
2πk B T
3/2
e −ε/k B T
(9.3-41)
which is called the Boltzmann probability distribution. We do not prove it at this time,
but this probability distribution applies to all forms of molecular energy, including
potential energy and the vibrational, rotational, and electronic energies. We write
(probability of a state of energy ε) ∝ e −ε/k B T
(9.3-42)
where the symbol ∝ means “is proportional to.”
There are several important physical facts about the Boltzmann probability
distribution:
1. At a finite temperature, molecular states of higher energy are less probable than
states of lower energy. States with energy much larger than k B T are quite improbable
compared with states of energy equal to zero.
2. A molecular state of high energy will be more probable at a high temperature than
at a low temperature.
3. As the temperature approaches infinity, all states approach equal probability.
4. If the temperature approaches zero, only the state of lowest energy will be populated.
E X A M P L E 9.7
Find the ratio of the probabilities of the two following velocities of neon atoms at 300 K:
first velocity: v x 500 m s −1 , v y −400 m s −1 , v z 250 m s −1
second velocity: v x 200 m s −1 , v y 350 m s −1 , v z −275 m s −1
Solution
Neon has several isotopes. We use the average molecular mass,
m
0.020183 kg mol −1
6.022 × 10 23 mol −1 3.35 × 10 −26 kg
403
b. Find a value v
x such that there is a 95% probability that a molecule will have a value of v x
between −v
x and v
x for N 2 molecules at 298.15 K. Find the ratio of this value to the standard
deviation σ v x .
c. Repeat this calculation for N 2 molecules at 1000.0 K.
The three-dimensional probability distribution g(v) can be written as
g(v) f (v x ) f (v y ) f (v z )
m
2πk B T
3/2
e
−m(v 2
x + v 2
y + v 2
z )/2k B T
g(v)
m
2πk B T
3/2
e −mv 2 /2k B T
(9.3-40)
This function is called the Maxwell probability distribution or the Maxwell–Boltzmann
probability distribution. In terms of the molecular kinetic energy,
g(v)
m
2πk B T
3/2
e −ε/k B T
(9.3-41)
which is called the Boltzmann probability distribution. We do not prove it at this time,
but this probability distribution applies to all forms of molecular energy, including
potential energy and the vibrational, rotational, and electronic energies. We write
(probability of a state of energy ε) ∝ e −ε/k B T
(9.3-42)
where the symbol ∝ means “is proportional to.”
There are several important physical facts about the Boltzmann probability
distribution:
1. At a finite temperature, molecular states of higher energy are less probable than
states of lower energy. States with energy much larger than k B T are quite improbable
compared with states of energy equal to zero.
2. A molecular state of high energy will be more probable at a high temperature than
at a low temperature.
3. As the temperature approaches infinity, all states approach equal probability.
4. If the temperature approaches zero, only the state of lowest energy will be populated.
E X A M P L E 9.7
Find the ratio of the probabilities of the two following velocities of neon atoms at 300 K:
first velocity: v x 500 m s −1 , v y −400 m s −1 , v z 250 m s −1
second velocity: v x 200 m s −1 , v y 350 m s −1 , v z −275 m s −1
Solution
Neon has several isotopes. We use the average molecular mass,
m
0.020183 kg mol −1
6.022 × 10 23 mol −1 3.35 × 10 −26 kg