Introduction
The kinetic molecular theory is used to determine the motion of a molecule of an ideal gas under a certain set of conditions. However, when looking at a mole of ideal gas, it is impossible to measure the velocity of each molecule at every instant of time. Therefore, the Maxwell-Boltzmann distribution is used to determine how many molecules are moving between velocities v and v + dv. Assuming that the one-dimensional distributions are independent of one another, that the velocity in the y and z directions does not affect the x velocity, for example, the Maxwell-Boltzmann distribution is given by
(1.1)
where
Additionally, the function can be written in terms of the scalar quantity speed c instead of the vector quantity velocity. This form of the function defines the distribution of the gas molecules moving at different speeds, between c1 and c2, thus
(1.2)
Finally, the Maxwell-Boltzmann distribution can be used to determine the distribution of the kinetic energy of for a set of molecules. The distribution of the kinetic energy is identical to the distribution of the speeds for a certain gas at any temperature.2
Plotting the Maxwell-Boltzmann Distribution Function
Figure 1 shows the Maxwell-Boltzmann distribution of speeds for a certain gas at a certain temperature, such as nitrogen at 298 K. The speed at the top of the curve is called the most probable speed because the largest number of molecules have that speed.
Figure 1: The Maxwell-Boltzmann distribution is shifted to higher speeds and is broadened at higher temperatures.
Figure 2 shows how the Maxwell-Boltzmann distribution is affected by temperature. At lower temperatures, the molecules have less energy. Therefore, the speeds of the molecules are lower and the distribution has a smaller range. As the temperature of the molecules increases, the distribution flattens out. Because the molecules have greater energy at higher temperature, the molecules are moving faster.
Figure 2: The Maxwell-Boltzmann distribution is shifted to higher speeds and is broadened at higher temperatures.
Figure 3 shows the dependence of the Maxwell-Boltzmann distribution on molecule mass. On average, heavier molecules move more slowly than lighter molecules. Therefore, heavier molecules will have a smaller speed distribution, while lighter molecules will have a speed distribution that is more spread out.
Figure 3: The speed probability density functions of the speeds of a few noble gases at a temperature of 298.15 K (25 °C). The y-axis is in s/m so that the area under any section of the curve (which represents the probability of the speed being in that range) is dimensionless.
Related Speed Expressions
Three speed expressions can be derived from the Maxwell-Boltzmann distribution: the most probable speed, the average speed, and the root-mean-square speed. The most probable speed is the maximum value on the distribution plot. This is established by finding the velocity when the following derivative is zero
(1.3)
which is
(1.4)
The average speed is the sum of the speeds of all the molecules divided by the number of molecules.
(1.5)
The root-mean-square speed is square root of the average speed-squared.
(1.6)
where
It always follows that for gases that follow the Maxwell-Boltzmann distribution (if thermallized)
Cmp<Cavg<Crms (1.7)
1. What is the Boltzmann distribution in statistical thermodynamics? | ![]() |
2. How is the Boltzmann distribution derived in statistical thermodynamics? | ![]() |
3. What is the significance of the Boltzmann distribution in physical chemistry? | ![]() |
4. How is the Boltzmann distribution related to CSIR-NET exam in physical chemistry? | ![]() |
5. Are there any government job opportunities for candidates with knowledge of the Boltzmann distribution and statistical thermodynamics? | ![]() |