13.5 Kinetic Theory and Statistical Physics

13.5 Kinetic Theory and Statistical Physics

So far we know that kinetic theory of gases is used to determine the pressure of the ideal gas using the gas laws assuming the thermal or random motion of molecules. For large number of molecule, we use statistical physics. It is the branch of physics that uses probability theory. Accordingly, atoms and molecules of a system may exist in different energy states (levels) $E_{1}, E_{2}, E_{3}, \dots$ etc due to different speeds. Boltzmann derived the Boltzmann kinetic equation. This equation describes the dynamic processes in gases having large number of molecules. Boltzmann constant is also related to the population of atoms in two levels.

where $(e^{-\Delta E/k_{B}T} = \text{Boltzmann factor})$ $N_{1}$ and $N_{2}$ are the population of lower and higher energy states, $\Delta E = (E_{2} - E_{1})$ is the energy difference between these states. The number density is directly proportional to the pressure. This equation is known as the Boltzmann distribution law and is important in describing the statistical mechanics of a large number of molecules. It states that the probability of finding the molecules in a particular energy state varies exponentially as the negative of the energy divided by $k_{B}T$; therefore, more particles reside in lower energy states than in higher ones.

In a gas container, a molecule undergoes billions of collisions every second. Each collision changes the speed of molecule thereby changing the kinetic energy, but kinetic theory concludes that average kinetic energy at temperature $T$ is:

where $k_{B}$ is effectively the gas constant per molecule.

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In 1860 James Clark Maxwell (1831–1879) derived an expression that describes the distribution of molecular speeds within the system at thermal equilibrium. About 60 years later, experiments were performed to confirm Maxwell's predictions.