The Classical Ideal Gas and Molecular Speeds

The Physical Picture

Imagine a box filled with a vast number of tiny, hard spheres bouncing around randomly. In the classical picture, these particles do not interact with each other except during instantaneous, elastic collisions. Because they are classical, they can occupy any position and have any velocity, provided the total energy is conserved. However, when this gas is in contact with a heat reservoir at temperature \( T \), the system does not have a fixed total energy; instead, it samples different energy states according to the Boltzmann factor. The key insight is that while the individual speeds of the molecules vary wildly, the distribution of these speeds is fixed by the temperature. There is no single "speed of the gas," but there is a most probable speed, an average speed, and a root-mean-square speed. These three distinct measures of speed all scale with the square root of temperature and inversely with the square root of mass. This statistical spread is what gives rise to the pressure exerted on the walls of the container: it is the cumulative effect of countless microscopic impacts, each carrying a momentum change determined by the molecule's velocity.

Single-Particle Partition Function

To begin, we consider a single particle of mass \( m \) confined to a volume \( V \). The energy of this particle is purely kinetic, given by \( E = \frac{1}{2}m v^2 = \frac{\mathbf{p}^2}{2m} \), where \( \mathbf{p} \) is the momentum vector. The canonical partition function \( Z_1 \) for this single particle is defined as the sum (or integral, in the classical limit) over all possible microstates. In phase space, a microstate is specified by a position \( \mathbf{r} \) and a momentum \( \mathbf{p} \). The integral over position simply yields the volume \( V \). The integral over momentum is a three-dimensional Gaussian integral.

Here, \( h \) is Planck's constant, introduced to make the partition function dimensionless and to account for the quantum nature of phase space cells, even in this classical approximation. The momentum integral separates into three identical one-dimensional integrals of the form \( \int_{-\infty}^{\infty} e^{-\beta p_x^2 / 2m} dp_x \). Using the standard Gaussian result \( \int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} \), with \( a = \frac{\beta}{2m} \), we find that each component contributes \( \sqrt{\frac{2\pi m}{\beta}} \). Multiplying the three components and the volume \( V \), we obtain:

This result can be written \( Z_1 = V/\lambda_{\mathrm{th}}^3 \), with thermal de Broglie wavelength \( \lambda_{\mathrm{th}} = h/\sqrt{2\pi m k_B T} \): the number of thermal-wavelength cells that fit in the box. It encapsulates the available phase space for a single particle at temperature \( T \).

Many-Particle System and Gibbs Correction

For \( N \) non-interacting particles, the total energy is the sum of individual energies. If the particles were distinguishable, the total partition function would simply be \( Z_N = (Z_1)^N \). However, in classical statistical mechanics, particles of the same species are indistinguishable. Counting permutations of identical particles leads to the Gibbs paradox, where entropy is not extensive. To correct for this overcounting, we divide by \( N! \). Thus, the canonical partition function for the ideal gas is:

Using Stirling's approximation \( \ln N! \approx N \ln N - N \) for large \( N \), we can calculate the Helmholtz free energy \( F = -k_B T \ln Z_N \). This yields the Sackur-Tetrode equation, which correctly predicts the extensive nature of entropy. More importantly for this lecture, the probability distribution of the momentum of any single particle is derived from the Boltzmann factor \( e^{-\beta p^2/2m} \). Since the momentum components \( p_x, p_y, p_z \) are independent, the joint probability density for the momentum vector is a product of three Gaussian distributions.

Deriving the Speed Distribution

We are interested in the speed \( v = |\mathbf{v}| \), not the velocity vector. The probability density for the velocity vector \( \mathbf{v} \) is proportional to \( e^{-\frac{1}{2}m v^2 / k_B T} \). To find the distribution of the scalar speed \( v \), we must integrate over the angular variables in velocity space. The volume element in spherical coordinates in velocity space is \( v^2 \sin\theta \, dv \, d\theta \, d\phi \). Integrating over the angles \( \theta \) and \( \phi \) gives a factor of \( 4\pi \). Therefore, the probability density function \( f(v) \) for the speed is proportional to \( v^2 e^{-\frac{m v^2}{2 k_B T}} \). Normalizing this distribution such that \( \int_0^\infty f(v) dv = 1 \) yields the Maxwell-Boltzmann speed distribution:

This function has a characteristic shape: it starts at zero for \( v=0 \) (because the volume of velocity space at low speeds is small), rises to a peak at the most probable speed \( v_p = \sqrt{\frac{2 k_B T}{m}} \), and then decays exponentially. The factor \( v^2 \) represents the increasing number of velocity states available at higher speeds, while the exponential term represents the decreasing probability of finding a particle with high kinetic energy. This distribution is universal for any classical ideal gas, depending only on the mass \( m \) and temperature \( T \).