Energy Distribution: 18 MCQs with answers

The distribution of particle speeds in a gas at thermal equilibrium

The Maxwell-Boltzmann distribution gives the probability distribution of speeds (and hence kinetic energies) of particles in a classical ideal gas at thermal equilibrium.

$e^{-E/k_B T}$

The Maxwell-Boltzmann distribution has the Boltzmann factor $e^{-E/k_B T}$, which gives the relative probability of a state with energy $E$ at temperature $T$.

Classical, distinguishable particles

Maxwell-Boltzmann statistics applies to classical particles that are distinguishable and do not obey quantum exclusion or condensation.

It applies to ideal gases in thermal equilibrium

The distribution describes the statistical spread of speeds in a classical ideal gas at thermal equilibrium, not a single speed.

$\sqrt{\frac{2k_B T}{m}}$

The most probable speed is found by maximizing the speed distribution function, yielding $v_p = \sqrt{2k_B T/m}$.

$\sqrt{\frac{8k_B T}{\pi m}}$

The average speed is calculated by integrating $v$ times the speed distribution, giving $\bar{v} = \sqrt{8k_B T/(\pi m)}$.

$\sqrt{\frac{3k_B T}{m}}$

The RMS speed is derived from the mean square speed, which is $\langle v^2 \rangle = 3k_B T/m$, so $v_{rms} = \sqrt{3k_B T/m}$.

The peak shifts to higher speeds and the distribution broadens

Higher temperature increases the typical kinetic energy, so the most probable speed increases and the spread of speeds widens.

$\sqrt{E}$

In 3D, the number of states with energy between $E$ and $E+dE$ scales as $\sqrt{E}$, leading to the $\sqrt{E}$ factor in the energy distribution.

$\sqrt{E} \, e^{-E/k_B T}$

The energy distribution is the product of the density of states ($\propto \sqrt{E}$) and the Boltzmann factor ($e^{-E/k_B T}$).

It is symmetric about the most probable speed

The distribution is skewed (asymmetric) with a longer tail toward higher speeds, not symmetric about the peak.

A higher most probable speed and a broader distribution

Since $v_p \propto 1/\sqrt{m}$, lighter molecules have higher typical speeds, and the distribution spreads over a wider range.

Integrating the speed distribution from $v_0$ to infinity

The fraction is the area under the probability density curve from $v_0$ to $\infty$, which requires integration.

$\frac{3}{2} k_B T$

Equipartition theorem gives $\frac{1}{2}k_B T$ per degree of freedom; with 3 translational degrees, the average kinetic energy is $\frac{3}{2}k_B T$.

False

At low T and high density, quantum effects become significant, and Fermi-Dirac or Bose-Einstein statistics are needed; MB is a classical approximation.

False

The RMS speed is always greater than the most probable speed because the distribution has a high-speed tail that increases the mean square value.

The gas is dilute and the temperature is high

When the average interparticle distance is much larger than the thermal de Broglie wavelength (dilute gas, high T), quantum effects are negligible and MB statistics applies.

The Boltzmann transport equation in equilibrium

In equilibrium, the Boltzmann transport equation reduces to the Maxwell-Boltzmann distribution as the stationary solution for a dilute gas.