26 May 2025

Maxwell Distribution, Equipartition, and Specific Heats

Equilibrium molecular speeds, active degrees of freedom, equipartition, and ideal-gas heat capacities.

bsc semester-iv mj-6 heat-and-thermodynamics kinetic-theory

Consider a dilute gas of identical, non-interacting molecules of mass $m$ in thermal equilibrium at absolute temperature $T$. Collisions randomize direction, so the velocity distribution is isotropic. The Cartesian components are also statistically independent. Boltzmann’s constant is $k_{\mathrm B}$ and $R=N_{\mathrm A}k_{\mathrm B}$.

Maxwell-Boltzmann velocity distribution

Independence gives

\[F(v_x,v_y,v_z)=\phi(v_x)\phi(v_y)\phi(v_z),\]

where $F\,dv_xdv_ydv_z$ is the probability of finding a molecule in the indicated velocity-space element. Isotropy requires $F$ to depend on the components only through

\[v^2=v_x^2+v_y^2+v_z^2.\]

The product of three one-variable functions can depend only on this sum if each factor is Gaussian. Hence

\[F=A\exp[-\alpha(v_x^2+v_y^2+v_z^2)].\]

Normalization and $\langle \tfrac12mv_x^2\rangle=\tfrac12k_{\mathrm B}T$ determine $A$ and $\alpha$:

\[\boxed{F(v_x,v_y,v_z)= \left(\frac{m}{2\pi k_{\mathrm B}T}\right)^{3/2} \exp\!\left(-\frac{mv^2}{2k_{\mathrm B}T}\right)}.\]

$F$ has units $(\mathrm{m\,s^{-1}})^{-3}$. Velocities with magnitude between $v$ and $v+dv$ occupy the spherical shell $4\pi v^2dv$ in velocity space. The speed probability density is therefore

\[\boxed{f(v)=4\pi\left(\frac{m}{2\pi k_{\mathrm B}T}\right)^{3/2} v^2\exp\!\left(-\frac{mv^2}{2k_{\mathrm B}T}\right)}, \qquad v\geq0,\]

with $\int_0^\infty f(v)\,dv=1$ and units $(\mathrm{m\,s^{-1}})^{-1}$.

For the nonzero maximum, $df/dv=0$ gives

\[\frac{df}{dv}\propto 2v\left(1-\frac{mv^2}{2k_{\mathrm B}T}\right) e^{-mv^2/(2k_{\mathrm B}T)},\]

so the most probable speed is

\[v_{\mathrm{mp}}=\sqrt{\frac{2k_{\mathrm B}T}{m}}.\]

Using

\[\int_0^\infty x^n e^{-ax^2}\,dx =\frac12a^{-(n+1)/2}\Gamma\!\left(\frac{n+1}{2}\right)\]

gives the mean and root-mean-square speeds:

\[\bar v=\int_0^\infty vf(v)\,dv =\sqrt{\frac{8k_{\mathrm B}T}{\pi m}},\] \[\langle v^2\rangle=\int_0^\infty v^2f(v)\,dv =\frac{3k_{\mathrm B}T}{m}, \qquad v_{\mathrm{rms}}=\sqrt{\frac{3k_{\mathrm B}T}{m}}.\]

Thus $v_{\mathrm{mp}}<\bar v<v_{\mathrm{rms}}$, and every characteristic speed varies as $\sqrt{T/m}$.

Degrees of freedom and equipartition

A degree of freedom is an independent coordinate needed to specify molecular motion. In classical equilibrium, every independent quadratic energy term contributes mean energy $k_{\mathrm B}T/2$ per molecule. For example,

\[\epsilon_{\mathrm{tr}}= \frac{p_x^2}{2m}+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}\]

contains three quadratic terms and contributes $3k_{\mathrm B}T/2$. A linear molecule has two rotational quadratic terms; a nonlinear molecule has three. Each vibrational normal mode has one kinetic and one potential quadratic term, so an active vibrational mode contributes $k_{\mathrm B}T$.

If $g$ quadratic terms are active, the mean molecular energy and the internal energy of $n$ moles are

\[\bar\epsilon=\frac{g}{2}k_{\mathrm B}T, \qquad U=N\bar\epsilon=\frac{g}{2}nRT.\]

For an ideal gas, $U$ depends only on $T$. Therefore

\[C_V=\left(\frac{\partial U}{\partial T}\right)_V =\frac{g}{2}nR.\]

Since $H=U+pV=U+nRT$,

\[C_P=\left(\frac{\partial H}{\partial T}\right)_p =C_V+nR=\frac{g+2}{2}nR, \qquad \gamma=\frac{C_P}{C_V}=\frac{g+2}{g}.\]

For one mole, a monatomic gas has $g=3$ and

\[C_{V,m}=\frac32R,\qquad C_{P,m}=\frac52R,\qquad \gamma=\frac53.\]

A rigid diatomic gas at ordinary temperatures has $g=5$ and

\[C_{V,m}=\frac52R,\qquad C_{P,m}=\frac72R,\qquad \gamma=\frac75.\]

Molar heat capacities have units $\mathrm{J\,mol^{-1}K^{-1}}$; division by molar mass gives mass-specific heat in $\mathrm{J\,kg^{-1}K^{-1}}$. A rotational or vibrational mode whose energy spacing is much larger than $k_{\mathrm B}T$ is effectively frozen and does not contribute its classical value.

The classical distribution requires non-relativistic particles, dilute-gas conditions, equilibrium, and negligible quantum degeneracy. Its $T\to0$ limit is therefore not physical: quantum statistics and discrete rotational or vibrational levels enter before that limit. Normalization is dimensionless, while every $r$th speed moment has units $(\mathrm{m\,s^{-1}})^r$.

Solved Problems

1. Obtain a general moment of the Maxwell speed distribution

Put $A=m/(2k_{\mathrm B}T)$. The normalized density becomes

\[f(v)=\frac{4}{\sqrt\pi}A^{3/2}v^2e^{-Av^2}.\]

For $r>-3$, use the Gaussian integral stated above:

\[\begin{aligned} \langle v^r\rangle &=\frac{4}{\sqrt\pi}A^{3/2} \int_0^\infty v^{r+2}e^{-Av^2}\,dv\\ &=\frac{2}{\sqrt\pi}A^{-r/2} \Gamma\!\left(\frac{r+3}{2}\right). \end{aligned}\]

For $r=0$ this gives $1$, checking normalization. For $r=1$ and $2$ it gives $\bar v=2/(\sqrt{\pi A})$ and $\langle v^2\rangle=3/(2A)$. Thus

\[v_{\mathrm{mp}}^2:\bar v^2:v_{\mathrm{rms}}^2 =2:\frac8\pi:3,\]

so $v_{\mathrm{mp}}<\bar v<v_{\mathrm{rms}}$. The factor $A^{-r/2}$ supplies the required speed-to-the-$r$ dimensions.

2. Heat capacities when only two vibration modes of a linear triatomic molecule are active

A linear triatomic molecule has three translations, two rotations, and $3N-5=4$ normal vibration modes. Suppose only two vibration modes satisfy $k_{\mathrm B}T$ large compared with their level spacings. Each active vibration supplies two quadratic terms, so

\[g=3+2+2(2)=9.\]

For one mole,

\[C_{V,m}=\frac92R,\qquad C_{P,m}=C_{V,m}+R=\frac{11}{2}R,\]

and therefore

\[\boxed{\gamma=\frac{C_{P,m}}{C_{V,m}}=\frac{11}{9}}.\]

The result lies between the rigid-molecule value $7/5$ and the value obtained when all four vibrations are active. It assumes ideal-gas behaviour and classical equipartition for the stated active modes.

Descriptive Questions

1. Why do isotropy and statistical independence force the Cartesian velocity distributions to be Gaussian?
2. Distinguish a velocity distribution from a speed distribution, including their domains and units.
3. Explain why a vibrational normal mode contributes twice as much classical energy as one translational coordinate.
4. Why can measured heat capacities depart from equipartition predictions at low temperature?

Numerical Problems

1. Find the most probable speed of oxygen molecules at $300\ \mathrm K$, taking $M=0.032\ \mathrm{kg\,mol^{-1}}$.

   Final answer: $v_{\mathrm{mp}}=\sqrt{2RT/M}=394.8\ \mathrm{m\,s^{-1}}$.

2. Find the rms speed of helium at $400\ \mathrm K$ for $M=0.004\ \mathrm{kg\,mol^{-1}}$.

   Final answer: $v_{\mathrm{rms}}=1.579\times10^3\ \mathrm{m\,s^{-1}}$.

3. Two moles of a monatomic ideal gas are heated through $50\ \mathrm K$ at constant volume. Calculate $\Delta U$.

   Final answer: $\Delta U=(3/2)nR\Delta T=1.247\times10^3\ \mathrm J$.

4. A classical ideal gas has $\gamma=1.40$. Infer its number $g$ of active quadratic degrees of freedom.

   Final answer: $g=2/(\gamma-1)=5$.

The Maxwell and equipartition Maxima worksheet verifies normalization, the first two moments, heat-capacity identities, and every numerical answer above.

References

1. Maxwell-Boltzmann distribution, Wikipedia.
2. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965, chapters 6-7.
3. F. W. Sears and G. L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, 3rd ed., Addison-Wesley, 1975, chapters 9-10.