PX284 - D2 - applications of the equipartition theorem

translational motion of a monatomic gas

considering a helium atom with three degrees of freedom:

E = \frac{1}{2} m v_{x}^{2} + \frac{1}{2} m v_{y}^{2} + \frac{1}{2} m v_{z}^{2}

∴ ⟨ E ⟩ = \frac{3}{2} k_{B} T

$C_{V} = \frac{3}{2} k_{B}$ per atom or $\frac{3}{2} N_{A} k_{B}$ per mole, where $N_{A}$ is avogrado's constant

mass on a spring

\begin{matrix} E = \frac{1}{2} k x^{2} + \frac{1}{2} m {\dot{x}}^{2} \\ ∴ ⟨ E ⟩ = k_{B} T \end{matrix}

two masses connected by a spring

\begin{matrix} E = \frac{1}{2} k (| {\vec{r}}_{1} - {\vec{r}}_{2} |^{2}) + \frac{1}{2} μ ({\dot{\vec{r}}}_{1} - {\dot{\vec{r}}}_{2})^{2} + \frac{1}{2} (m_{1} + m_{2}) {\dot{\vec{r}}}_{C O M}^{2} \\ ∴ ⟨ E ⟩ = \frac{3}{2} k_{B} T \end{matrix}

where, $μ = \frac{m_{1} m_{2}}{m_{1} + m_{2}}$

lattice vibrations

a 3D arrangement of masses and springs
for a cubic lattice, there are 3 springs per ion:

⟨ E ⟩ = 3 k_{B} T

$C_{V} = 3 k_{B}$ per atom or $2 N_{A} k_{B}$ per mole

diatomic molecules

PX154 - C9 - heat capacity of the ideal gas

translational motion:

\begin{matrix} E = \frac{1}{2} m v_{x}^{2} + \frac{1}{2} m v_{y}^{2} + \frac{1}{2} m v_{z}^{2} \\ ⟹ ⟨ E_{t r a n s} ⟩ = \frac{3}{2} k_{B} T \end{matrix}

vibrational motion:

⟨ E_{v i b} ⟩ \approx k_{B} T

rotational motion:

\begin{matrix} E_{r o t} = \frac{1}{2} \frac{L_{1}^{2}}{I_{1}} + \frac{1}{2} \frac{L_{2}^{2}}{I_{2}} \\ ⟨ E_{r o t} ⟩ = k_{B} T \end{matrix}

where, $L$ is the angular momentum, $I$ is the moment of inertia, and $1$ and $2$ represent rotation along the $z$ and the $y$ -axes respectively for a molecule along the $x$ -axis

the average energy of the molecule:

⟨ E ⟩ = ⟨ E_{t r a n s} ⟩ + ⟨ E_{v i b} ⟩ + ⟨ E_{r o t} ⟩ = \frac{3}{2} k_{B} T + k_{B} T + k_{B} T = \frac{7}{2} k_{B} T

$C_{V} = \frac{7}{2} k_{B}$ per molecule or $\frac{7}{2} N_{A} k_{B}$ per mole