Law of Equipartition of Energy — Explained

Detailed Explanation

The Law of Equipartition of Energy is a powerful concept derived from classical statistical mechanics, providing a fundamental understanding of how energy is distributed within a system of particles in thermal equilibrium.

It forms a crucial part of the Kinetic Theory of Gases, allowing us to connect the microscopic behavior of atoms and molecules to the macroscopic thermodynamic properties of gases, such as internal energy and specific heat capacities.

\n\n### Conceptual Foundation\nAt its core, the Kinetic Theory of Gases postulates that gases consist of a large number of rapidly moving particles (atoms or molecules) that are in constant, random motion.

The temperature of a gas is a direct measure of the average kinetic energy of these particles. When we talk about the internal energy of a gas, we are referring to the sum of all forms of energy possessed by its constituent particles, including translational kinetic energy, rotational kinetic energy, and vibrational energy.

The Law of Equipartition of Energy provides a systematic way to quantify this internal energy.\n\n### Key Principles and Laws\nStatement of the Law: The Law of Equipartition of Energy states that for a system in thermal equilibrium, the total energy is equally distributed among all its independent degrees of freedom, and the average energy associated with each degree of freedom is $1/2 k_B T$, where $k_B$ is Boltzmann's constant and $T$ is the absolute temperature of the system.

\n\n**Degrees of Freedom ($f$): A degree of freedom refers to an independent way a molecule can possess energy. These can be categorized into three types:\n1. Translational Degrees of Freedom ($f_t$):** These correspond to the motion of the molecule's center of mass along the three independent spatial axes (x, y, z).

Every molecule, regardless of its structure, possesses 3 translational degrees of freedom.\n * Energy associated with each translational degree of freedom: $1/2 m v_x^2$, $1/2 m v_y^2$, $1/2 m v_z^2$.

The average energy for each is $1/2 k_B T$.\n2. **Rotational Degrees of Freedom ($f_r$):** These correspond to the rotation of the molecule about axes passing through its center of mass. The number of rotational degrees of freedom depends on the molecule's geometry:\n * **Monoatomic gases (e.

g., He, Ne, Ar):** These are single atoms, considered as point masses. They have negligible moment of inertia about any axis passing through the atom itself. Hence, they have 0 rotational degrees of freedom.

\n * **Diatomic gases (e.g., O$_2$, N$_2$, H$_2$):** These molecules are linear. They can rotate about two axes perpendicular to the molecular axis. Rotation about the molecular axis itself has negligible moment of inertia.

Thus, they have 2 rotational degrees of freedom.\n * **Polyatomic gases (non-linear, e.g., H$_2$O, CH$_4$):** These molecules are non-linear. They can rotate about three mutually perpendicular axes. Thus, they have 3 rotational degrees of freedom.

\n * Energy associated with each rotational degree of freedom: $1/2 I_x \omega_x^2$, $1/2 I_y \omega_y^2$, $1/2 I_z \omega_z^2$. The average energy for each is $1/2 k_B T$.\n3. **Vibrational Degrees of Freedom ($f_v$):** These correspond to the oscillation of atoms within a molecule relative to each other.

Each vibrational mode contributes two degrees of freedom: one for kinetic energy and one for potential energy. Vibrational modes are generally 'excited' only at high temperatures because they require more energy.

At room temperature, for most diatomic and polyatomic gases, vibrational modes are considered 'frozen out' (not active). \n * For a molecule with N atoms, the total degrees of freedom are $3N$. \n * For linear molecules, $f_v = 3N - f_t - f_r = 3N - 3 - 2 = 3N - 5$.

\n * For non-linear molecules, $f_v = 3N - f_t - f_r = 3N - 3 - 3 = 3N - 6$.\n * Energy associated with each vibrational mode: $1/2 \mu (dr/dt)^2 + 1/2 k r^2$. The average energy for each vibrational mode (which has two degrees of freedom) is $2 \times (1/2 k_B T) = k_B T$.

\n\n### Derivations and Applications\nUsing the Law of Equipartition of Energy, we can calculate the total internal energy ($U$) of one mole of an ideal gas and subsequently its specific heat capacities.

\n\n**Internal Energy ($U$) for one mole of gas:**\nIf a molecule has $f$ degrees of freedom, the average energy per molecule is $f \times (1/2 k_B T)$. For one mole of gas, containing Avogadro's number ($N_A$) of molecules, the total internal energy is:\n

$$U = N_A \times f \times \frac{1}{2} k_B T = \frac{f}{2} (N_A k_B) T$$

\nSince $N_A k_B = R$ (the universal gas constant), the internal energy for one mole is:\n

$$U = \frac{f}{2} R T$$

\nFor $n$ moles of gas, $U = \frac{f}{2} nRT$.

\n\nSpecific Heat Capacities:\n1. **Molar Specific Heat at Constant Volume ($C_v$):** This is the heat required to raise the temperature of one mole of gas by $1 K$ at constant volume. From the first law of thermodynamics, $dU = dQ - dW$.

At constant volume, $dW = P dV = 0$, so $dQ = dU$. Thus, $C_v = (\frac{dU}{dT})_V$.\n

$$C_v = \frac{d}{dT} \left( \frac{f}{2} RT \right) = \frac{f}{2} R$$

\n2. **Molar Specific Heat at Constant Pressure ($C_p$):** This is the heat required to raise the temperature of one mole of gas by $1 K$ at constant pressure.

Using Mayer's relation, $C_p - C_v = R$.\n

$$C_p = C_v + R = \frac{f}{2} R + R = \left( \frac{f}{2} + 1 \right) R = \frac{f+2}{2} R$$

\n3. **Ratio of Specific Heats (Adiabatic Index, $\gamma$):**\n

$$\gamma = \frac{C_p}{C_v} = \frac{(f+2)/2 R}{f/2 R} = \frac{f+2}{f} = 1 + \frac{2}{f}$$

\n\nSummary of Degrees of Freedom and Specific Heats for Ideal Gases (at moderate temperatures where vibrations are frozen out):\n* **Monoatomic Gas (e.

g., He, Ne, Ar):**\n * $f_t = 3$, $f_r = 0$, $f_v = 0$. Total $f = 3$.\n * $U = \frac{3}{2} RT$\n * $C_v = \frac{3}{2} R$\n * $C_p = \frac{5}{2} R$\n * $\gamma = \frac{5}{3} \approx 1.67$\n* **Diatomic Gas (rigid rotor, e.

g., O$_2$, N$_2$, H$_2$ at room temperature):**\n * $f_t = 3$, $f_r = 2$, $f_v = 0$. Total $f = 5$.\n * $U = \frac{5}{2} RT$\n * $C_v = \frac{5}{2} R$\n * $C_p = \frac{7}{2} R$\n * $\gamma = \frac{7}{5} = 1.

4$\n* **Diatomic Gas (non-rigid rotor, with vibrations, e.g., at high temperatures):**\n *$f_t = 3$,$f_r = 2$,$f_v = 2$(one vibrational mode). Total$f = 7$.\n *$U = \frac{7}{2} RT$\n *$C_v = \frac{7}{2} R$\n *$C_p = \frac{9}{2} R$\n *$\gamma = \frac{9}{7} \approx 1.