12.5. LAW OF EQUIPARTITION OF ENERGY

The kinetic energy of a single molecule is

$\epsilon = \frac{1}{2} mv_x^2 + \frac{1}{2} mv_y^2 + \frac{1}{2} mv_z^2$ (12.22)

The kinetic energy ($\epsilon$) of a molecule is given by the formula where $m$ is the mass of the molecule and $v_x$, $v_y$, and $v_z$ are the velocities of the molecule in the x, y, and z directions, respectively. This equation indicates how energy is distributed in a molecule's movement across three dimensions.

Imagine a basketball moving in a three-dimensional space. The way the ball moves forward, sideways, and upwards corresponds to its different velocity components (x, y, z). The faster it rolls in those directions, the more kinetic energy it has.

For a gas in thermal equilibrium at temperature T, the average value of energy denoted by ⟨ε⟩ is

$\langle\epsilon\rangle = \frac{3}{2} k_B T$ (12.23)

In thermal equilibrium, the average kinetic energy ($\langle\epsilon\rangle$) of molecules in a gas is directly proportional to the temperature ($T$) multiplied by Boltzmann's constant ($k_B$). This relationship shows that as temperature increases, the average kinetic energy of the gas molecules also increases, thus reflecting more vigorous motion.

Think about how the temperature of a pot of water increases as you heat it. The greater the heat (or temperature), the more vigorously the water molecules move around, demonstrating the direct relationship between temperature and kinetic energy.

A molecule free to move in space needs three coordinates to specify its location. This can also be expressed in another way. We say that it has one degree of freedom for motion in a line, two for motion in a plane, and three for motion in space (12.24)

Degrees of freedom refer to the number of independent ways in which a system can move. For a molecule in three-dimensional space, it has three degrees of freedom corresponding to its ability to move in three perpendicular directions (x, y, z). This concept helps in understanding how the energy is distributed in a gas.

Consider a bird flying in an open sky. It can move forward and backward (along the x-axis), up and down (along the y-axis), and turn around (along the z-axis). Each of these movements represents a degree of freedom.

Molecules of a monatomic gas like argon have only translational degrees of freedom. But what about a diatomic gas such as O2 or N2? A molecule of O2 has three translational degrees of freedom and can also rotate about its center of mass (12.25)

Monatomic gases possess only translational degrees of freedom, meaning they can only move in space. Diatomic gases, such as oxygen (O2), have both translational and rotational degrees of freedom, allowing them to move as well as rotate. This leads to different contributions to the energy of the molecule.

Visualize a single solid ball representing a monatomic gas which can only roll along the ground versus a toy model of a figure-eight roller coaster representing diatomic gas that can roll and also twist or turn in different directions. The complexity of motion increases with more degrees of freedom.

One vibrational mode contributes two 'squared terms': kinetic and potential energies (12.26)

In addition to translational and rotational modes, some molecules can vibrate, creating a vibrational energy mode. For each vibrational mode, since there are motions in both kinetic and potential forms, it contributes more to the overall energy, effectively making it higher than other modes.

Think of a guitar string. As you pluck it, the string vibrates back and forth (kinetic energy), and also gets pulled taut and relaxed (potential energy). The energy involved in both the stretching and moving of the string reflects the dual contribution of vibrational energy modes.

In equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having an average energy equal to ½ k_BT. This is known as the law of equipartition of energy (12.26)

The law of equipartition of energy states that in thermal equilibrium, energy is distributed equally among all available degrees of freedom of a system. Each degree of freedom contributes an energy of ½ k_B T, which reflects energy conservation and distribution principles in statistical mechanics.

Imagine a classroom during a group project. If each group member is contributing equally to the task (equilibrium), everyone’s contribution would be similar. If one person works much harder, others would need to catch up to balance the contributions just as energy balances out in physical systems.

Accordingly, each translational and rotational degree of freedom contributes ½ k_B T to the energy, while each vibrational frequency contributes k_B T (12.26)

Each degree of freedom for translation and rotation contributes an energy amount of ½ k_B T, while each vibrational degree contributes k_B T due to both kinetic and potential contributions. This helps in calculating specific heats for different types of gases based on their molecular structure.

Think of a cooking pot on a stove. The heat adds energy, causing the water to move (kinetic energy) and vibrate (potential energy). If you think of each movement type as contributing differently to cooking, it mirrors how different motions contribute to the energy of molecules.

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