Kinetic Theory of Gases and Degree of Freedom

The kinetic theory of gases is a fundamental theory in physics and chemistry that explains the behavior of gases by considering them as a large number of small particles (atoms or molecules), all of which are in constant, random motion. The theory provides a microscopic explanation for macroscopic phenomena such as pressure, temperature, and volume of a gas.

Basic Assumptions of the Kinetic Theory of Gases

The kinetic theory of gases is based on several key assumptions:

1. Gas consists of a large number of particles: These particles are molecules or atoms which are in constant motion.
2. Particle size is negligible: The actual volume of the gas particles is much smaller than the volume occupied by the gas.
3. Particles are in constant, random motion: This motion leads to collisions between particles and with the walls of the container.
4. Collisions are perfectly elastic: This means that there is no net loss of kinetic energy when particles collide.
5. There are no intermolecular forces: Except during collisions, the particles do not exert forces on each other.
6. The average kinetic energy of the particles is proportional to the temperature: This is given in absolute temperature (Kelvin scale).

Degree of Freedom

The degree of freedom of a gas molecule refers to the number of independent ways in which it can possess energy. The concept is crucial in understanding the distribution of energy in a gas.

Types of Degrees of Freedom

1. Translational: Movement of the center of mass of the molecule in space. There are three translational degrees of freedom, corresponding to motion along the x, y, and z axes.
2. Rotational: Rotation of the molecule about its center of mass. Non-linear molecules have three rotational degrees of freedom, while linear molecules have two.
3. Vibrational: Internal vibrations of the atoms within a molecule. Each vibrational mode has two degrees of freedom, one for kinetic energy and one for potential energy.

Equipartition Theorem

The equipartition theorem states that the total energy of a system is equally distributed among its degrees of freedom. For each degree of freedom, the average energy is given by:

$$ \langle E \rangle = \frac{1}{2} k_B T $$

where $\langle E \rangle$ is the average energy per degree of freedom, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature.

Formulas in the Kinetic Theory of Gases

The kinetic theory of gases allows us to derive several important formulas:

1. Pressure: The pressure exerted by a gas is due to collisions of particles with the walls of the container.

$$ P = \frac{1}{3} \frac{Nm\bar{v}^2}{V} $$

where $P$ is the pressure, $N$ is the number of molecules, $m$ is the mass of a molecule, $\bar{v}^2$ is the mean square velocity, and $V$ is the volume of the gas.

1. Average Kinetic Energy: The average kinetic energy of a gas particle is related to the temperature.

$$ \langle KE \rangle = \frac{3}{2} k_B T $$

for a monatomic gas with three translational degrees of freedom.

1. Root Mean Square Speed: The root mean square speed of the gas particles can be found using the formula:

$$ v_{rms} = \sqrt{\frac{3k_B T}{m}} $$

1. Ideal Gas Law: The ideal gas law is a combination of Boyle's law, Charles's law, and Avogadro's law.

$$ PV = nRT $$

where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the ideal gas constant, and $T$ is the temperature.

Examples

Example 1: Calculating Pressure

Consider a container with a volume of $1 \text{ m}^3$ containing $2 \times 10^{23}$ helium atoms at a temperature of $300 \text{ K}$. The mass of a helium atom is approximately $4 \times 10^{-27} \text{ kg}$. Calculate the pressure exerted by the helium gas.

Using the formula for pressure:

$$ P = \frac{1}{3} \frac{Nm\bar{v}^2}{V} $$

We need to find $\bar{v}^2$, which is related to the temperature:

$$ \bar{v}^2 = \frac{3k_B T}{m} $$

Substituting the values:

$$ P = \frac{1}{3} \frac{(2 \times 10^{23})(4 \times 10^{-27} \text{ kg})\left(\frac{3(1.38 \times 10^{-23} \text{ J/K})(300 \text{ K})}{4 \times 10^{-27} \text{ kg}}\right)}{1 \text{ m}^3} $$

After calculating, we find the pressure $P$.

Example 2: Degrees of Freedom

Calculate the average energy per degree of freedom for nitrogen gas ($N_2$) at $300 \text{ K}$.

Nitrogen is a diatomic molecule, so it has 5 degrees of freedom at room temperature (3 translational and 2 rotational). Using the equipartition theorem:

$$ \langle E \rangle = \frac{1}{2} k_B T $$

For each degree of freedom, the energy is:

$$ \langle E \rangle = \frac{1}{2} (1.38 \times 10^{-23} \text{ J/K})(300 \text{ K}) $$

Multiplying by the number of degrees of freedom (5 for $N_2$), we get the total average energy for a nitrogen molecule.

Table: Degrees of Freedom for Different Molecules

Molecule Type	Translational	Rotational	Vibrational	Total
Monatomic	3	0	0	3
Diatomic	3	2	0*	5
Non-linear	3	3	0*	6

* At room temperature, vibrational degrees of freedom are typically not excited. At higher temperatures, vibrational modes may become significant.

In conclusion, the kinetic theory of gases and the concept of degrees of freedom provide a detailed understanding of the microscopic behavior of gas particles and how it relates to macroscopic properties like pressure and temperature. These principles are fundamental to thermodynamics and statistical mechanics.