Complete Kinetic Theory of Gases Summary

Kinetic Pressure, Temperature Interpretation, Gas Speeds, Internal Energy, and Specific Heats

Kinetic Theory
Gas Speeds
Internal Energy
Specific Heats
Gas Mixtures

1. Kinetic Theory of Gases

Kinetic Pressure
\[P = \frac{1}{3} \rho v_{rms}^2\]

where:

  • \(P\) = pressure of the gas
  • \(\rho\) = density of the gas
  • \(v_{rms}\) = root mean square speed of the molecules
Kinetic Interpretation of Temperature

From the point of view of kinetic theory, the average kinetic energy of a molecule is:

\[K_{av} = \frac{1}{2} m v_{rms}^2 = \frac{3}{2} kT\]

For an ideal gas, the absolute temperature is a measure of the average translational kinetic energy of the molecules.

Where:

  • \(K_{av}\) = average kinetic energy per molecule
  • \(m\) = mass of a single molecule
  • \(k\) = Boltzmann constant = \(1.38 \times 10^{-23} \, \text{J/K}\)
  • \(T\) = absolute temperature

2. Different Speeds of Gas Molecules

The motion of molecules in a gas is characterized by any of the following three speeds.

(a) Root Mean Square Speed (\(v_{rms}\))
\[v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + ... + v_N^2}{N}}\]

Alternative expressions:

\[v_{rms} = \sqrt{\frac{3PV}{\text{mass of gas}}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3P}{\rho}}\]

Where:

  • \(M\) = molecular mass
  • \(m\) = mass of a single molecule
  • \(\rho\) = density of gas
  • \(R\) = universal gas constant
  • \(k\) = Boltzmann constant
(b) Most Probable Speed (\(v_{mp}\))
\[v_{mp} = \sqrt{\frac{2RT}{M}} = \sqrt{\frac{2kT}{m}} = \sqrt{\frac{2P}{\rho}}\]

This is the speed at which the maximum number of molecules are moving.

(c) Average Speed (\(v_{av}\))
\[v_{av} = \frac{v_1 + v_2 + ... + v_N}{N}\]

Alternative expressions:

\[v_{av} = \sqrt{\frac{8RT}{\pi M}} = \sqrt{\frac{8kT}{\pi m}} = \sqrt{\frac{8P}{\pi \rho}}\]

Note: The formula in the PDF appears to have a typo with extra \(\pi\) in numerator.

2.1 Relationship Between Different Speeds

\[v_{mp} : v_{av} : v_{rms} = 1 : \sqrt{\frac{8}{3\pi}} : \sqrt{\frac{3}{2}} \approx 1 : 1.128 : 1.224\]

Key Points:

  • For a given gas at a fixed temperature: \(v_{mp} < v_{av} < v_{rms}\)
  • All three speeds are directly proportional to \(\sqrt{T}\) and inversely proportional to \(\sqrt{M}\)
  • Lighter gas molecules move faster than heavier ones at the same temperature

3. Internal Energy of an Ideal Gas

For an ideal gas, the internal energy \(U\) depends upon temperature \(T\) only and is directly proportional to it:

\[U \propto T\]

3.1 Degrees of Freedom

Degree of freedom (\(f\)) is defined as the number of possible independent ways in which a system can have energy. The independent motions can be translational, rotational, or vibrational or any combination of them.

Type of Molecule Degrees of Freedom Explanation
Monatomic 3 All translational (x, y, z directions). Rotational energy is insignificant due to small moment of inertia.
Diatomic 5 3 translational + 2 rotational. Examples: H₂, O₂, N₂
Polyatomic 6 or more 3 translational + 3 rotational + vibrational modes

3.2 Law of Equipartition of Energy

According to this law, the energy of an ideal gas is equally distributed in each degree of freedom.

  • The average kinetic energy per degree of freedom per molecule is \(\frac{1}{2}kT\)
  • The kinetic energy per degree of freedom per mole is \(\frac{1}{2}RT\)
  • In general, the kinetic energy per molecule is \(\frac{1}{2}fkT\) and that per mole is \(\frac{1}{2}fRT\)

Example: For a monatomic gas with 3 degrees of freedom:

\[U = \frac{3}{2}RT \quad \text{(per mole)}\]

4. Specific Heat and Heat Capacity

Heat capacity is the amount of heat required to raise the temperature of a substance by 1°C (or 1K):

\[C = \frac{Q}{\Delta T}\]

The SI unit of heat capacity is JK⁻¹.

Specific heat is the heat capacity per unit mass:

\[Q = mc\Delta T\]

where \(c\) is called the specific heat of the substance.

4.1 Molar Specific Heats of an Ideal Gas

Molar Specific Heat at Constant Volume (\(C_V\))
\[C_V = \frac{dU}{dT} = \frac{R}{\gamma - 1}\]

Where:

  • \(U\) = internal energy
  • \(R\) = universal gas constant
  • \(\gamma\) = adiabatic exponent (\(C_P/C_V\))
Molar Specific Heat at Constant Pressure (\(C_P\))
\[C_P = C_V + R\]

This relationship is known as Mayer's relation.

4.2 Adiabatic Exponent

\[\gamma = \frac{C_P}{C_V}\]

4.3 Values for Different Types of Gases

Type of Gas Degrees of Freedom (f) \(C_V\) \(C_P\) \(\gamma\)
Monatomic 3 \(\frac{3}{2}R\) \(\frac{5}{2}R\) \(\frac{5}{3} \approx 1.67\)
Diatomic 5 \(\frac{5}{2}R\) \(\frac{7}{2}R\) \(\frac{7}{5} = 1.4\)
Polyatomic 6 \(3R\) \(4R\) \(\frac{4}{3} \approx 1.33\)

5. Mixture of Gases

Following results are helpful for a mixture of gases with \(n_1, n_2, \ldots\) moles of different gases.

5.1 Properties of Gas Mixtures

\[U_{mix} = U_1 + U_2 + \ldots\]

Total internal energy of the mixture equals the sum of internal energies of individual gases.

\[M_{mix} = \frac{n_1 M_1 + n_2 M_2 + \ldots}{n_1 + n_2 + \ldots}\]

Effective molar mass of the mixture.

\[P_{mix} = P_1 + P_2 + \ldots\]

Dalton's law of partial pressures: Total pressure equals sum of partial pressures.

5.2 Specific Heats of Mixtures

\[(C_V)_{mix} = \frac{n_1 C_{V1} + n_2 C_{V2} + \ldots}{n_1 + n_2 + \ldots}\]

Molar specific heat at constant volume for the mixture.

\[(C_P)_{mix} = (C_V)_{mix} + R\]

Mayer's relation applies to mixtures as well.

\[\gamma_{mix} = \frac{(C_P)_{mix}}{(C_V)_{mix}}\]

Adiabatic exponent for the mixture.

\[\frac{n_1 + n_2 + n_3 + \ldots}{\gamma_{mix} - 1} = \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} + \ldots\]

Alternative formula to calculate \(\gamma_{mix}\).