Kinetic Theory

Learning Objectives

• Understand the Kinetic Theory of Gases and how it explains the behavior of gases through rapidly moving atoms or molecules, providing a molecular interpretation of pressure and temperature.
• Calculate and apply the Ideal Gas Law equation $PV = \mu RT = Nk_B T$, relating pressure, volume, and temperature of an ideal gas, where $\mu$ is the number of moles, $N$ is the number of molecules, $R$ is the universal gas constant, and $k_B$ is Boltzmann's constant.
• Apply the Law of Equipartition of Energy to determine that in thermal equilibrium, the total energy is distributed equally among all possible energy modes, with each mode having an average energy of $\frac{1}{2}k_B T$.
• Compute the Mean Free Path using the formula $l = \frac{1}{\sqrt{2\pi nd^2}}$, where $n$ is the number density and $d$ is the molecular diameter.
• Determine the Specific Heat Capacities of Gases using the law of equipartition of energy, with formulas such as $C_v = \frac{3}{2}R$ for monatomic gases and $C_v = \frac{5}{2}R$ for diatomic gases.
• Interpret the Kinetic Interpretation of Temperature which relates the average kinetic energy of gas molecules to temperature, showing that temperature is a measure of the average kinetic energy.
• Derive the Pressure of an Ideal Gas from kinetic theory, using the formula $P = \frac{1}{3}nmv^2$, where $n$ is the number density, $m$ is the molecular mass, and $v^2$ is the mean squared speed.
• Analyze Gas Laws and Real Gas Behaviour, including Boyle’s law, Charles’ law, and deviations from ideal behavior at high pressure/low temperature.
• Apply Avogadro Hypothesis and Dalton’s Law of Partial Pressures to understand that equal volumes of gases at the same temperature and pressure contain equal molecules and that the mixture pressure is the sum of partial pressures.
• Calculate RMS Speed and Molecular Speed using $v_{rms} = \sqrt{\frac{3k_B T}{m}}$ and compare speeds of different gases.
• Explore the Degrees of Freedom of Molecules for translational, rotational, and vibrational modes in monoatomic, diatomic, and polyatomic molecules.

Chapter Notes

Kinetic Theory of Gases

• Kinetic Theory: Describes gases as composed of rapidly moving atoms or molecules. It provides a molecular interpretation of pressure and temperature and aligns with gas laws and Avogadro's hypothesis.
• Molecular Interpretation: Pressure and temperature are explained through molecular motion, consistent with gas laws.

Ideal Gas Law

• Equation: $PV = \mu RT = Nk_B T$
  • $P$: Pressure
  • $V$: Volume
  • $\mu$: Number of moles
  • $N$: Number of molecules
  • $R$: Universal gas constant
  • $k_B$: Boltzmann's constant
• Constants: $R = 8.314 \, \text{J mol}^{-1} \text{K}^{-1}$, $k_B = 1.38 \times 10^{-23} \, \text{J K}^{-1}$

Law of Equipartition of Energy

• Statement: In thermal equilibrium, energy is equally distributed among all possible energy modes, with each mode having an average energy of $\frac{1}{2}k_B T$.
• Applications: Applies to translational, rotational, and vibrational modes.

Mean Free Path

• Definition: Average distance between successive molecular collisions.
• Formula: $l = \frac{1}{\sqrt{2} \pi n d^2}$
  • $n$: Number density
  • $d$: Molecular diameter

Specific Heat Capacities of Gases

• Monatomic Gases: $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$
• Diatomic Gases: $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$
• Polyatomic Gases: Depend on the number of vibrational modes.

Kinetic Interpretation of Temperature

• Relation: Average kinetic energy of gas molecules is proportional to temperature.
• Formula: $\frac{1}{2}mv^2 = \frac{3}{2}k_B T$

Pressure of an Ideal Gas

• Derived Formula: $P = \frac{1}{3}nmv^2$
  • $n$: Number density
  • $m$: Molecular mass
  • $v^2$: Mean squared speed

Gas Laws and Real Gas Behaviour

• Boyle’s Law: $PV = \text{constant}$ at constant temperature.
• Charles’ Law: $V \propto T$ at constant pressure.
• Real Gases: Deviate from ideal behavior at high pressure/low temperature.

Avogadro Hypothesis and Dalton’s Law of Partial Pressures

• Avogadro's Hypothesis: Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
• Dalton’s Law: Total pressure of a gas mixture is the sum of partial pressures of individual gases.

RMS Speed and Molecular Speed Calculations

• RMS Speed: $v_{rms} = \sqrt{\frac{3k_B T}{m}}$

Degrees of Freedom of Molecules

• Monatomic: 3 translational
• Diatomic: 3 translational, 2 rotational
• Polyatomic: 3 translational, 3 rotational, vibrational modes depend on the molecule