Kinetic Theory of Gases — MHT-CET Physics

Formulas (2)

• One Law at a Time · Combined gas law
  $$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$
• PV = nRT and Counting Molecules · Ideal gas equation
  $$PV = nRT = NkT, \qquad \rho = \frac{PM}{RT}$$

Watch out for (3)

• Equal percentages in Boyle's law
  → One Law at a Time
• Using degrees Celsius in P ∝ T
  → One Law at a Time
• Leaving T out of a molecule count
  → PV = nRT and Counting Molecules

Formulas (3)

• Pressure as Two Thirds of the Kinetic Energy Density · Kinetic pressure
  $$P = \frac{1}{3}\rho\langle v^2\rangle = \frac{2}{3}\frac{E}{V}$$
• R.M.S. Speed, Temperature and Molar Mass · R.M.S. speed
  $$v_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$$
• R.M.S. Speed Through an Adiabatic Expansion · Adiabatic cooling
  $$\frac{V_2}{V_1} = \left(\frac{T_1}{T_2}\right)^{1/(\gamma - 1)}, \qquad \frac{T_1}{T_2} = \left(\frac{v_1}{v_2}\right)^2$$

Watch out for (5)

• Writing P = ⅓ of the energy density
  → Pressure as Two Thirds of the Kinetic Energy Density
• Heating a gas at constant volume changes its mean free path
  → Pressure as Two Thirds of the Kinetic Energy Density
• Scaling speed with temperature in °C
  → R.M.S. Speed, Temperature and Molar Mass
• Scaling speed with T instead of √T
  → R.M.S. Speed, Temperature and Molar Mass
• Using the speed ratio as the temperature ratio
  → R.M.S. Speed Through an Adiabatic Expansion

Formulas (2)

• Average Kinetic Energy Depends Only on Temperature · Average kinetic energy
  $$\bar{E} = \frac{3}{2}kT, \qquad \text{each degree of freedom } \tfrac{1}{2}kT$$
• Degrees of Freedom, C_v, C_p and γ · Equipartition
  $$C_v = \frac{f}{2}R, \qquad C_p = C_v + R, \qquad \gamma = 1 + \frac{2}{f}$$

Watch out for (4)

• Letting the heavier gas have more kinetic energy
  → Average Kinetic Energy Depends Only on Temperature
• Halving the Celsius temperature
  → Average Kinetic Energy Depends Only on Temperature
• Averaging the specific heats of a mixture by gas, not by mole
  → Degrees of Freedom, C_v, C_p and γ
• Counting one vibrational degree of freedom
  → Degrees of Freedom, C_v, C_p and γ