Kinetic Theory — JEE Mains Physics

Formulas (3)

• One fixed amount of gas: P₁V₁/T₁ = P₂V₂/T₂ · Fixed amount of gas
  $$PV = nRT = NkT \qquad \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$
• Counting moles: mixtures and joined vessels · Moles are conserved
  $$n = \frac{PV}{RT} \qquad \sum_i \frac{P_iV_i}{T_i} = \text{constant}$$
• Gas-law graphs and pressure that changes through the gas · Hottest state on a path
  $$T = \frac{PV}{nR} \qquad \frac{d(PV)}{dV} = 0$$

Watch out for (6)

• A Celsius temperature in a ratio
  → One fixed amount of gas: P₁V₁/T₁ = P₂V₂/T₂
• Leaving out the air pressure at depth
  → One fixed amount of gas: P₁V₁/T₁ = P₂V₂/T₂
• Adding pressures instead of moles
  → Counting moles: mixtures and joined vessels
• k with moles, or R with molecules
  → Counting moles: mixtures and joined vessels
• Reading a V–T slope as pressure
  → Gas-law graphs and pressure that changes through the gas
• Expecting a Celsius graph to pass through the origin
  → Gas-law graphs and pressure that changes through the gas

Formulas (3)

• Pressure from molecular impacts, and PV = (2/3)E · Kinetic-theory pressure
  $$P = \frac{1}{3}\rho v_{rms}^{2} \qquad PV = \frac{2}{3}E_{trans}$$
• Average kinetic energy is (3/2)kT · Mean translational kinetic energy
  $$\overline{KE} = \frac{3}{2}kT \qquad E_{trans} = \frac{3}{2}nRT$$
• Same temperature, same mean kinetic energy · Equal temperatures
  $$\tfrac{1}{2}m_1\overline{v_1^{2}} = \tfrac{1}{2}m_2\overline{v_2^{2}} = \tfrac{3}{2}kT$$

Watch out for (6)

• Taking PV as the kinetic energy
  → Pressure from molecular impacts, and PV = (2/3)E
• Squaring the mean speed
  → Pressure from molecular impacts, and PV = (2/3)E
• Doubling the Celsius temperature
  → Average kinetic energy is (3/2)kT
• Per molecule or per mole
  → Average kinetic energy is (3/2)kT
• Letting the mass ratio decide the answer
  → Same temperature, same mean kinetic energy
• Equal energy is not equal speed
  → Same temperature, same mean kinetic energy

Formulas (3)

• The rms speed scales as √(T/M) · rms speed
  $$v_{rms} = \sqrt{\frac{3RT}{M}} \qquad \frac{v_2}{v_1} = \sqrt{\frac{T_2M_1}{T_1M_2}}$$
• The rms, average and most probable speeds · Three molecular speeds
  $$v_p = \sqrt{\frac{2RT}{M}} \qquad \bar{v} = \sqrt{\frac{8RT}{\pi M}} \qquad v_{rms} = \sqrt{\frac{3RT}{M}}$$
• Mean free path and collision frequency · Mean free path
  $$\lambda = \frac{1}{\sqrt{2}\,\pi d^{2}n} = \frac{kT}{\sqrt{2}\,\pi d^{2}P} \qquad Z = \frac{\bar{v}}{\lambda}$$

Watch out for (6)

• Speed goes as √T, not T
  → The rms speed scales as √(T/M)
• Celsius in the ratio
  → The rms speed scales as √(T/M)
• Using the rms speed when the average speed is asked
  → The rms, average and most probable speeds
• Molar mass in grams
  → The rms, average and most probable speeds
• 1/d instead of 1/d²
  → Mean free path and collision frequency
• Heating at constant pressure versus in a sealed vessel
  → Mean free path and collision frequency

Formulas (1)

• Cv, Cp and γ from the degrees of freedom · Heat capacities of an ideal gas
  $$C_v = \frac{f}{2}R \qquad C_p = C_v + R \qquad \gamma = 1 + \frac{2}{f}$$

Reference tables (1)

Watch out for (4)

• Counting a vibrational mode once
  → Counting degrees of freedom
• Linear and non-linear triatomics
  → Counting degrees of freedom
• Expecting γ to grow with f
  → Cv, Cp and γ from the degrees of freedom
• Cp − Cv = R is per mole
  → Cv, Cp and γ from the degrees of freedom

Formulas (2)

• Internal energy U = n(f/2)RT · Internal energy of an ideal gas
  $$U = n\frac{f}{2}RT = \frac{f}{2}PV \qquad \Delta U = nC_v\Delta T$$
• A mixture as one equivalent gas, and the temperature after mixing · Equivalent gas
  $$f_{mix} = \frac{n_1f_1 + n_2f_2}{n_1 + n_2} \qquad \gamma_{mix} = 1 + \frac{2}{f_{mix}}$$

Watch out for (4)

• k for molecules, R for moles
  → Internal energy U = n(f/2)RT
• "Neglect vibration" sets f = 5
  → Internal energy U = n(f/2)RT
• Averaging γ directly
  → A mixture as one equivalent gas, and the temperature after mixing
• Weighting temperatures by moles alone
  → A mixture as one equivalent gas, and the temperature after mixing