Kinetic Theory

• Gas Laws & Ideal Gas Equation

  • Boyle's Law: $P \propto \frac{1}{V}$ (at constant $T$)
  • Charles's Law: $V \propto T$ (at constant $P$)
  • Gay-Lussac's Law: $P \propto T$ (at constant $V$)
  • Avogadro's Law: Equal volume contains equal number of molecules
  • Ideal Gas Equation: $PV = nRT = N k_B T$
    • Universal Gas Constant ($R$): $8.314 \text{ J/mol}\cdot\text{K}$
    • Boltzmann Constant ($k_B$): $k_B = \frac{R}{N_A} = 1.38 \times 10^{-23} \text{ J/K}$

• Assumptions of Kinetic Theory

  • Gases consist of huge number of tiny, identical molecules
  • Large inter-molecular distances $\rightarrow$ Negligible inter-molecular forces
  • Continuous, random, straight-line motion
  • Collisions are perfectly elastic (KE and momentum conserved)
  • Collision duration is negligible compared to time between collisions

• Pressure of an Ideal Gas

  • Expression: $P = \frac{1}{3} \rho v_{rms}^2 = \frac{1}{3} \frac{N m}{V} v_{rms}^2$
  • Root Mean Square Speed ($v_{rms}$): $v_{rms} = \sqrt{\frac{3 k_B T}{m}} = \sqrt{\frac{3 RT}{M}}$
  • Average Speed ($v_{avg}$): $v_{avg} = \sqrt{\frac{8 k_B T}{\pi m}}$
  • Most Probable Speed ($v_{mp}$): $v_{mp} = \sqrt{\frac{2 k_B T}{m}}$
  • Speed Ratio: $v_{mp} : v_{avg} : v_{rms} = 1 : 1.128 : 1.225$

• Kinetic Energy and Temperature

  • Average KE per molecule: $E = \frac{3}{2} k_B T$
  • Total KE per mole: $E = \frac{3}{2} R T$
  • Absolute Temperature: Absolute zero ($0\text{ K}$) is where molecular motion completely ceases

• Degrees of Freedom ($f$)

  • Definition: Total number of independent coordinates needed to specify motion
  • Monatomic Gas (e.g., $\text{He}, \text{Ar}$): $f = 3$ (Translational only)
  • Diatomic Gas (e.g., $\text{O}_2, \text{N}_2$):
    • Room Temperature: $f = 5$ ($3\text{ Trans} + 2\text{ Rot}$)
    • High Temperature: $f = 7$ ($3\text{ Trans} + 2\text{ Rot} + 2\text{ Vib}$)
  • Polyatomic Gas (e.g., $\text{H}_2\text{O}, \text{NH}_3$): $f = 6$ ($3\text{ Trans} + 3\text{ Rot}$)

• Law of Equipartition of Energy

  • Statement: Energy per degree of freedom per molecule = $\frac{1}{2} k_B T$
  • Total Internal Energy per mole ($U$): $U = \frac{f}{2} R T$

• Specific Heat Capacities

  • Mayer's Relation: $C_p - C_v = R$
  • Adiabatic Index ($\gamma$): $\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$
  • Monatomic Gas: $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$, $\gamma = 1.67$
  • Diatomic Gas (Rigid): $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$, $\gamma = 1.40$
  • Polyatomic Gas: $C_v = 3R$, $C_p = 4R$, $\gamma = 1.33$

• Mean Free Path ($\lambda$)

  • Definition: Average distance travelled between two consecutive collisions
  • Formula: $\lambda = \frac{1}{\sqrt{2} \pi n d^2}$
    • $n$: Number density ($\frac{N}{V}$)
    • $d$: Diameter of the gas molecule
  • Dependence: Inversely proportional to density and pressure