Chapter 19: The Kinetic Theory of Gases - Notes

Avogadro's Number

• Avogadro's Number ($N_A$):
  • One mole of a substance contains $N_A$ elementary units (atoms or molecules).
  • $N_A$ is experimentally found to be $6.02 \times 10^{23} \text{ mol}^{-1}$. (Equation 19.1.1)
• Molar Mass (M):
  • The mass per mole ($M$) of a substance is related to the mass (m) of an individual molecule by $M = mN_A$. (Equation 19.1.4)
• Number of Moles (n):
  • The number of moles (n) in a sample of mass $M<em>{sam}$ consisting of N molecules is related to the molar mass M and Avogadro's number $N</em>A$ by: $n = \frac{M_{sam}}{M} = \frac{mN}{M}$. (Equation 19.1.3)

Ideal Gases

• Ideal Gas Law:
  • An ideal gas is defined by the relationship between pressure (p), volume (V), and temperature (T): $pV = nRT$. (Equation 19.2.1)
    • n is the number of moles.
    • R is the gas constant, approximately 8.31 J/mol.K.
  • Alternative form: $pV = NkT$, where k is the Boltzmann constant. (Equation 19.2.5)
• Isothermal Process:
  • A process occurring at constant temperature.
  • Isothermal Work: The work done by an ideal gas during an isothermal change from volume $V<em>i$ to volume $V</em>f$ is given by: $W = nRT \ln{\frac{V<em>f}{V</em>i}}$. (Equation 19.2.10)
• P-V Diagrams:
  • Isotherms are curves on a p-V diagram representing constant-temperature processes.
  • The area under a curve on a p-V diagram represents the work done by the gas.
  • Constant-volume process: no work is done.
  • Constant-pressure process: work done is $W = p\Delta V$.
• Isothermal Change in Internal Energy:
  • For an isothermal process, the change in internal energy ($\Delta E$) is zero.
  • The energy transferred as heat (Q) is equal to the work done (W).

Pressure, Temperature, and RMS Speed

• Pressure and Molecular Collisions:
  • Pressure on the walls of a gas container is due to molecular collisions with the walls.
• RMS Speed:
  • The pressure exerted by n moles of an ideal gas is: $p = \frac{nMv<em>{rms}^2}{3V}$, where $v</em>{rms}$ is the root-mean-square speed of the molecules, M is the molar mass, and V is the volume. (Equation 19.3.4)
  • The rms speed can be written in terms of temperature: $v_{rms} = \sqrt{\frac{3RT}{M}}$. (Equation 19.3.5)

Translational Kinetic Energy

• Average Translational Kinetic Energy:
  • The average translational kinetic energy is related to temperature: $K_{avg} = \frac{3}{2}kT$. (Equation 19.4.2)
  • At a given temperature, all ideal gas molecules have the same average translational kinetic energy.

Mean Free Path

• Mean Free Path ($\lambda$):
  • The average distance a molecule travels between collisions.
  • $\lambda = \frac{1}{\sqrt{2}\pi d^2 (N/V)}$, where N/V is the number of molecules per unit volume and d is the molecular diameter. (Equation 19.5.1)

Maxwell Speed Distribution

• Maxwell Speed Distribution P(v):
  • $P(v)dv$ gives the fraction of molecules with speeds in the interval dv at speed v:
  • $P(v) = 4\pi \left( \frac{M}{2\pi RT} \right)^{3/2} v^2 e^{-\frac{Mv^2}{2RT}}$. (Equation 19.6.1)
• Measures of Speed Distribution:
  • Average speed: $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$. (Equation 19.6.5)
  • Most probable speed: $v_P = \sqrt{\frac{2RT}{M}}$. (Equation 19.6.8)
  • RMS speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$. (Equation 19.6.9)

Molar Specific Heats of an Ideal Gas

• Internal Energy (Eint):
  • The internal energy of an ideal gas is a function of temperature only.
  • For a monatomic ideal gas: $E_{int} = \frac{3}{2}nRT$. (Equation 19.7.2)
• Molar Specific Heat at Constant Volume (Cv):
  • $C_V = \frac{Q}{n\Delta T}$. (Equation 19.7.3)
• Molar Specific Heat at Constant Pressure (Cp):
  • $C_p = \frac{Q}{n\Delta T}$. (Equation 19.7.10)
  • $C<em>p = C</em>V + R$. (Equation 19.7.13)

Degree of Freedom and Molar Specific Heats

• Equipartition of Energy:
  • Each degree of freedom has an average energy of $\frac{1}{2}kT$ per molecule (or $\frac{1}{2}RT$ per mole).
• Degrees of Freedom:
  • Monatomic gas: 3 translational degrees of freedom.
  • Diatomic gas: 3 translational and 2 rotational degrees of freedom at moderate temperatures; can also have 2 vibrational degrees of freedom at high temperatures.

Adiabatic Expansion of an Ideal Gas

• Adiabatic Process:
  • A process with no heat exchange (Q = 0).
  • $pV^{\gamma} = \text{constant}$, where $\gamma = \frac{C<em>p}{C</em>V}$. (Equation 19.9.1)
• Free Expansion:
  • Adiabatic expansion into a vacuum where no work is done, so the internal energy and temperature do not change.