Sections#

Review & Summary#

Kinetic Theory of Gases#

The kinetic theory of gases relates the macroscopic properties of gases (pressure, temperature) to the microscopic properties of gas molecules (speed, kinetic energy).

Avogadro’s Number#

One mole of a substance contains \(N_A\) (Avogadro’s number) elementary units (usually atoms or molecules), where \(N_A = 6.02 \times 10^{23}\) mol\(^{-1}\). One molar mass \(M\) of any substance is the mass of one mole. It is related to the mass \(m\) of individual molecules by \(M = N_A m\). The number of moles \(n\) in a sample of mass \(M_{\mathrm{sam}}\) consisting of \(N\) molecules is

(162)#\[ n = \frac{N}{N_A} = \frac{M_{\mathrm{sam}}}{M} \]

Ideal Gas#

An ideal gas is one for which pressure \(p\), volume \(V\), and temperature \(T\) are related by

(163)#\[ pV = nRT \]

where \(n\) is the number of moles and \(R = 8.31\) J/(mol·K) is the gas constant. Equivalently, \(pV = NkT\), where \(k = R/N_A = 1.38 \times 10^{-23}\) J/K is the Boltzmann constant and \(N\) is the number of molecules.

Pressure, Temperature, and RMS Speed#

The pressure exerted by \(n\) moles of an ideal gas, in terms of molecular speeds, is \(p = (nM/V) v_{\mathrm{rms}}^2/3\), where \(v_{\mathrm{rms}} = \sqrt{\overline{v^2}}\) is the root-mean-square speed and \(M\) is the molar mass. This gives

(164)#\[ v_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}} \]

The average translational kinetic energy \(\overline{K}\) per molecule is

(165)#\[ \overline{K} = \frac{3}{2} kT \]

Mean Free Path#

The mean free path \(\lambda\) of a gas molecule is its average path length between collisions:

(166)#\[ \lambda = \frac{1}{\sqrt{2}\,\pi d^2 (N/V)} \]

where \(N/V\) is the number of molecules per unit volume and \(d\) is the molecular diameter.

Maxwell Speed Distribution#

The Maxwell speed distribution \(P(v)\) is such that \(P(v)\,dv\) gives the fraction of molecules with speeds in the interval \(dv\) at speed \(v\). Three measures are \(v_{\mathrm{avg}}\), \(v_P\) (most probable), and \(v_{\mathrm{rms}}\).

Molar Specific Heats#

For an ideal gas, the molar specific heat at constant volume is \(C_V = \frac{f}{2}R\) and at constant pressure \(C_P = C_V + R = \frac{f+2}{2}R\), where \(f\) is the number of degrees of freedom. For monatomic: \(f=3\); diatomic: \(f=5\).

The internal energy for \(n\) moles of an ideal gas at temperature \(T\) is

(167)#\[ E_{\mathrm{int}} = n C_V T \]

Adiabatic Expansion#

For a reversible adiabatic process (\(Q=0\)), \(pV^\gamma = \mathrm{constant}\) and \(TV^{\gamma-1} = \mathrm{constant}\), where \(\gamma = C_P/C_V\) is the adiabatic exponent. The work done by the gas is \(W = -\Delta E_{\mathrm{int}}\).