Short Notes: Kinetic Theory

The following points are important for use and conversions:

• n is the number of moles, N is the total number of molecules and N = nN_A, where N_A is Avogadro's number.
• When using PV = NkT, use k (Boltzmann constant). When using PV = nRT, use R.
• To use v_rms = √(3RT/M) ensure M is in kg mol^-1.

12.2 Kinetic Theory Postulates and Velocity Relations

Kinetic theory of gases explains macroscopic properties (pressure, temperature, internal energy) in terms of microscopic motion of molecules. Main postulates for an ideal monoatomic gas:

• Gas consists of a large number of identical molecules moving in random directions with a distribution of speeds.
• Molecules are treated as point particles with negligible volume compared to container volume.
• Between collisions molecules move in straight lines; collisions (with each other and with the walls) are elastic.
• No intermolecular forces act except during collisions.

From these postulates the pressure exerted by a gas can be related to molecular motion. The kinetic-theory result is:

\[ P = \tfrac{1}{3}\,\rho\,v_{\text{rms}}^{2} \]

where ρ is mass density and v_rms is the root-mean-square speed of molecules.

The three common characteristic speeds obtained from the Maxwell-Boltzmann distribution are:

• Root-mean-square speed (v_rms): \[ v_{\text{rms}} = \sqrt{\dfrac{3RT}{M}} = \sqrt{\dfrac{3P}{\rho}} \]
• Average (mean) speed (v_avg): \[ v_{\text{avg}} = \sqrt{\dfrac{8RT}{\pi M}} \]
• Most probable speed (v_mp): \[ v_{\text{mp}} = \sqrt{\dfrac{2RT}{M}} \]

The numerical ratios are:

\[ v_{\text{rms}} : v_{\text{avg}} : v_{\text{mp}} = \sqrt{3} : \sqrt{\dfrac{8}{\pi}} : \sqrt{2} \approx 1.73 : 1.60 : 1.41 \]

Maxwell-Boltzmann distribution gives the probability density of molecular speeds in an ideal gas and leads to these characteristic speeds. These speeds are temperature dependent and decrease with increasing molar mass.

Worked example: Compute v_rms of O₂ (M = 32 g mol^-1) at T = 300 K.

Use M in kg mol^-1: M = 0.032 kg mol^-1.

\[ v_{\text{rms}} = \sqrt{\dfrac{3RT}{M}} \]

\[ v_{\text{rms}} = \sqrt{\dfrac{3 \times 8.314 \times 300}{0.032}} \approx 4.84 \times 10^{2}\ \text{m s}^{-1} \]

12.3 Kinetic Energy and Temperature

Temperature is a measure of the average kinetic energy of molecules. For an ideal gas the connection is simple and fundamental.

• Average translational kinetic energy per molecule: \[ \langle KE \rangle = \tfrac{3}{2}kT \]
• Total translational kinetic energy for n moles: \[ KE_{\text{total}} = \tfrac{3}{2}nRT \]
• Internal energy of an ideal gas: \[ U = \tfrac{f}{2}\,nRT \]

Here f is the number of degrees of freedom available for energy storage (see next section). For a monoatomic ideal gas, only translational motion contributes and f = 3, so

\[ U = \tfrac{3}{2} nRT \]

Temperature thus directly measures the average microscopic kinetic energy; this is why two gases at the same temperature have the same average translational kinetic energy per molecule regardless of molecular mass.

12.4 Degrees of Freedom

Degrees of freedom (f) are independent ways in which a molecule can store energy (translational, rotational, vibrational). For typical ideal-gas treatment in the temperature range where vibrational modes are frozen out:

Notes:

• At higher temperatures vibrational degrees of freedom become active and f increases; then internal energy and heat capacity increase accordingly.
• For an ideal gas the molar specific heats are related to f: \[ C_{V,m} = \tfrac{f}{2}R \quad\text{and}\quad C_{P,m} = C_{V,m} + R = \tfrac{f+2}{2}R. \]
• The ratio of specific heats is \[ \gamma = \dfrac{C_{P,m}}{C_{V,m}} = \dfrac{f+2}{f}. \]

12.5 Mean Free Path

Mean free path (λ) is the average distance a molecule travels between successive collisions. It depends on number density, molecular diameter and temperature/pressure.

• Expression in terms of number density (n): \[ \lambda = \dfrac{1}{\sqrt{2}\,\pi\,n\,d^{2}} \]
• Expression using pressure and temperature: \[ \lambda = \dfrac{kT}{\sqrt{2}\,\pi\,d^{2}P} \]
• Dependence: mean free path decreases as pressure increases and increases as temperature increases (at constant pressure the density decreases so λ increases).

Derivation sketch using the ideal gas relation:

Number density is related to pressure and temperature by

\[ n = \dfrac{P}{kT} \]

Substitute into the expression for λ:

\[ \lambda = \dfrac{1}{\sqrt{2}\,\pi\,d^{2}\,n} = \dfrac{1}{\sqrt{2}\,\pi\,d^{2}}\cdot\dfrac{kT}{P} = \dfrac{kT}{\sqrt{2}\,\pi\,d^{2}P} \]

Typical values: at atmospheric pressure and room temperature λ is of the order of 10^-7 to 10^-8 m for molecular diameters ~10^-10 m; in rarified gases and high-vacuum systems λ can be much larger, affecting flow regimes (continuum vs molecular flow).