Kinetic Theory of Gases: Study Notes

Kinetic Theory of Gases

Chapter Outline

• Empirical gas laws

• Ideal gas law

• Avogadro’s number

• Pressure, Temperature, & RMS speed

• Mean free path

• Distribution of molecular speeds

• Molecular model of an ideal gas

• Molar specific heat of an ideal gas

• The equipartition of energy

• Degrees of freedom and molar specific heats

Empirical Gas Laws

Key Empirical Relationships

• Boyle’s Law (1662): At constant temperature, the pressure and volume of a gas are inversely related.

• Charles’s Law (1787): At constant pressure, the volume of a gas is proportional to its temperature.

• Gay-Lussac’s Law (1802): At constant volume, the pressure of a gas is proportional to its temperature.

• Avogadro’s Law (1811): At constant pressure and temperature, the volume of a gas is proportional to the number of particles (V/N = const).

Example: If the temperature of a gas is doubled at constant pressure, its volume also doubles.

pVT Diagram: Ideal Gas

State Variables and Surfaces

• The state of an ideal gas can be represented as a point on a three-dimensional surface defined by pressure (p), volume (V), and temperature (T).

• Isotherms, isobars, and isochors are curves of constant temperature, pressure, and volume, respectively.

Example: Moving along an isotherm means changing pressure and volume while keeping temperature constant.

Ideal Gas Law

Equation of State

• The ideal gas law combines the empirical laws into a single equation:

$PV = nRT$

• R is the universal gas constant: $R = 8.314 \ \frac{J}{mol \cdot K}$

• At standard temperature and pressure (STP), 1 mole of any gas occupies 22.4 L.

Molecular Form

• The ideal gas law can also be written in terms of the number of molecules:

$PV = Nk_bT$

• Boltzmann’s constant: $k_b = 1.38 \times 10^{-23} \ \frac{J}{K}$

• P, V, and T are called thermodynamic variables.

The Mole and Avogadro’s Number

Definition and Significance

• One mole is the number of atoms in 12 g of carbon-12.

• Avogadro’s number ($N_A$) is the number of particles in one mole:

$N_A = 6.022 \times 10^{23} \ mol^{-1}$

• Avogadro’s number is determined experimentally.

Example: Avogadro’s number of table tennis balls would cover the Earth to a depth of about 40 km.

Van der Waals Equation

Real Gas Corrections

• The van der Waals equation accounts for intermolecular forces and finite molecular volume:

$\left[P + a\left(\frac{n}{V}\right)^2\right](V - nb) = nRT$

• a: Corrects for attractive forces between molecules (reduces pressure).

• b: Corrects for the finite volume of molecules (reduces available volume).

• At low density, a and b are negligible, and the equation reduces to the ideal gas law.

Dalton’s Law

Partial Pressures in Gas Mixtures

• The total pressure of a mixture of gases is the sum of the partial pressures of each component:

$p_{total} = \sum_{i=1}^m p_i$

• The partial pressure of a component is proportional to its molar fraction:

$p_i = \frac{n_i}{n_{total}} p_{total}$

• Vapor pressure is the partial pressure of a vapor in equilibrium with its liquid or solid phase.

Table: Vapor Pressure of Water at Various Temperatures

T (°C)	Vapor Pressure (Pa)
0	610.5
3	757.9
5	872.3
8	1073
10	1228
13	1497
15	1705
18	2063
20	2338
23	2809
25	3167
30	4243
35	5623
40	7376

Kinetic Theory of Gases

Microscopic Model

• Relates macroscopic properties (pressure, temperature) to microscopic properties (molecular motion).

• Focuses on ideal gases, where intermolecular forces are negligible except during collisions.

Molecular Model of an Ideal Gas

Assumptions

• Large number of molecules with large average separation compared to their size.

• Molecules obey Newton’s laws and move randomly in all directions.

• Collisions are elastic (no energy loss).

• Forces between molecules are negligible except during collisions.

Pressure of an Ideal Gas

Microscopic Derivation

• The force exerted on a wall by a single molecule colliding with it:

$\Delta p_x = -mv_x - (mv_x) = -2mv_x$

$f_{wall} = \frac{-2mv_x}{2d/v_x} = \frac{mv_x^2}{d}$

• For N molecules, the average total force and pressure are:

$F_{wall} = \frac{m}{d}(v_{x1}^2 + v_{x2}^2 + ... + v_{xN}^2)$

$P = \frac{F}{A} = \frac{Nm\langle v_x^2 \rangle}{V}$

$P = \frac{2}{3}\left(\frac{N}{V}\right)\left(\frac{1}{2}m\langle v^2 \rangle\right)$

• Increasing average kinetic energy or density increases pressure.

Temperature and Kinetic Energy

Relationship to Molecular Motion

• Comparing pressure and the ideal gas law gives:

$PV = \frac{2}{3}N\left(\frac{1}{2}m\langle v^2 \rangle\right)$

$PV = Nk_bT$

$\frac{1}{2}m\langle v^2 \rangle = \frac{3}{2}k_bT$

• Temperature is proportional to the average kinetic energy of molecules.

RMS Speeds

Root Mean Square Speed

• The root mean square (rms) speed of molecules:

$v_{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3k_bT}{m}} = \sqrt{\frac{3RT}{M}}$

• RMS speed increases with temperature and decreases with molecular mass.

Table: Some RMS Speeds

Gas	Molar Mass (g/mol)	vrms at 20°C (m/s)
H2	2.00	1920
He	4.00	1360
N2	28.0	511
O2	32.0	482
CO2	44.0	394
SO2	64.1	338

Equipartition of Energy

Energy Distribution

• Each degree of freedom contributes $(1/2)k_bT$ to the average energy per molecule.

• For a monatomic ideal gas (3 translational degrees):

$E_{trans} = N\left(\frac{1}{2}m\langle v^2 \rangle\right) = \frac{3}{2}Nk_bT$

Mean Free Path

Definition and Calculation

• The mean free path (l) is the average distance a molecule travels between collisions.

• For molecules of diameter d and number density $n_v$:

$l = \frac{1}{\sqrt{2}\pi d^2 n_v}$

$f = \sqrt{2}\pi d^2 n_v v_{avg}$

• Typical values for air at room temperature:

$d = 2 \times 10^{-10} m$ $D = 3 \times 10^{-9} m$ $l = 2 \times 10^{-7} m$ $f = 2 \times 10^{9} 1/s$

Molecular Speed Distribution

Maxwell-Boltzmann Distribution

• The probability that a molecule has speed between v and v+dv:

$p(v) = 4\pi \left(\frac{m}{2\pi k_b T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_b T}}$

• The number of molecules with speeds in this range:

$N_v = 4\pi N \left(\frac{m}{2\pi k_b T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_b T}}$

Temperature Increase and Speed Distribution

Effects of Temperature

• As temperature increases, the speed distribution shifts to higher values.

• Molecules in the high-speed tail can escape from a liquid (evaporation) or undergo fusion (in stars).

Average, RMS, and Most Probable Velocities

Definitions and Relationships

• Average speed:

$\bar{v} = \int v p(v) dv = \sqrt{\frac{8kT}{\pi m}}$

• RMS speed:

$v_{rms} = \sqrt{\frac{3kT}{m}}$

• Most probable speed:

$v_{mp} = \sqrt{\frac{2kT}{m}}$

• Relationship: $v_{rms} > \bar{v} > v_{mp}$

Specific Heats

Definitions

• CV: Molar specific heat at constant volume

• CP: Molar specific heat at constant pressure (CP > CV)

$Q = nC_V \Delta T \ (V = const.)$ $Q = nC_P \Delta T \ (P = const.)$

Constant Volume for Monoatomic Gases

Internal Energy and Specific Heat

• For a monoatomic ideal gas (e.g., helium):

$E_{int} = N\left(\frac{1}{2}m\langle v^2 \rangle\right)$ $E_{int} = \frac{3}{2}Nk_bT = \frac{3}{2}nRT$ $c_v = \frac{1}{n}\frac{dE_{int}}{dT} = \frac{3}{2}R$

• Thus, $C_V = \frac{3}{2}R = 12.5 \ \frac{J}{mol \cdot K}$

Complex Molecules

Degrees of Freedom

• Each degree of freedom (translational, rotational, vibrational) contributes $(1/2)k_bT$ to the internal energy.

• Examples: Helium (He) has 3 translational degrees; Nitrogen (N2) and Methane (CH4) have additional rotational degrees.

Diatomic Molecules

• For diatomic molecules (e.g., N2):

$E_{int} = 3N\left(\frac{1}{2}k_bT\right) + 2N\left(\frac{1}{2}k_bT\right)$ $E_{int} = \frac{5}{2}nRT$ $C_V = \frac{5}{2}R$

Temperature Variation of Specific Heat

Excitation of Modes

• As temperature increases, more rotational and vibrational modes are excited, increasing $C_V$.

• For hydrogen, vibrational modes are not excited at room temperature due to large energy gaps.

Example: The specific heat of a gas increases in steps as more degrees of freedom become accessible with rising temperature.

Additional info: These notes provide a comprehensive summary of the kinetic theory of gases, including both macroscopic and microscopic perspectives, and are suitable for college-level physics students preparing for exams.