Postulates of Kinetic Theory
The kinetic theory explains the macroscopic properties of gases based on the microscopic motion of molecules.
Basic Assumptions
- A gas consists of a large number of molecules in constant random motion.
- The volume of individual molecules is negligible compared to the volume of the gas.
- There are no intermolecular forces except during collisions.
- Collisions between molecules and with walls are perfectly elastic.
- The duration of a collision is negligible compared to time between collisions.
- Pressure is due to molecules colliding with the walls.
Pressure of an Ideal Gas
P = (1/3) (N/V) m bar(v^2) = (1/3) rho bar(v^2)
Where bar(v^2) is the mean square speed of molecules.
Root Mean Square Speed
v_(rms) = sqrt(bar(v^2)) = sqrt(3P/rho) = sqrt(3kT/m) = sqrt(3RT/M)
Where k = R/N_A is Boltzmann's constant (1.38 x 10^(-23) J/K).
Gas Laws
Boyles Law
At constant temperature, P prop 1/V, i.e., PV = constant.
Charless Law
At constant pressure, V prop T, i.e., V/T = constant.
Gay-Lussacs Law (Pressure Law)
At constant volume, P prop T, i.e., P/T = constant.
Avogadros Law
Equal volumes of all gases at same temperature and pressure contain equal numbers of molecules.
Ideal Gas Equation
PV = nRT = (N/N_A) RT = NkT
Where:
n= number of molesR = 8.314 J/mol K(universal gas constant)k = 1.38 x 10^(-23) J/K(Boltzmann constant)N_A = 6.022 x 10^23 /mol(Avogadro constant)
Real Gases and van der Waals Equation
Real gases deviate from ideal behaviour at high pressure and low temperature.
(P + a n^2/V^2)(V - n b) = nRT
Where a accounts for intermolecular attraction, and b accounts for molecular volume.
Mean Free Path
The average distance a molecule travels between successive collisions.
lambda = 1/(sqrt(2) pi d^2 n_v)
Where n_v = number density (N/V), d = molecular diameter.
lambdadepends on temperature and pressure:lambda = kT/(sqrt(2) pi d^2 P).
Degrees of Freedom
The number of independent coordinates required to specify the position and configuration of a molecule.
| Molecule Type | Translational | Rotational | Vibrational | Total |
|---|---|---|---|---|
| Monatomic (He, Ar) | 3 | 0 | 0 | 3 |
| Diatomic (H2, O2, N2) | 3 | 2 | 1 | 6 |
| Triatomic linear (CO2) | 3 | 2 | 4 | 9 |
| Triatomic non-linear (H2O) | 3 | 3 | 3 | 9 |
At moderate temperatures, vibrational degrees are usually frozen, so effective DOF for diatomic = 5.
Law of Equipartition of Energy
Each degree of freedom contributes (1/2) kT of energy per molecule (or (1/2) RT per mole).
Energy per Mole
- Monatomic:
U = 3/2 RT - Diatomic (no vibration):
U = 5/2 RT - Diatomic (with vibration):
U = 7/2 RT
Specific Heats
C_v = dU/dT, C_p = C_v + R, gamma = C_p/C_v
| Gas Type | C_v | C_p | gamma |
|---|---|---|---|
| Monatomic | 3R/2 | 5R/2 | 5/3 = 1.67 |
| Diatomic (rigid) | 5R/2 | 7R/2 | 7/5 = 1.4 |
Worked Examples
Example 1: Find rms speed of oxygen molecules at 300 K. (M = 32 g/mol, R = 8.314).
Solution: v_(rms) = sqrt(3RT/M) = sqrt(3*8.314*300/0.032) = sqrt(233831) = 483.6 m/s.
Example 2: Calculate the number of molecules in 1 cm^3 of ideal gas at STP.
Solution: At STP, 1 mol (6.022 x 10^23 molecules) occupies 22.4 L = 22400 cm^3.
Number per cm^3 = 6.022 x 10^23/22400 = 2.69 x 10^19 molecules/cm^3 (Loschmidt number).
Common Mistakes
- rms speed vs average speed:
v_(rms) = sqrt(3kT/m),v_(avg) = sqrt(8kT/(pi m)). They differ. - Degrees of freedom and temperature: Vibrational DOF contribute only at high temperatures.
- Dalton's law of partial pressures: In a mixture, total pressure = sum of partial pressures.
- Real vs ideal gas: Real gases approach ideal behaviour at low pressure and high temperature.
ISC Exam Focus
- Theory (70%): Kinetic theory postulates, derivation of pressure, degrees of freedom, equipartition.
- Application (30%): RMS speed calculations, mean free path, specific heat calculations.
- ISC frequently asks: "Derive expression for pressure of an ideal gas" or "Find rms speed of ...".
- Degrees of freedom and specific heat relations are important.
Self-Test Questions
Q1: State the postulates of kinetic theory of gases. Answer: Large number of molecules, negligible volume, no intermolecular forces, elastic collisions, random motion.
Q2: Find the rms speed of hydrogen at 300 K. (M = 2 g/mol, R = 8.314 J/mol K).
Answer: v_(rms) = sqrt(3*8.314*300/0.002) = sqrt(3741300) = 1934 m/s.
Q3: Calculate the degrees of freedom of a monatomic gas and its gamma value.
Answer: DOF = 3. C_v = 3R/2, C_p = 5R/2, gamma = 5/3 = 1.67.
Q4: Define mean free path. What happens to mean free path when temperature is increased?
Answer: lambda = kT/(sqrt(2) pi d^2 P). At constant pressure, mean free path increases with temperature.
Q5: State the law of equipartition of energy.
Answer: Each degree of freedom contributes (1/2) kT of energy per molecule.
Q6: A gas has C_v = 5R/2. What type of gas is it? Find its gamma.
Answer: Diatomic (rigid, without vibration). gamma = (5R/2 + R)/(5R/2) = 7/5 = 1.4.
