Heat & Thermodynamics – Part II 

Topics covered: Kinetic Theory of Gases, Gas Laws, Thermodynamic Processes, Laws of Thermodynamics, Heat Engines & Refrigerators (JEE Main focus).

1. Assumptions of Kinetic Theory of Gases

• A gas consists of a large number of identical, tiny, spherical, neutral and elastic molecules.
• Molecules move randomly in all directions with different speeds.
• Volume of molecules is negligible compared to volume of the gas.
• No intermolecular force except during collisions.
• Collisions between molecules and with walls are perfectly elastic.
• Pressure is due to collisions of molecules with container walls.

2. Pressure of an Ideal Gas

P = (1/3) ρ c²

where ρ = density of gas, c² = mean square speed

Root Mean Square Speed:

v_rms = √(3P / ρ)

v_rms = √(3RT / M)

Graham’s Law of Diffusion:

Rate ∝ 1 / √ρ

3. Molecular Speeds

• Most probable speed: v_mp = √(2RT / M) = 0.816 v_rms
• Average speed: v_av = √(8RT / πM) = 0.92 v_rms
• Relation: v_rms > v_av > v_mp

4. Kinetic Interpretation of Temperature

Average translational kinetic energy of one molecule:

⟨KE⟩ = (3/2) kT

For one mole of gas:

KE = (3/2) RT

Average kinetic energy depends only on temperature.

5. Degrees of Freedom (f)

• Monoatomic gas: f = 3
• Diatomic gas (rigid): f = 5
• Diatomic gas (with vibration): f = 7

6. Law of Equipartition of Energy

Energy per degree of freedom = (1/2) kT

Internal energy of μ moles of gas:

U = (f/2) μRT

7. Gas Laws

• Boyle’s Law: PV = constant (T constant)
• Charles’ Law: V ∝ T (P constant)
• Gay-Lussac’s Law: P ∝ T (V constant)
• Avogadro’s Law: Equal volumes at same P and T contain equal molecules
• Dalton’s Law: P = P₁ + P₂ + …

8. Thermodynamic System

• State variables: P, V, T, U
• P–V diagram represents different states
• Area under P–V curve = work done
• Change in internal energy depends only on initial and final states

9. Thermodynamic Processes

Isobaric Process (P constant)

W = P (V_f − V_i)

Q = μC_pΔT

Isochoric Process (V constant)

W = 0

Q = μC_vΔT

Isothermal Process (T constant)

PV = constant

W = μRT ln(V_f/V_i)

Adiabatic Process

PV^γ = constant

T V^γ−1 = constant

γ = C_p / C_v

10. Work Done in Thermodynamic Processes

• General: W = ∫PdV
• Isobaric: W = P(V_f − V_i)
• Isochoric: W = 0
• Isothermal: W = μRT ln(V_f/V_i)
• Adiabatic: W = (P_iV_i − P_fV_f) / (γ − 1)

11. Laws of Thermodynamics

Zeroth Law: If A and B are in equilibrium with C, then A and B are in equilibrium.

First Law:

ΔQ = ΔU + ΔW

Second Law:

• Kelvin statement: No engine can convert all heat into work.
• Clausius statement: Heat cannot flow from cold to hot without external work.

12. Specific Heats of a Gas

C_p − C_v = R

γ = C_p / C_v

γ = (f + 2) / f

13. Heat Engine

Efficiency:

η = W / Q₁ = 1 − (Q₂ / Q₁) = 1 − (T₂ / T₁)

Perfect engine: η = 1 (not possible)

14. Refrigerator / Heat Pump

Coefficient of Performance (COP):

β = Q₂ / W = Q₂ / (Q₁ − Q₂)

Perfect refrigerator: β → ∞ (not possible)

15. Carnot Engine

• Consists of two isothermal and two adiabatic processes
• Most efficient engine possible between two temperatures

Carnot efficiency:

η = 1 − (T_L / T_H)