Unit 8: Thermodynamics and Kinetic Theory

This is a 3–4 MCQ-per-year unit on NEET. It is content-rich but formula-heavy. The dominant question types are: (i) calorimetry mixing problems, (ii) thermal conduction in series/parallel rods, (iii) kinetic-theory ratios involving RMS, average, most probable speeds, (iv) first-law calculations on standard processes (isothermal, adiabatic), and (v) Carnot efficiency.

Concept Map

• Heat and Temperature scales
• Thermal expansion — linear, area, volume
• Calorimetry — specific heat, latent heat
• Heat transfer — conduction (Fourier), convection (qualitative), radiation (Stefan, Wien, Newton's cooling)
• Kinetic Theory of Gases — postulates, pressure formula, kT, RMS, equipartition, degrees of freedom, $C_p$, $C_v$, $\gamma$, mean free path
• First Law of Thermodynamics and processes (isothermal, adiabatic, isobaric, isochoric)
• Cyclic processes, heat engines, refrigerator
• Carnot cycle and second law

Topic 1: Temperature Scales and Thermal Expansion

Sub-topic A: Temperature Scales

Absolute zero: $T_K = 0$ K. All thermodynamic temperatures use Kelvin.

Sub-topic B: Thermal Expansion

For an isotropic solid, $\alpha : \beta : \gamma = 1 : 2 : 3$.

Anomalous expansion of water: water has maximum density at 4 °C; below this, expansion occurs as $T$ decreases. This is why lakes freeze top-down.

Topic 2: Calorimetry

Sub-topic A: Specific Heat and Heat Capacity

Heat needed to change temperature of mass $m$ by $\Delta T$:

where $s$ is specific heat capacity (J/(kg·K)). For water, $s_w \approx 4186\ \text{J/(kg K)} = 1\ \text{cal/(g °C)}$.

Heat capacity $C = ms$.

Sub-topic B: Latent Heat

For a phase change at constant temperature:

with $L_\text{fusion}$ (solid ↔ liquid) and $L_\text{vap}$ (liquid ↔ vapour). For water: $L_f = 80\ \text{cal/g}$, $L_v = 540\ \text{cal/g}$.

Sub-topic C: Mixing (Principle of Calorimetry)

Heat lost by hot body = heat gained by cold body (no external heat loss). Solve for final $T$:

For three substances: balance net heat.

Topic 3: Heat Transfer

Sub-topic A: Conduction — Fourier's Law

Rate of heat flow through a rod of cross-section $A$, length $L$, conductivity $K$, temperature difference $\Delta T$:

Thermal resistance: $R = L/(KA)$. So $dQ/dt = \Delta T/R$ — Ohm's-law analogy.

Series rods: $R_\text{eq} = R_1 + R_2$. Parallel rods: $1/R_\text{eq} = 1/R_1 + 1/R_2$.

Sub-topic B: Convection

Heat transfer by bulk motion of fluid. Two types:

• Natural convection: density-driven (warm air rises).
• Forced convection: external pumping (fan).

Quantitatively, $dQ/dt = h A \Delta T$ (Newton).

Sub-topic C: Radiation

All bodies emit thermal radiation. For a black body:

Stefan-Boltzmann law:

For a body with emissivity $\epsilon$: $P = \epsilon \sigma A T^4$.

Net radiation in surroundings at $T_0$:

Wien's displacement law: peak emission wavelength

Hotter bodies emit at shorter wavelengths.

Newton's law of cooling (for small $\Delta T$):

This is the linearised version of Stefan's law. Solution: $T(t) = T_0 + (T_i - T_0) e^{-kt}$.

Topic 4: Kinetic Theory of Gases

Sub-topic A: Postulates

• Gas consists of identical molecules in random motion.
• Molecules are point-like (volume negligible compared to container).
• No intermolecular forces except during elastic collisions.
• Collisions with walls are elastic and instantaneous.
• Velocity distribution obeys Maxwell-Boltzmann.

Sub-topic B: Pressure of an Ideal Gas

where $n$ is number density and $\overline{v^2}$ is mean-square speed.

Sub-topic C: Temperature and Kinetic Energy

For monatomic ideal gas, average translational KE per molecule:

with Boltzmann's constant $k_B = 1.38 \times 10^{-23}\ \text{J/K}$.

For $n$ moles, total translational KE is $\tfrac{3}{2} n R T$, where $R = N_A k_B = 8.314\ \text{J/(mol K)}$.

Sub-topic D: Speed Distributions

For Maxwell-Boltzmann:

Ratio: $v_p : \bar v : v_\text{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1 : 1.128 : 1.224$.

So $v_\text{rms} > \bar v > v_p$.

Sub-topic E: Equipartition Theorem

Each quadratic degree of freedom contributes $\tfrac{1}{2} k_B T$ to the average energy.

Mayer's relation: $C_p - C_v = R$.

Sub-topic F: Mean Free Path

At STP, $\lambda \approx 10^{-7}\ \text{m}$.

Topic 5: First Law of Thermodynamics

Sub-topic A: Statement

where $W$ is work done BY the gas (some books use $Q = \Delta U + W$). Be consistent with sign convention.

Internal energy $U$ is a state function; $Q$ and $W$ are path functions.

For an ideal gas, $U = U(T)$ only (depends only on temperature, not on $V$ or $P$).

Sub-topic B: Work Done by a Gas

Sign: $W > 0$ for expansion, $W < 0$ for compression.

Sub-topic C: Specific Processes

Sub-topic D: Adiabatic Relations

For a reversible adiabatic of ideal gas:

Adiabatic curve is steeper than isothermal on a PV diagram.

Sub-topic E: Cyclic Processes

For a cycle: $\Delta U = 0$, so $Q = W$. Net work = area enclosed by the cycle on the PV diagram. Sign: positive if traversed clockwise (heat engine), negative if counter-clockwise (refrigerator).

Topic 6: Heat Engines and the Second Law

Sub-topic A: Heat Engine

Operates between hot reservoir at $T_H$ and cold reservoir at $T_C$. Per cycle: absorbs $Q_H$, does work $W$, rejects $Q_C$.

Efficiency:

Sub-topic B: Refrigerator / Heat Pump

Operates in reverse — work input $W$ extracts $Q_C$ from cold reservoir and dumps $Q_H = Q_C + W$ into hot reservoir.

Coefficient of performance (refrigerator):

For a heat pump: $\beta_\text{HP} = Q_H/W = \beta_\text{ref} + 1$.

Sub-topic C: Carnot Cycle

Reversible cycle: two isotherms (at $T_H$ and $T_C$) + two adiabats. Achieves maximum efficiency between given reservoirs:

Carnot refrigerator: $\beta = T_C/(T_H - T_C)$.

No engine can exceed Carnot efficiency (second law). For 100% efficiency we'd need $T_C = 0$ — impossible.

Sub-topic D: Second Law

Several equivalent statements:

• Kelvin-Planck: No process can convert heat entirely into work in a cycle.
• Clausius: Heat cannot flow spontaneously from a colder to a hotter body.
• Entropy of an isolated system never decreases.

NEET Pattern MCQ Tips

• Calorimetry mixing: ice + water + warmer water, find final $T$.
• Conduction through composite: series/parallel rods.
• Stefan / Newton cooling: $P \propto T^4$ or $dT/dt = -k(T - T_0)$.
• Kinetic theory: ratios of $v_p, \bar v, v_\text{rms}$; effect of doubling $T$ on $v_\text{rms}$ ($\sqrt{2}$ times).
• $\gamma$ for ideal gas: $5/3, 7/5, 9/7$.
• Carnot efficiency: insert $T_H$ and $T_C$ in kelvin.
• Adiabatic vs isothermal: distinguish work formulas, slope comparisons.
• Assertion-Reason: any heat engine efficiency $<$ Carnot.

Common Confusions and Traps

• $\gamma = 7/5$ for diatomic gas at room temperature (5 DoF, vibration frozen).
• $C_p - C_v = R$ per mole, not per molecule (per molecule it would be $k_B$).
• For monatomic gas $C_v = (3/2)R$ — students sometimes use the diatomic value.
• In adiabatic, $\Delta U = -W$ (work done by gas equals negative of internal energy change).
• $PV = nRT$: $R = 8.314\ \text{J/(mol K)}$; or use the per-molecule $k_B T$.
• Stefan's law: rate $\propto T^4$ in kelvin, never centigrade.
• For Newton's cooling, the rate $dT/dt$ is proportional to $(T - T_0)$ in either kelvin or celsius (only differences matter).
• A heat engine never has $\eta = 1$ — second law.
• The most probable speed is the smallest of $v_p, \bar v, v_\text{rms}$.

Quick Revision Card

• Linear: $\beta = 2\alpha$; volume: $\gamma_\text{vol} = 3\alpha$.
• $Q = ms\Delta T$, $Q = mL$.
• Conduction: $dQ/dt = KA\Delta T/L$, $R = L/(KA)$.
• Stefan: $P = \epsilon\sigma A T^4$; Wien: $\lambda_m T = b$.
• Newton's cooling: $dT/dt = -k(T - T_0)$.
• Pressure: $P = \tfrac{1}{3}\rho\overline{v^2}$.
• Avg KE per molecule (monatomic) $= \tfrac{3}{2}k_BT$.
• Speeds: $v_p = \sqrt{2k_BT/m}$, $\bar v = \sqrt{8k_BT/\pi m}$, $v_\text{rms} = \sqrt{3k_BT/m}$.
• Mayer: $C_p - C_v = R$; $\gamma = C_p/C_v$.
• Monatomic: $\gamma = 5/3$; diatomic: $\gamma = 7/5$.
• Adiabatic: $PV^\gamma =$ const.
• Carnot: $\eta = 1 - T_C/T_H$; $\beta_\text{ref} = T_C/(T_H - T_C)$.
• First law: $\Delta U = Q - W$ (with $W$ = work by gas).

Worked NEET Examples

Example 1: Mixing Hot and Cold Water

200 g water at 80 °C mixed with 100 g at 20 °C. Final temperature:

$200 \times 80 - 200T = 100T - 2000$, $18000 = 300T$, $T = 60$ °C.

Example 2: Conduction Through Composite Rod

Two rods of equal length and cross-section but conductivities $K_1$ and $K_2$ in series:

Heat current must be equal in steady state. If hot end at $T_1$, cold at $T_2$, junction at $T_j$:

Effective conductivity for series: $K_\text{eff} = 2 K_1 K_2/(K_1 + K_2)$ (harmonic-like mean).

For parallel arrangement (rods side by side, same length, different areas): $K_\text{eff} = (K_1 A_1 + K_2 A_2)/(A_1 + A_2)$.

Example 3: Black Body Radiation

A black body at 500 K emits at rate $P_0$. At 1000 K it emits at $P = P_0 (1000/500)^4 = 16 P_0$.

Example 4: Carnot Efficiency

Hot reservoir at 400 K, cold at 300 K. Carnot efficiency:

So per 4 J absorbed at hot, 1 J becomes work, 3 J rejected to cold.

Example 5: RMS Speed of Oxygen

For O₂ at T = 300 K, molecular mass $M = 32$ g/mol = 0.032 kg/mol:

Derivations

Pressure of an Ideal Gas

Consider a cube of side $L$ containing $N$ molecules each of mass $m$. A molecule with $v_x$ component impacts a face, transferring momentum $2 m v_x$. Time between impacts on the same face: $2L/v_x$. So average force from one molecule: $2 m v_x/(2L/v_x) = m v_x^2/L$.

For $N$ molecules: total force = $N m \overline{v_x^2}/L$. Pressure on face of area $L^2$:

By isotropy, $\overline{v_x^2} = \overline{v^2}/3$:

Connection to Temperature

Compare with ideal gas law $PV = nRT = N k_B T$:

So average translational KE per molecule = $(3/2) k_B T$.

Adiabatic Process for Ideal Gas

First law: $dU = -dW$ (since $dQ = 0$). For ideal gas $dU = n C_v dT$ and $dW = P dV$.

From $PV = nRT$: $P dV + V dP = nR dT$, so $dT = (P dV + V dP)/(nR)$.

Substituting: $C_v (P dV + V dP)/R + P dV = 0$, so $C_v P dV + C_v V dP + R P dV = 0$, i.e. $(C_v + R) P dV + C_v V dP = 0$, or $C_p P dV + C_v V dP = 0$.

Rearranging: $\gamma\,dV/V + dP/P = 0$, giving $PV^\gamma =$ constant.

Carnot Efficiency

Carnot cycle has two isothermals (at $T_H, T_C$) and two adiabats. Heat absorbed at $T_H$: $Q_H = nRT_H\ln(V_2/V_1)$. Heat rejected at $T_C$: $Q_C = nRT_C\ln(V_3/V_4)$. From adiabatic legs: $T_H V_2^{\gamma-1} = T_C V_3^{\gamma-1}$ and $T_H V_1^{\gamma-1} = T_C V_4^{\gamma-1}$, so $V_2/V_1 = V_3/V_4$. Therefore $Q_C/Q_H = T_C/T_H$ and

Indicator Diagrams and Cyclic Work

For any cyclic process, work done by the gas equals the area enclosed on the P-V diagram. Direction:

• Clockwise traversal: positive work (heat engine).
• Counter-clockwise: negative work (refrigerator).

In a rectangular cycle on P-V plane: ABCDA with corners $(V_1, P_1), (V_2, P_1), (V_2, P_2), (V_1, P_2)$. Work = $(V_2 - V_1)(P_1 - P_2)$.

Two Specific Heat Relations

Mayer's Relation Derivation

For ideal gas, $U = nC_v T$ depends only on T. At constant pressure: $Q = nC_p \Delta T$, $W = P\Delta V = nR\Delta T$, $\Delta U = nC_v\Delta T$. First law: $Q = \Delta U + W$, so

Effective γ for Mixtures

For a mix of $n_1$ moles of gas with $C_{v1}$ and $n_2$ moles with $C_{v2}$:

with similar formula for $C_p^\text{mix}$. Then $\gamma_\text{mix} = C_p^\text{mix}/C_v^\text{mix}$. NEET often tests this with mono+diatomic mixtures.

Heat Engine and Pump Efficiencies

• Refrigerator COP: $\beta = Q_C/W = Q_C/(Q_H - Q_C)$. For Carnot, $\beta = T_C/(T_H - T_C)$. Higher COP at lower $\Delta T$.
• Heat pump COP (for heating): $\beta_\text{HP} = Q_H/W = 1 + \beta_\text{ref}$.

Entropy (Conceptual)

The second law can be phrased: in any process, the total entropy of the universe increases (for irreversible) or stays the same (reversible).

For reversible heat exchange: $dS = dQ/T$.

For an ideal gas isothermal expansion: $\Delta S = nR\ln(V_f/V_i)$ — positive (irreversible if free expansion, reversible if isothermal but slow).

Phase Diagrams (Qualitative)

A typical P-T diagram for water shows three regions (solid, liquid, gas), separated by phase boundaries meeting at the triple point (273.16 K, 611 Pa). Above the critical point (647 K, 22 MPa), no distinction between liquid and gas.

Formula Sheet