Maharashtra HSC · Class XII · Physics

Chapter 4: Thermodynamics

Full revision notes for Chapter 4 - Thermodynamics.

4.1

First Law of Thermodynamics

dQ = dU + dW

Heat $dQ$ added to a system increases its internal energy $dU$ and does work $dW$ on the surroundings. Sign convention: $dQ > 0$ heat into the system; $dW > 0$ work done by the system. (The sign of $W$ is a convention - fix it once and stay consistent.)

Internal energy of an ideal gas

U = \frac{f}{2} nRT \qquad dU = n C_V dT

$f$ is the degrees of freedom, $C_V$ the molar specific heat at constant volume. For an ideal gas $U$ depends on temperature alone. Heat capacities satisfy $C_P - C_V = R$ and $\gamma = C_P/C_V$ .

4.2

Thermodynamic Processes

Isobaric (constant P)

Work done is $W = P\,\Delta V$ ; heat changes both $U$ and $W$ .

Isochoric (constant V)

No work is done, $W = 0$ , so $Q = \Delta U = n C_V \Delta T$ .

Isothermal (constant T)

For an ideal gas $\Delta U = 0$ , so $Q = W$ , and the reversible work is

W = nRT \ln\frac{V_2}{V_1}

Adiabatic (Q = 0)

No heat exchange; the process follows $PV^\gamma = \text{constant}$ and $TV^{\gamma-1} = \text{constant}$ , with reversible work

W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}

4.3

Heat Engines, Refrigerators and Carnot

Heat engine efficiency

\eta = \frac{W}{Q_1} = 1 - \frac{Q_2}{Q_1}

$Q_1$ heat absorbed from the hot reservoir, $Q_2$ heat rejected to the cold reservoir and $W = Q_1 - Q_2$ the usable work. The Carnot efficiency is the maximum,

\eta_{Carnot} = 1 - \frac{T_2}{T_1}

Refrigerator / heat pump coefficient of performance

COP = \frac{Q_2}{W}

Useful cooling extracted $Q_2$ per unit work input. For a Carnot (reversible) refrigerator,

COP_{Carnot} = \frac{T_2}{T_1 - T_2}

Entropy and the second law

The entropy change of a reversible process is $dS = dQ_{rev}/T$ . The total entropy of a closed system never decreases: an irreversible process increases it. This sets the direction of every natural process.