Thermodynamics formulas for JEE Main & Advanced

18 formula entries for Thermodynamics, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

First Law of Thermodynamics

Δ U = Q - W

When to use. For any thermodynamic process to relate heat, work, and internal energy change.

Trap. Using wrong sign convention: $W$ is work done BY the system; if work is done ON the system, $W$ is negative.

Third Law of Thermodynamics

\lim_{T \to 0} S = 0

When to use. To define absolute entropy scale and understand behavior near absolute zero.

Trap. Thinking entropy becomes zero for all substances at 0 K; only perfect crystals.

Work Done in Thermodynamic Processes

Isothermal:

W = n R T \ln (\frac{V_{f}}{V_{i}})

; Adiabatic:

W = \frac{P_{i} V_{i} - P_{f} V_{f}}{γ - 1}

; Isobaric:

W = P Δ V

; Isochoric:

W = 0

When to use. To calculate work in any given thermodynamic process.

Trap. Using isothermal formula for adiabatic processes; they have different $P - V$ relations.

Internal Energy of Ideal Gas

Δ U = n C_{v} Δ T

When to use. To find change in internal energy when temperature changes, especially in adiabatic or isochoric processes.

Trap. Using $C_{p}$ instead of $C_{v}$ for internal energy change.

Molar Specific Heats

C_{p} - C_{v} = R

;

γ = \frac{C_{p}}{C_{v}}

When to use. To relate heat capacities and compute $γ$ for adiabatic processes.

Trap. Using $C_{p} - C_{v} = R$ for real gases; it's approximate.

Adiabatic Process Relations

P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ}

;

T_{1} V_{1}^{γ - 1} = T_{2} V_{2}^{γ - 1}

When to use. To relate pressure, volume, and temperature in adiabatic processes.

Trap. Using isothermal relation $P V = constant$ for adiabatic processes.

Heat Engine Efficiency

η = 1 - \frac{Q_{c}}{Q_{h}}

; for Carnot engine:

η_{Carnot} = 1 - \frac{T_{c}}{T_{h}}

When to use. To calculate efficiency of any heat engine or compare with Carnot efficiency.

Trap. Using $T_{c} / T_{h}$ directly without converting to Kelvin.

Coefficient of Performance (Refrigerator & Heat Pump)

{COP}_{R} = \frac{Q_{c}}{Q_{h} - Q_{c}} = \frac{T_{c}}{T_{h} - T_{c}}

(Carnot);

{COP}_{H P} = \frac{Q_{h}}{Q_{h} - Q_{c}} = \frac{T_{h}}{T_{h} - T_{c}}

(Carnot)

When to use. To evaluate performance of refrigerators and heat pumps.

Trap. Confusing COP of refrigerator with heat pump; they have different numerators.

Entropy

Δ S = \int \frac{d Q_{rev}}{T}

; for isothermal:

Δ S = \frac{Q}{T}

When to use. To determine spontaneity of processes and calculate entropy changes.

Trap. Using $d Q$ instead of $d Q_{rev}$ ; entropy change is defined for reversible paths.

Carnot Cycle Analysis

η = 1 - \frac{T_{c}}{T_{h}}

When to use. To find maximum possible efficiency of an engine operating between two temperatures.

Trap. Thinking Carnot cycle is the only reversible cycle; it's the most efficient between two temperatures.

Entropy Change Calculations

Δ S = n C_{v} \ln (\frac{T_{f}}{T_{i}}) + n R \ln (\frac{V_{f}}{V_{i}})

for ideal gas

When to use. To compute entropy change for ideal gas processes, phase changes, or mixing.

Trap. Using irreversible path directly; must use reversible path.

Degrees of Freedom and $γ$

C_{v} = \frac{f}{2} R

;

γ = 1 + \frac{2}{f}

When to use. To compute specific heats and $γ$ for different gases.

Trap. Using $f = 3$ for diatomic gases at high temperatures; vibrational modes may be active.

Phase Change and Latent Heat

Q = m L

(where

L

is latent heat)

When to use. To calculate heat transfer during melting, boiling, or condensation.

Trap. Forgetting that $Δ U$ is not zero during phase change; internal energy changes.

Mixing of Gases and Temperature

n_{1} C_{v 1} (T_{f} - T_{1}) + n_{2} C_{v 2} (T_{f} - T_{2}) = 0

When to use. To find final temperature when gases are mixed in an insulated container.

Trap. Using $C_{p}$ instead of $C_{v}$ for internal energy.

Polytropic Processes

W = \frac{P_{2} V_{2} - P_{1} V_{1}}{1 - n}

(for

n \neq 1

); for

n = 1

, isothermal

When to use. For processes that are not strictly isothermal, adiabatic, etc., but follow $P V^{n} = constant$ .

Trap. Using $n = 0$ for isobaric; $n = \infty$ for isochoric; $n = γ$ for adiabatic; confusing these.

Heat Capacity in Polytropic Processes

C = C_{v} + \frac{R}{1 - n}

When to use. To find heat capacity for a given polytropic process.

Trap. Using this formula for $n = 1$ (isothermal); $C$ becomes infinite.

Otto Cycle (Petrol Engine)

η = 1 - \frac{1}{r^{γ - 1}}

When to use. To analyze spark-ignition internal combustion engines.

Trap. Using $r = V_{2} / V_{1}$ instead of $V_{1} / V_{2}$ .

Diesel Cycle

η = 1 - \frac{1}{r^{γ - 1}} (\frac{ρ^{γ} - 1}{γ (ρ - 1)})

When to use. To analyze compression-ignition engines.

Trap. Using Otto cycle formula for Diesel cycle.

Values to remember

$γ = 5 / 3$ for monatomic gas Work Done in Thermodynamic Processes
$γ = 7 / 5$ for diatomic gas Work Done in Thermodynamic Processes
$C_{v} = \frac{3}{2} R$ for monatomic Internal Energy of Ideal Gas
$C_{v} = \frac{5}{2} R$ for diatomic Internal Energy of Ideal Gas
$R = 8.314 J/mol \cdot K$ Molar Specific Heats
$L_{f} = 3.36 \times 10^{5} J/kg$ (water) Phase Change and Latent Heat
$L_{v} = 2.26 \times 10^{6} J/kg$ (water) Phase Change and Latent Heat

Read the full chapter: Thermodynamics for JEE, with what JEE asks and the common traps.