Thermodynamics for JEE: the 12 formulas + when to apply each

Thermodynamics is a small chapter that punches above its weight in JEE — 2 to 3 questions per Main, often a 4-marker in Advanced. The trick is not memorizing more formulas; it is knowing which formula applies under which condition.

The state functions

Five state functions appear in every problem. Only their changes matter — never the absolute value.

• Internal energy $U$: depends only on temperature for an ideal gas.
• Enthalpy $H = U + PV$: the natural variable for constant-pressure processes.
• Entropy $S$: measure of disorder. Always increases for a spontaneous process in an isolated system.
• Gibbs free energy $G = H - TS$: the natural variable for constant-T, constant-P processes.
• Helmholtz free energy $A = U - TS$: appears occasionally in Advanced.

The 12 formulas

First law

$Delta U = q + w$

with the sign convention $w = -P_{ext} Delta V$ for expansion against constant external pressure.

Work in each process type

• Isothermal reversible (ideal gas): w = -nRT lnrac{V_2}{V_1}
• Isothermal irreversible: $w = -P_{ext}(V_2 - V_1)$
• Adiabatic reversible: $w = Delta U = nC_v Delta T$
• Adiabatic irreversible: still $w = Delta U$ because $q = 0$ — but you must compute $Delta T$ from $P_{ext}Delta V = -nC_v Delta T$.

Enthalpy vs internal energy

$Delta H = Delta U + Delta n_g RT$

where $Delta n_g$ is the change in moles of gas. This is the single most under-used formula in JEE — always check whether the reaction involves a change in gaseous moles before using $Delta H$ and $Delta U$ interchangeably.

Heat capacities

For an ideal gas:

• $C_p - C_v = R$
• $C_p / C_v = gamma$ (5/3 for monoatomic, 7/5 for diatomic at low T)

Adiabatic relations

$PV^gamma = ext{const}, quad TV^{gamma-1} = ext{const}, quad TP^{(1-gamma)/gamma} = ext{const}$

Entropy change

• Isothermal expansion of ideal gas: Delta S = nR lnrac{V_2}{V_1}
• Heating at constant pressure: Delta S = nC_p lnrac{T_2}{T_1}
• Phase change at constant T: Delta S = rac{Delta H_{transition}}{T}

Gibbs free energy and spontaneity

$Delta G = Delta H - T Delta S$

At equilibrium, $Delta G = 0$. For a reaction:

$Delta G = Delta G^circ + RT ln Q$

At equilibrium, $Q = K_{eq}$ and $Delta G = 0$, so $Delta G^circ = -RT ln K_{eq}$.

Hess's law

$Delta H$ for a reaction is the sum of $Delta H$ for any sequence of steps that takes you from reactants to products. Always check that the equations add stoichiometrically before adding their enthalpies.

The four common process types

JEE problems are almost always one of these four:

Memorize this table. About 80% of thermodynamics MCQs in Main reduce to looking up the right row.

Where JEE trips students up

Trap 1: confusing reversible and irreversible work. Reversible work is the maximum (in magnitude) for expansion and the minimum for compression. Irreversible work against constant $P_{ext}$ is always less.

Trap 2: forgetting that $Delta H eq q$ in general. $Delta H = q$ only at constant pressure. In a sealed bomb calorimeter, you measure $Delta U$, not $Delta H$.

Trap 3: sign of $Delta S$ in mixed phase changes. Going from gas to liquid decreases entropy. Always check whether the problem is asking for system, surroundings, or universe entropy.

A 30-second method for spontaneity questions

If a question asks whether a process is spontaneous and gives you $Delta H$ and $Delta S$:

• $Delta H < 0$, $Delta S > 0$ → spontaneous at all $T$.
• $Delta H > 0$, $Delta S < 0$ → non-spontaneous at all $T$.
• $Delta H < 0$, $Delta S < 0$ → spontaneous at low $T$ (when $|TDelta S| < |Delta H|$).
• $Delta H > 0$, $Delta S > 0$ → spontaneous at high $T$.

That's the chapter. Learn the table, learn the four traps, and you'll have a full-mark chapter in your hands.