Isothermal vs AdiabaticJEE Main

Two ways to expand the same gas — interactive Physics simulation for IIT-JEE.

Concept

Expand a gas the same amount two ways. Isothermal (slow, $T$ constant) keeps $PV$ fixed. Adiabatic (fast, no heat) keeps $PV^\gamma$ fixed, so it falls more steeply and cools the gas. The work done — area under each P–V curve — differs.

Key formula

PV = \text{const}, \quad W_{iso}=nRT\ln\frac{V_2}{V_1}; \qquad PV^\gamma=\text{const}, \quad W_{adia}=\frac{P_1V_1-P_2V_2}{\gamma-1}

Derivation

Isothermal: with $T$ fixed, $\Delta U = 0$ , so by the first law $Q = W = nRT\ln(V_2/V_1)$ .

Adiabatic: $Q = 0$ , so $W = -\Delta U = nC_v(T_1 - T_2)$ , which rearranges to $(P_1V_1 - P_2V_2)/(\gamma-1)$ . Because $PV^\gamma$ is constant, the adiabat is steeper than the isotherm and the gas does less work while expanding.

Scenarios to explore

Isothermal vs Adiabatic — Compare work done in two ways of expanding a gas.

Real-world applications

Diesel engine compression (adiabatic heating ignites fuel).
Cloud formation — rising air expands adiabatically and cools.
Sound propagation is an adiabatic, not isothermal, process (Laplace's correction).

JEE exam tips

For the same expansion, $W_{iso} > W_{adia}$ — the isotherm encloses more area.
Adiabatic relations: $TV^{\gamma-1}=$ const and $T^\gamma P^{1-\gamma}=$ const.

Common mistakes

Using $V$ in litres for SI work — convert to m³ first.
Applying $PV =$ const to an adiabatic step (it is $PV^\gamma$ ).
Forgetting an adiabatic expansion lowers temperature.

Exam traps to avoid

An adiabat is always steeper than the isotherm through the same point (slope ratio $\gamma$ ).
'No heat exchange' (adiabatic) is not the same as 'constant temperature' (isothermal).