Adiabatic Process

In an adiabatic process, no heat is exchanged with the surroundings ($Q = 0$). From the First Law of Thermodynamics:

$\Delta U = Q - W = -W$

Any work done by the gas comes entirely at the expense of its internal energy, so the temperature changes.

Adiabatic Relations for an Ideal Gas

The ratio of heat capacities $\gamma = C_p / C_v$ governs adiabatic behaviour. Two useful relations:

Temperature–volume: $TV^{\gamma-1} = \text{const} \implies T_2 = T_1\!\left(\frac{V_1}{V_2}\right)^{\!\gamma-1}$

Pressure–volume: $PV^{\gamma} = \text{const} \implies P_2 = P_1\!\left(\frac{V_1}{V_2}\right)^{\!\gamma}$

Work Done by the Gas

Since $Q = 0$, the work equals the drop in internal energy:

$W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} = \frac{nR(T_1 - T_2)}{\gamma - 1}$

Heat Capacity Ratios

Your Task

Implement the two functions below.