Entropy and the Second Law

Entropy ($S$) is a state function measuring the dispersal of energy — loosely, the "disorder" of a system. For any reversible process:

$dS = \frac{dQ_{\text{rev}}}{T} \implies \Delta S = \int \frac{dQ_{\text{rev}}}{T}$

Isothermal Entropy Change

At constant temperature, the integral simplifies to:

$\Delta S = \frac{Q}{T}$

Entropy Change of an Ideal Gas

For a general process taking an ideal gas from $(T_1, V_1)$ to $(T_2, V_2)$:

$\Delta S = nC_v \ln\!\frac{T_2}{T_1} + nR\ln\!\frac{V_2}{V_1}$

The first term captures the temperature change; the second captures the volume change.

Phase Transitions

At a phase transition (e.g. melting or boiling) the process is isothermal at temperature $T$:

$\Delta S = \frac{\Delta H}{T}$

where $\Delta H$ is the enthalpy of the transition (latent heat per mole).

The Second Law

The total entropy of an isolated system never decreases:

$\Delta S_{\text{universe}} \geq 0$

Equality holds for reversible processes; strict inequality holds for irreversible ones.

Your Task

Implement the three functions below.