Isothermal Process

An isothermal process occurs at constant temperature. For an ideal gas, since $T$ is constant, the product $PV$ is also constant:

$PV = nRT = \text{const}$

Work Done in an Isothermal Process

When a gas expands isothermally from volume $V_1$ to $V_2$, the work done by the gas is found by integrating $P\,dV$ with $P = nRT/V$:

$W = \int_{V_1}^{V_2} P\,dV = nRT \int_{V_1}^{V_2} \frac{dV}{V} = nRT \ln\!\left(\frac{V_2}{V_1}\right) = P_1 V_1 \ln\!\left(\frac{V_2}{V_1}\right)$

If $V_2 > V_1$ (expansion), $W > 0$ — the gas does positive work. If $V_2 < V_1$ (compression), $W < 0$.

Heat and Internal Energy

For an ideal gas, internal energy depends only on temperature. Since $T$ is constant:

$\Delta U = 0 \implies Q = W$

All heat absorbed from the surroundings is converted entirely into work.

Boyle's Law

At constant temperature, pressure and volume are inversely proportional:

$P_1 V_1 = P_2 V_2 \implies P_2 = \frac{P_1 V_1}{V_2}$

So doubling the volume halves the pressure — and vice versa.

Your Task

Implement the two functions below.