Reversible isotherm

The reversible isotherm is the transformation at constant temperature: $T_B = T_A = T$. The gas remains in contact with a single thermostat at temperature $T$, and precisely because the temperature never varies the gas does not need to jump between different thermostats: a single heat bath is enough to maintain equilibrium.

Since for an ideal gas the internal energy depends only on temperature, at constant $T$ it does not change: $\Delta E_\text{int} = 0$. The first law then reduces to a direct equality between the heat absorbed and the work done,

$Q = L_\text{gas}$

all the heat that enters is entirely converted into work, with no reserves stored in the gas.

The change in entropy of the gas follows from the general formula by cancelling out the temperature term, since $\ln(T_B/T_A) = 0$:

$\Delta S_\text{gas} = nR\,\ln\frac{V_B}{V_A}$

The value of the heat is obtained by imposing reversibility. The entropy of the thermostat changes by $\Delta S_\text{term} = -Q/T$ (it loses the heat gained by the gas), and the condition $\Delta S_\text{tot} = 0$ becomes:

$nR\,\ln\frac{V_B}{V_A} - \frac{Q}{T} = 0 \;\Rightarrow\; Q = nRT\,\ln\frac{V_B}{V_A}$

Principle — Reversible isotherm

$\begin{aligned} T_B &= T_A = T \\ \Delta E_\text{int} &= 0 \\ Q &= L_\text{gas} = nRT\,\ln\frac{V_B}{V_A} \\ \Delta S_\text{gas} &= nR\,\ln\frac{V_B}{V_A} \end{aligned}$

Links

Topics: Thermodynamics Concepts: Thermodynamic transformations · Entropy · First law of thermodynamics · Internal energy · Second law of thermodynamics Skills: Entropy balance · Solving a thermodynamic cycle Objects: Ideal gas

Related exercises: Worked exercise — reversible vs irreversible isothermal expansion · Worked exercise — free expansion is not reversible · Gas in reversible adiabatic expansion