= Isothermal expansion
An <isothermal expansion> increases the <volume (thermodynamics)> of a <thermodynamic system> at fixed <temperature>. For a reversible <ideal gas> path, the <gas> delivers positive <work> and absorbs positive <heat>, with $Q=W_{\rm by,rev}=nRT\ln(V_f/V_i)$ and $\Delta U=0$. The <heat bath> supplies the expansion <energy>; without sufficient <heat transfer>, doing <work> instead lowers the <temperature>. This transfer is the hot step of a <Carnot cycle>, and it is why an expanding <gas> need not cool when its surroundings continuously replenish its <internal energy>.
An insulated <free expansion> into an evacuated compartment is different. In the idealization with no <heat> exchange, no external <work> and no container <energy> change, $Q=W_{\rm by}=0$. Since <internal energy of an ideal gas> depends only on <temperature>, its final equilibrium <temperature> equals its initial value. The transient need not have a uniform <temperature> or <pressure>, so equal endpoint <temperatures> do not make it a reversible isotherm. Do not apply the logarithmic <isothermal ideal-gas work> formula to this irreversible process. Its <entropy> still increases by $\Delta S=nR\ln(V_f/V_i)$; <entropy change in free expansion> evaluates this <state function> difference using a hypothetical reversible path.<Thermodynamics and Statistical Mechanics by Richard Fitzpatrick/Free Expansion of Gas>[Equations 6.144–6.148 and the negligible-container-heat-capacity assumption]{ref}
For a <real gas>, even equal-<temperature> equilibrium endpoints need not have equal <internal energy>. An insulated <free expansion> can therefore change the <temperature>, while a controlled <isothermal expansion> must provide the <heat> needed for both its <internal energy> change and its delivered <work>. The <internal energy at constant temperature> article gives the <Van der Waals equation> example.<Thermodynamics and Statistical Mechanics by Richard Fitzpatrick/Van der Waals Gas>[Equations 6.163–6.166: attractive interactions and free-expansion cooling]{ref}
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