As this equation relates changes in thermodynamic state variables, it is valid for any quasistatic change, regardless of whether it is irreversible or reversible. For the above defined path we have that d U = 0 and thus T d S = P d V, and hence the increase in entropy for the Joule expansion is Δ Δ S = ∫ ∫ i f d S = ∫ ∫ V 0 2 V 0 P d V T = ∫ ∫ V 0 2 V 0 n R d V V = n R ln ⁡ ⁡ 2.