Combining the Gibbs relations with the ideal-gas model gives practical formulas for entropy differences.

For an ideal gas,

$$ds=\frac{c_p(T)}{T},dT-R\frac{dp}{p},$$

or equivalently

$$ds=\frac{c_v(T)}{T},dT+R\frac{dv}{v}.$$

If specific heats are approximately constant,

$$s_2-s_1=c_p\ln\frac{T_2}{T_1}-R\ln\frac{p_2}{p_1},$$

and

$$s_2-s_1=c_v\ln\frac{T_2}{T_1}+R\ln\frac{v_2}{v_1}.$$

For an isothermal ideal-gas compression with $p_2/p_1=4$,

$$\Delta s=-R\ln4<0.$$

The gas's entropy decreases, but the second law is not violated: the surroundings must receive enough entropy through heat transfer that total entropy generation remains nonnegative.

Setting $\Delta s=0$ in these relations produces the standard isentropic ideal-gas pressure-temperature-volume relations.