(a) State and prove Clausius inequality theorem.

(b) Nitrogen is compressed from an initial state of 1 bar and \(25^{\circ} \mathrm{C}\) to a final state of \(5 \mathrm{bar}\) and \(25^{\circ} \mathrm{C}\) by three different reversible processes in a closed system.
(i) Heating at constant volume followed by cooling at constant pressure;
(ii) Isothermal expansion;
(iii) Adiabatic compression followed by cooling at constant volume.
Nitrogen is assumed to behave an ideal gas with \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\), and the molar volume of nitrogen \(=0.02479 \mathrm{~m}^{3} / \mathrm{mol}\). Estimate the heat, work requirement, \(\Delta U\) and \(\Delta H\) of nitrogen for the each process.