1.) Derive the cat constant temperature, constant volume and constant pressure.  Note: These are three separate expressions.

2.) Calculate the change in entropy when 50 g of water at 80 ^oC is poured into 100 g of water at 10 ^oC in an insulated vessel given that C_{p, m} - 75.5 JK^-1 mol^-1.

3.) Calculate the change in entropies of the system and the surroundings, and the total change in entropy, when a sample of nitrogen gas of mass 14 g at 298 K and 1 bar doubles its volume in (a) an isothermal reversible expansion, (b) an isothermal irreversible expansion against p_ex = 0, and (c) an adiabatic reversible expansion.

4.) Calculate (a) the maximum work and (b) the maximum non-expansion work that can be obtained from the freezing of supercooled water at - 5 ^oC and 1.0 atm.  The densities of water and ice are 0.999 g cm³ and 0.917 g cm³, respectively at  - 5 ^oC.

5.) Consider a system in thermal equilibrium, with its surroundings at temperature, T.  Use the Clausius inequality to derive a differential expression for the Helmhlotz energy (A) and Gibbs energy (G).