. (40 points) Consider an insulated container of volume V2. N idea gas molecules are initially confined within volume V, by a piston and the remaining volume V2 – Vi is in vacuum. Let T,, P1, E1, S, Al, Hi, G, be the temperature, pressure, energy, entropy, Helmholtz free energy, enthalpy, and Gibbs free energy of the ideal gas at this state, respectively V1 V2-V Using Sackur-Tetrode equation for the entropy of ideal gas where kB R/NA is Boltzmann’s constant (kB-1.38e- 23JK) with R the universal constant and N the Avogadro constant, h is the Planck constant (h-6.626e- 34Js) do the following a) Imagine that the piston is suddenly removed so that the gas has volume V2. After some time the system settles down to equilibrium again. What are the temperature T2, pressure p,, energy E,, entropy S2, Helmholtz free energy A2, enthalpy H2, and Gibbs free energy G in the new equilibrium state? Express them in terms of difference from state 1 using VV, unless the difference is 0. Mark the initial and final states in the p-V plot and the T-S plot. b) Suppose we move the piston infinitely slowly (a reversible process) to let the gas expand to the full volume V. The gas container is thermally insulated during this process. What is the work done to the system? What are T2, p2, E,, Аг, Нг, G, in the final equilibrium state? Express them in terms of the thermodynamic functions of state 1 and V./V. Mark the initial and final states in the p-V plot and the T-S plot.
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