**Question 4:** Suppose that we want to think of energy as a function of temperature and volume, E(T, V). Show that the total differential dE may be written:\[ dE = C_V dT + \left( T \frac{\alpha}{\kappa_T} - P \right) dV. \]In this equation:- \( C_V \) is the heat capacity at constant volume.- \( \alpha \) is the coefficient of thermal expansion.- \( \kappa_T \) is the isothermal compressibility.- \( T \) represents temperature.- \( P \) represents pressure.- \( dT \) and \( dV \) are infinitesimal changes in temperature and volume, respectively. This expression demonstrates how changes in temperature and volume can affect the energy of a system.

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Part A: 10 moles of ideal gas initially at 10 atm and 250 K is expanded isothermally and reversibly to a final pressure of 2 atm. Calculate W, AU, and Q in Joules. Part B: The same gas in problem A, starting in the same initial state, is expanded isothermally against a constant external pressure of 2 atm, to a final pressure of 2 atm. Calculate W, AU, and Q in Joules for this irreversible process. Part C: The same gas in part B, starting in the same initial state, is expanded isothermally to a final pressure of 2 atm. The process is carried out in 4 stages. First the gas is expanded against a constant pressure of 8 atm, to 8 atm. Then the gas is expanded against a constant pressure of 6 atm, to 6 atm. Then the gas is expanded against a constant pressure of 4 atm, to 4 atm. Finally, the gas is expanded against a constant pressure of 2 atm, to 2 atm. Calculate w, AU, and Q in Joules for the 4-step process. How do your answers compare to those you obtained for problems 5 and 6 above?
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Question 4: Suppose that we want to think of energy as a function of temperature and volume, E(T,V). Show that the total differential dE may be written: dE = C,dT + Kr