Find the minimum attainable pressure of an ideal gas in process T = T 0 + α V 2 , where T 0 and α are positive constants and V is the volume of one mole of gas.

Step-by-step solution

Given, T = T 0 + α V 2 ...(i) For 1 mol of a gas, P V = R T of V = R T P Substituting this value in Eq. (i), we get T = T 0 + α R T P 2 = T 0 + α R 2 T 2 P 2 or T P 2 = T 0 P 2 + α R 2 T 2 or P = α R T T − T 0 − 1 / 2 ...(ii) After differentiating, we get d P d T = α R T − T 0 − 1 / 2 − 1 2 T T − T 0 − 3 / 2 For minimum pressure, d P d T = 0 ∴ 0 = α R T - T 0 - 1 / 2 - 1 2 T T - T 0 - 3 / 2 After solving, T = 2T 0 From Eq. (ii), P min = α R 2 T 0 2 T 0 − T 0 − 1 / 2 = 2 R α T 0