Calculate the final temperature of the reversible adiabatic (answer provided but need a solution)

Question

Consider two identical bodies (not necessarily an ideal gas), each having moles and molar heat capacity \(c_p\). The two bodies are placed in thermal contact in an adiabatic enclosure and have initial temperature \(T_1\) and \(T_2\) respectively.

Assume that the entire process occurs at constant pressure.
a) Compute the final temperature (show your work).

\(Ans: T_f=(\frac{T_1+T_2}{2})\)

Now consider these two bodies being brought into thermal equilibrium by a reversible Carnot engine operating between them (the bodies are still in an adiabatic enclosure). The size of the cycle is small so that the temperature of the bodies behave as reservoirs during one cycle.
 

b) What is the final temperature after this process?

\(T_f= \sqrt{T_1T_2}\)

Answer 1

(a) Let say , final temperature = T_f

\(\because\) System is in an adiabatic enclosure, there is no loss of heat into surrounding 

Heat loss by body (A) = Heat gain by body (B)

C_pΔT_A = C_pΔT_B

C_p(T₁ - T_f) = Cp (T_f - T₂)

= T₁ - T_f = T_f - T₂

= 2T_f = T₁ + T₂ 

= T_f = \(\cfrac{T_1+T_2}2\)