When two identical bodies with internal energy U=NCT are connected to a carnot engine and used as a source of work, they exchange heat until they reach a common final temperature Tf. The carnot engine operates in a reversible cycle and has a maximum efficiency, given by the Carnot efficiency, which is given by:

η = 1 - Tc/Th

where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir.



We can use the first law of thermodynamics to determine the final temperature Tf of the two bodies. The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system:

ΔU = Q - W

where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.

Since the two bodies have the same internal energy, we can write:

ΔU = U2 - U1 = NCT - NCT = 0

Therefore, the heat added to one body is equal to the work done by the other body:

Q1 = -W2

Now, let us consider the carnot engine. The carnot engine operates between two reservoirs, one at temperature Th and the other at temperature Tc. The carnot engine absorbs heat Qh from the hot reservoir and rejects heat Qc to the cold reservoir. The work done by the carnot engine is given by:

W = Qh - Qc

Since the carnot engine operates in a reversible cycle, the efficiency is given by:

η = 1 - Tc/Th

Therefore, we can write:

Qh = η Th W

Qc = (1-η) Tc W

Since the two bodies are used as a source of work, we can write:

Q1 = Qh

Q2 = Qc

Substituting the expressions for Qh and Qc in terms of W, we get:

Q1 = η Th W

Q2 = (1-η) Tc W

Substituting the expression for Q1 in terms of W, we get:

η Th W = -W2

W2 = -η Th W

Substituting the expression for W2 in terms of Q2, we get:

Q2 = (1-η) Tc (-η Th W)

Substituting the expressions for Q1 and Q2 in terms of Tf, we get:

NCT1 = η Th (-W2) + NCTf

NCT2 = (1-η) Tc (-η Th W) + NCTf

Since the two bodies have the same internal energy, we can equate NCT1 and NCT2:

η Th (-W2) + NCTf = (1-η) Tc (-η Th W) + NCTf

Simplifying the expression, we get:

η Th W2 = (1-η) Tc η Th W

W2 = (1-η)/η Tc Th W

Substituting the expression for W2 in terms of W, we get:

W = η/(2-η) NCT (T