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State and prove Carnot theorem
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Carnot Theorem:

The Carnot theorem states: No heat engine operating on a cycle between two heat reservoirs at different fixed temperatures can be more efficient than a reversible engine. Two reversible heat engines operating between two heat reservoirs at different fixed temperatures will have the same efficiency.

Proof of Statement (1):

Consider two heat reservoir at fixed temperatures T1 and T2 (T1 > T2). A reversible engine R and irreversible engine (I) are operating between the same two thermal reservoir as shown in Fig.

The engine I takes in heat Q1, rejects heat Q2 and does the work W1

W1 = Q1 − Q2

While the reversible engine takes in heat Q1’ rejects heat Q2’ and does the work W1’

W1’ = Q1’ − Q2’

Both the engines are so adjusted that they produce equal amount of work i.e. W1 = W1’ Q1 − Q2= Q1’ − Q2’…………….(i)

Assume that efficiency of irreversible engine in greater than the efficiency of reversible engine.

ηI > ηR

W1/ Q1 > W1’/ Q1’

Q1’> Q1 …………….

(ii) since W1 = W1’

from eqns (i) and (ii)

Q2’> Q2

It follows from equation (ii) and (iii) that more efficient reversible engine will abstract less amount of heat from source and rejects less amount of heat to the sink compared to less efficient reversible engine provided both produce equal amount of work.

The reversible engine can now be operated as a heat pump since the engine is reversible, it is possible and the work developed by the irreversible engine is used to drive the pump as shown in fig.

The magnitude of heat and work transfer for the reversible heat engine will remain the same but their directions will be reversed when it works as a heat pump.

Combining engine I and heat pump R into one system, We observe that the sole effect of the combined system is the transfer of heat energy Q2’- Q2 equal to Q1’- Q1 from low temperature heat reservoir to high temperature reservoir without any external effect violates the Clausius statement of second law of thermodynamics.

Therefore basic assumption ηI > ηR is wrong.

Therefore the efficiency of an irreversible engine cannot be greater than that of the reversible engine if both operate between the fixed temperature heat reservoirs.

ηR ≥ ηI

Proof of statement (2):

Let both the engines I and R be reversible engines. Assuming that either of the reversible engine with higher efficiency to operated upon as an engine and the other less efficient reversible engine as heat pump.

We can show that in either case the second law of thermodynamics is violated. Hence either of the reversible can be more efficient than the other.

We conclude that, "All reversible engines will have the same efficiency when operating between two fixed temperature heat reservoirs."