Question:

Consider a Carnot heat engine operating between a hot reservoir at a temperature T1 and a cold reservoir at a temperature T2, where T1 > T2.

(a) Derive an expression for the efficiency of the Carnot engine in terms of the temperatures T1 and T2.

(b) Assume that the Carnot engine absorbs Q1 amount of heat from the hot reservoir and exhausts Q2 amount of heat to the cold reservoir. Express the efficiency of the Carnot engine in terms of Q1 and Q2.

(c) Prove that the efficiency of the Carnot engine is independent of the working substance used.

(d) Two Carnot engines, one operating between temperatures T1 and T2, and the other operating between temperatures T1 and T3, are interconnected, such that the heat supplied by the hot reservoir to the first engine is the heat exhausted by the second engine to the cold reservoir. Derive an expression for the efficiency of this combined system in terms of T1, T2, and T3.

Answer:

(a) The efficiency of a Carnot engine is given by the equation:

Efficiency (η) = 1 - (T2 / T1)

To derive this expression, we can use the Second Law of Thermodynamics, which states that the efficiency of a heat engine cannot exceed the efficiency of a reversible engine operating between the same two temperatures.

For a reversible engine, the work done per cycle is given by:

Work (W) = Q1 - Q2

where Q1 is the heat absorbed from the hot reservoir and Q2 is the heat exhausted to the cold reservoir.

The efficiency of the reversible engine is given by:

Efficiency_reversible (η_rev) = W / Q1 = (Q1 - Q2) / Q1 = 1 - (Q2 / Q1)

We know that for a reversible engine, the ratio of the heat transfers is given by:

Q1 / T1 = Q2 / T2

Simplifying this equation, we get:

Q2 = Q1 * (T2 / T1)

Substituting this expression back into the equation for the efficiency of the reversible engine:

Efficiency_reversible (η_rev) = 1 - (Q2 / Q1) = 1 - (T2 / T1)

Therefore, we have derived the expression for the efficiency of the Carnot engine in terms of the temperatures T1 and T2.

(b) The efficiency of the Carnot engine can also be expressed in terms of the heat transferred, Q1 and Q2.

Using the equation derived in part (a), we have:

Efficiency (η) = 1 - (T2 / T1)

Similarly, we can express the ratio of the heat transfers as:

Q1 / T1 = Q2 / T2

Rearranging this equation, we can express Q2 in terms of Q1:

Q2 = Q1 * (T2 / T1)

Substituting this expression into the equation for efficiency, we have:

Efficiency (η) = 1 - (Q2 / Q1) = 1 - (Q1 * (T2 / T1) / Q1) = 1 - (T2 / T1)

Therefore, we have shown that the efficiency of the Carnot engine can be expressed in terms of the heat transfers, Q1 and Q2.

(c) To prove that the efficiency of the Carnot engine is independent of the working substance used, we can use the fact that the efficiency of the Carnot engine is solely determined by the temperatures of the reservoirs.

As derived in part (a), the efficiency of the Carnot engine is given by:

Efficiency (η) = 1 - (T2 / T1)

This expression does not depend on any specific properties or details of the working substance. It solely depends on the temperatures T1 and T2.

Therefore, the efficiency of the Carnot engine is indeed independent of the working substance used.

(d) To find the efficiency of the combined system of two interconnected Carnot engines, we can consider the heat transfers in each individual engine.

Let the first engine (engine 1) operate between temperatures T1 and T2, and let the second engine (engine 2) operate between temperatures T1 and T3.

By the definition of a Carnot engine, we know that:

Q1_1 / T1 = Q2_1 / T2   (1)   // Heat transfers for engine 1
Q1_2 / T1 = Q2_2 / T3   (2)   // Heat transfers for engine 2

Given that the heat supplied by the hot reservoir to engine 1 is the heat exhausted by engine 2 to the cold reservoir, we have:

Q1_1 = Q2_2   (3)

To find the efficiency of the combined system, we need to determine the ratio of the work done to the heat absorbed from the hot reservoir, using the expression derived in part (a):

Efficiency_combined (η_combined) = 1 - (T2 / T1)

Substituting the heat transfers for engine 1 and engine 2 from equations (1) and (2), we get:

Efficiency_combined (η_combined) = 1 - (Q2_1 / T1) = 1 - (Q1_2 / T3)

Using equations (3) and (2), we can eliminate Q2_2 and Q1_2, and obtain an expression for the efficiency of the combined system:

Efficiency_combined (η_combined) = 1 - (Q2_1 / T1) = 1 - ((Q1_1 * T3) / (T1 * T2))
                                  = 1 - (Q1_1 / (T1 * T2 / T3))
                                  = 1 - (Q1_1 / (T1 * T3 / T2))

Since Q1_1 / T1 = Q2_1 / T2 from equation (1), we can rewrite the expression as:

Efficiency_combined (η_combined) = 1 - (Q2_1 / T2) = 1 - (Q2_1 / (T3 * (T2 / T3)))
                                  = 1 - (Q2_1 / (T3 * T2)) = 1 - (Q1_1 / (T1 * T2))

Therefore, the expression for the efficiency of the combined system of interconnected Carnot engines is the same as the expression for the efficiency of a single Carnot engine, which is 1 - (T2 / T1).