Question

A certain heat engine operates between 1000 K and 500 K, where 1000 K is the temperature for the heat source and 500 K is the temperature for the cold sink. (a) What is the theoretical (or maximum) efficiency of the engine? (b) Calculate the maximum work that can be done by for each 1.0 kJ of heat supplied by the hot source. (c) How much heat is discharged into the cold sink in a reversible process for each 1.0 kJ supplied by the hot source?

Answer 1

The maximum theoretical efficiency of a engine operating between 1000 K and 500 K is 50%, meaning for each 1.0 kJ of heat supplied, 0.5 kJ can be converted to work, and 0.5 kJ is discharged to the cold sink.

Step-by-step explanation:

The theoretical (or maximum) efficiency of a heat engine operating between two temperatures can be calculated using the formula for Carnot efficiency, which is E = 1 - (Tc / Th), where E is the efficiency, Tc is the cold sink temperature, and Th is the hot source temperature. In this case:

• Th = 1000 K
• Tc = 500 K

Putting these values into the formula gives:

E = 1 - (500 K / 1000 K) = 1 - 0.5 = 0.5 or 50%

(b) The maximum work that can be done by the engine for each 1.0 kJ of heat supplied from the hot source is calculated by multiplying the efficiency by the heat input:

W = E * Qh = 0.5 * 1.0 kJ = 0.5 kJ (or 500 J).

(c) The amount of heat discharged into the cold sink in a reversible process for each 1.0 kJ supplied by the hot source is the remainder of the heat that is not converted into work:

Qc = Qh - W = 1.0 kJ - 0.5 kJ = 0.5 kJ (or 500 J).