A newly proposed device for generating electricity from the sun is a heat engine in which the hot reservoir is created by focusing sunlight on a small spot on one side of the engine. The cold reservoir is ambient air at 20°C. The designer claims that the efficiency will be 50%.
What minimum hot-reservoir temperature, in degrees C, would be required to produce this efficiency?

The minimum temperature of hot reservoir is 586K or 313°C

Explanation:

The Carnot cycle is defined as ideal reversible process thermodynamic process which has four successive step. During the expansion and compression of  the substance it can done upto desired point and then reversed up.

Here the energy is used from the hot reservoir to do work and deposited into the cold reservoir. As is it reversible the efficiency of the carnot cycle is the theoretical maximum of the heat engine.

The efficiency of carnot cycle is

η = 1 -    eq 1

Where Tc is the temperature of cold reservoir = 20° = 20°+273K = 293K

           Th is the temperature of hot reservoir.

            η  is the efficency 50% = 0.5

The minimum temperature of the reservoir is related to the maximum efficiency,

Substituting values in eqn 1

0.05 = 1-

Th = =586 K

The minimum temperature of hot reservoir is 586K.

ie 586K-273 = 313°C