A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink.

Maximum efficiency of an engine working between temperatures T₂ and T₁ is given by the fraction of the heat absorbed by an engine which can be converted into work is known as efficiency of the heat engine.

Mathematically,
In First case:
Efficiency, η = (T₂ – T₁) / T₂
T₂ = 500 K (temperature of the source)
T₁ = 100 K (temperature of the sink)
Therefore,
η = (T₂ – T₁) / T₂
= (500 – 100) / 500
= 4 / 5

In second case:
Efficiency, η = (T₂ – T₁) / T₂
T₂ = 900 K (temperature of the source)
T₁ = T K (temperature of the sink)
Therefore,
η = (T₂ – T₁) / T₂
= (900 – T) / 900

It is given that the Carnot engine has the same efficiency. Hence equating the efficiency of both the cases we will get:
(900 – T) / 900 = 4 / 5
5 (900 – T) = 3600
4500 – 3600 = 5T
900 / 5 = T
T = 180 K

Hence the temperature T of the sink is equal to 180 K