Consider the direct heat transfer from a heat reservoir at T 1 to another heat reservoir at

Question:

Consider the direct heat transfer from a heat reservoir at T₁ to another heat reservoir at temperature T₂, where T₁ > T₂ > T_σ. It is not obvious why the lost work of this process should depend on T_σ, the temperature of the surroundings, because the surroundings are not involved in the actual heat-transfer process. Through appropriate use of the Carnot-engine formula, show for the transfer of an amount of heat equal to |Q| that