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Why does a larger temperature difference increase the maximum efficiency of a Carnot engine?

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Engraved portrait of Rudolf Julius Emanuel Clausius, a bearded man in formal attire with sideburns, facing forward. — thermodynamics Rudolf Clausius (1822 - 1888) is considered one of the central founders of the science of thermodynamics.

The maximum efficiency of a Carnot engine is determined by the temperatures of the heat reservoirs between which it operates. Specifically, the efficiency ( $η$ ) is given by the relationship $η = 1 - T_{2} / T_{1}$ , where $T_{1}$ is the temperature of the hot reservoir and $T_{2}$ is the temperature of the cold reservoir.12

A larger temperature difference increases efficiency because it allows the engine to minimize the amount of waste heat ( $Q_{2}$ ) that must be exhausted to the environment. According to the second law of thermodynamics, a heat engine cannot convert all absorbed heat ( $Q_{1}$ ) into work because the net entropy change of the system must be greater than or equal to zero ( $Δ S \geq 0$ ).312

To maximize work ( $W = Q_{1} - Q_{2}$ ), one must minimize $Q_{2}$ relative to $Q_{1}$ . The theoretical limit occurs when $Δ S = 0$ , which requires $Q_{2} / T_{2} = Q_{1} / T_{1}$ . Consequently, as the ratio $T_{2} / T_{1}$ decreases—achieved by increasing the temperature difference—the fraction of heat that must be discarded as waste decreases, thereby increasing the maximum possible efficiency.12