CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 8, Exercise 1 Solution
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Calculating Carnot Efficiency

PROBLEM
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An engine operates between a hot reservoir at 600 K and a cold
reservoir at 300 K. Calculate its maximum possible (Carnot)
efficiency.

SOLUTION
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Using the Carnot efficiency formula:

    eta = 1 - Tc / Th

Substitute the given values (both already in Kelvin, an absolute
scale, as the formula requires):

    Tc = 300 K
    Th = 600 K

    eta = 1 - 300/600
    eta = 1 - 0.5
    eta = 0.5, or 50%

ANSWER: The engine's maximum possible efficiency is 50%.

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WHY THIS WORKS AS AN ANSWER

The Carnot efficiency formula depends only on the RATIO of the two
absolute temperatures, not on their individual values - which is why
this problem (600 K and 300 K, a 2:1 ratio) gives a clean, exact 50%
answer. This 50% is a genuine, hard theoretical ceiling: no real
engine operating between exactly these two temperatures, no matter
how well-built or expensive, can exceed it. Any real engine will fall
somewhat short of 50% due to friction, heat losses, and the fact that
no real process is perfectly reversible the way Carnot's own
idealised cycle assumes.