Q.Show that for a Carnot's engine the efficiency η = 1 − T₂/T₁ where T₂ < T₁.

Carnot Efficiency: The Ultimate Limit

Imagine you have a heat engine — a device that takes in heat from a hot source, does some useful work (like turning a wheel or generating electricity), and dumps the leftover heat into a cold sink. A steam engine, a car engine, a power plant — they all do this.

Your first question should be: How much of the heat I put in can I actually turn into work? That fraction is the efficiency. If you put in 100 J of heat and get 40 J of work out, the efficiency is 0.4 (or 40%). The rest — 60 J — is wasted heat.

Now, can you ever get 100% efficiency? Turn all the heat into work, with zero waste? Intuition says no. Heat flows spontaneously from hot to cold, not the other way. To extract work, you must let some heat fall to the cold sink — that's the price of doing business. The question is: what's the best you can possibly do?

That's where Carnot comes in.

The Carnot Engine: A Perfect, Idealized Machine

In 1824, Sadi Carnot imagined a perfectly reversible engine — one that operates without any friction, heat loss, or wasteful turbulence. It's a theoretical ideal, not something you can build. But it sets the absolute upper limit on efficiency for any heat engine working between two temperatures.

The Carnot engine works in a cycle of four reversible steps: two isothermal (constant temperature) and two adiabatic (no heat exchange). The details of the cycle matter less than the result.

Here:

• $T_h$ = absolute temperature of the hot reservoir (source), in Kelvin
• $T_c$ = absolute temperature of the cold reservoir (sink), in Kelvin

That's it. The efficiency depends only on the two temperatures. Nothing else — not the working substance, not the design, not the size.

What This Tells You

First, notice the fraction $T_c/T_h$. If the cold sink is at absolute zero (0 K), then $T_c/T_h = 0$ and efficiency = 1 (100%). But you can never reach absolute zero — that's the third law of thermodynamics. So 100% efficiency is impossible.

Second, the bigger the temperature difference, the higher the efficiency. A hot source at 600 K and a cold sink at 300 K gives $\eta = 1 - 300/600 = 0.5$ (50%). If you raise the hot source to 900 K (same cold sink), $\eta = 1 - 300/900 \approx 0.667$ (66.7%). Hotter source = better efficiency.

Third, no real engine can beat this. A steam turbine, a car engine, a jet engine — all have efficiencies lower than the Carnot limit for their operating temperatures. The Carnot efficiency is the ceiling.

Why Only Temperatures? …