Problem — Carnot efficiency as T_f tends to zero

Problem

What does the Carnot efficiency become if the cold reservoir’s temperature tends to zero kelvin? Is that actually achievable? Which principle of thermodynamics forbids it?

Solution

Limit of the efficiency. The Carnot efficiency is

$\eta = 1 - \frac{T_f}{T_c}$

As $T_f \to 0$ the ratio $T_f/T_c \to 0$, so

$\lim_{T_f \to 0} \eta = \ev{1}$

that is, $100\%$: all the absorbed heat would be converted into work, with nothing rejected to the cold side.

Why it’s impossible. A unit efficiency would mean an engine that fully converts heat into work without giving up anything — a direct violation of the Kelvin statement of the second law. The deeper point, though, is that $T_f = 0\;\text{K}$ is unreachable: the third law of thermodynamics states that absolute zero cannot be reached in a finite number of processes. Each cooling step extracts only a fraction of the remaining heat, so zero stays an asymptotic limit.

So $\eta = 1$ is an unreachable ideal: no real engine can have access to a $0\;\text{K}$ reservoir, and the coldest environment available ($\sim 300\;\text{K}$, or at best space at $\sim 3\;\text{K}$) sets a much lower limit.

Collegamenti

Argomenti: Macchine termiche Concetti: Rendimento · Ciclo di Carnot · Secondo principio della termodinamica Competenze: Analisi di casi limite e fantafisica