Problem — True or false on engines and refrigerators

Problem

True or false?

• (a) A heat engine can convert all the heat absorbed entirely into work.
• (b) All reversible engines operating between the same temperatures have the same efficiency.
• (c) In a refrigerator, the work spent is required by the second law.
• (d) The Carnot efficiency depends on the working fluid.
• (e) The COP of a refrigerator can be greater than $1$.

Solution

(a) FALSE. The Kelvin-Planck statement of the second law forbids a cycle whose sole effect is to convert heat entirely into work. That would mean $\eta = 1$, i.e. $Q_f = 0$: impossible, some heat must always be rejected to the cold reservoir.

$\eta = 1 - \frac{Q_f}{Q_c} < 1$

(b) TRUE. This is Carnot’s theorem: all reversible engines operating between the same two temperatures have the same efficiency $\eta_{\text{C}} = 1 - T_f/T_c$, regardless of construction details.

(c) TRUE. Heat does not flow spontaneously from cold to hot (Clausius statement). To pump heat uphill thermally, work must be done from outside.

(d) FALSE. $\eta_{\text{C}} = 1 - T_f/T_c$ depends only on the reservoir temperatures, not on the working fluid. It is precisely this universality that makes Carnot an absolute limit.

(e) TRUE. The COP of a refrigerator is $\beta = Q_f/W = T_f/(T_c - T_f)$ in the Carnot limit, and when $T_c - T_f$ is small compared with $T_f$, $\beta \gg 1$ (far more heat is extracted than work spent).

$\ev{\text{(a) F} \quad \text{(b) T} \quad \text{(c) T} \quad \text{(d) F} \quad \text{(e) T}}$

Collegamenti

Argomenti: Heat engines · Thermodynamics Concetti: Efficiency · Second law of thermodynamics · Carnot cycle · Refrigerator and heat pump Competenze: Ranking reasoning Oggetti: Heat engine · Refrigerator