Carnot Cycle: Efficiency, Work, and the Second Law Limit
The Carnot cycle is an ideal reversible heat-engine cycle operating between a hot reservoir at
temperature \(T_h\) and a cold reservoir at \(T_c\) (Kelvin). Its efficiency is the
maximum possible for any engine between those two temperatures.
Key results
\[
\eta_{\text{Carnot}} = 1 - \frac{T_c}{T_h}
\]
\[
W_{net} = Q_h - Q_c = Q_h\,\eta_{\text{Carnot}}
\]
\[
\text{(reversible)}\quad \Delta S = \frac{Q_h}{T_h} = \frac{Q_c}{T_c}
\]
The second law implies \(\eta \le \eta_{\text{Carnot}}\) for any real engine operating between \(T_h\) and \(T_c\).
Carnot cycle steps (reversible)
-
Isothermal expansion at \(T_h\): the working fluid absorbs heat \(Q_h\) and its entropy increases:
\[
\Delta S = \frac{Q_h}{T_h}.
\]
-
Adiabatic (isentropic) expansion: no heat transfer (\(Q=0\)), entropy constant, temperature drops from \(T_h\) to \(T_c\).
-
Isothermal compression at \(T_c\): the fluid rejects heat \(Q_c\) and entropy decreases by the same magnitude:
\[
\Delta S = \frac{Q_c}{T_c}.
\]
-
Adiabatic (isentropic) compression: no heat transfer (\(Q=0\)), entropy constant, temperature rises back to \(T_h\).
Reversibility enforces \(\dfrac{Q_c}{Q_h}=\dfrac{T_c}{T_h}\), which leads directly to \(\eta=1-\dfrac{T_c}{T_h}\).
Why the TS diagram is a rectangle
On a temperature–entropy diagram, reversible heat transfer satisfies:
\[
\delta Q_{rev} = T\,dS.
\]
Along an isotherm, \(T\) is constant, so \(Q = \int T\,dS = T\,\Delta S\).
Along a reversible adiabat, \(dS=0\) (vertical line).
Therefore the Carnot cycle forms a rectangle in TS:
top edge at \(T_h\), bottom edge at \(T_c\), and vertical sides at constant \(S\).
Area equals net work
\[
Q_{net} = \oint \delta Q_{rev} = \oint T\,dS
\]
\[
\text{For a cycle: }\Delta U = 0 \Rightarrow W_{net}=Q_{net}
\]
\[
\Rightarrow W_{net} = (T_h-T_c)\,\Delta S
\]
The calculator shades the TS rectangle so you can see the “work = enclosed area” idea directly.
Worked example
Given \(T_h=600\,\text{K}\), \(T_c=300\,\text{K}\), \(Q_h=1000\,\text{J}\):
\[
\eta = 1 - \frac{300}{600} = 0.5 = 50\%
\]
\[
W_{net} = Q_h \eta = 1000(0.5)=500\ \text{J}
\]
\[
Q_c = Q_h - W_{net} = 500\ \text{J}
\]
\[
\Delta S = \frac{Q_h}{T_h} = \frac{1000}{600}\approx 1.667\ \text{J/K}
\]
Real engines vs. Carnot benchmark
Real engines have irreversibilities (friction, finite temperature gradients, turbulence, non-ideal components),
so they always satisfy:
\[
\eta_{real} < \eta_{\text{Carnot}}.
\]
The Carnot efficiency is a benchmark: it tells you the best you can possibly do between
\(T_h\) and \(T_c\), no matter the working fluid.
A common performance metric is \(\eta_{real}/\eta_{\text{Carnot}}\), which the calculator can show if you enter \(\eta_{real}\).
