The Second Law of Thermodynamics — Question 10

Question 10

Measurement intervals can straddle the second-law feasibility boundary

A cyclic device has zero net internal-energy and entropy changes. Take heat magnitudes $Q_h,Q_c>0$ for absorption from a hot reservoir and rejection to a cold one. Then $W=Q_h-Q_c$ and $S_{\mathrm{gen}}=Q_c/T_c-Q_h/T_h\ge0$. Heat measurements give $Q_h=1000\pm10$ J and $Q_c=590\pm10$ J, where the intervals are hard bounds. Reservoir temperatures are exactly 500 K and 300 K. No correlation between heat errors is specified.

1. Calculate nominal entropy generation and its full allowed interval.

2. Can these data conclusively establish a second-law violation? State the boundary between compatible and incompatible heat pairs.

Question 10 - Solution

Measurement intervals can straddle the second-law feasibility boundary

(a) Method and calculation

$$S_{{\mathrm{gen}},0}=590/300-1000/500=-1/30\,\mathrm{J/K}.$$$$S_{\min}=580/300-1010/500=-.0866667,\quad S_{\max}=600/300-990/500=.0200\quad\mathrm{J/K}.$$

(b) Interpretation and further calculation

$$S_{\mathrm{gen}}\ge0\ \Leftrightarrow\ Q_c\ge(T_c/T_h)Q_h=.600Q_h.$$$$S_{\min}<0<S_{\max}.$$The nominal point is incompatible, but feasible pairs remain within the stated bounds. Better precision or additional information is needed to establish a violation of the assumed device model.

Physical check

An uncertainty interval is not a probability distribution, so it does not assign a probability of violation.