The First Law of Thermodynamics — Question 7

Question 7

Measurement intervals give a rigorous sign test for stored energy

Use $\Delta U=Q-W$, where $Q$ is heat into the system and $W$ is work done by it. A stationary closed system has measured heat input $Q=500\pm20$ J and measured work output $W=480\pm30$ J. Treat both uncertainty ranges as independent hard bounds, not statistical standard deviations.

1. Find the tight possible interval for $\Delta U$. Can its sign be established?

2. Keeping the same heat measurement and 480 J central work reading, determine whether any nonnegative work uncertainty could guarantee $\Delta U>0$. What condition on the central work reading would guarantee positivity if its uncertainty remained 30 J?

Question 7 - Solution

Measurement intervals give a rigorous sign test for stored energy

(a) Method and calculation

$$Q\in[480,520],\quad W\in[450,510]\quad(\mathrm J).$$$$\Delta U\in[Q_{\min}-W_{\max},Q_{\max}-W_{\min}]=[-30,70]\,\mathrm J.$$Both signs and zero are allowed.

(b) Interpretation and further calculation

For work half-width $\delta W\ge0$,$$\Delta U_{\min}=480-(480+\delta W)=-\delta W\le0.$$Thus improving work precision alone cannot guarantee strict positivity. With central reading $W_c$ and half-width 30 J,$$480-(W_c+30)>0\ \Longleftrightarrow\ W_c<450\,\mathrm J.$$

Physical check

A positive central estimate of 20 J is not a proof of positive stored-energy change when the allowed interval straddles zero.