The First Law of Thermodynamics — Question 10

Question 10

A thermal energy balance determines an unknown heat leak

Use $\Delta U=Q-W$, where $Q$ is heat into the system and $W$ is work done by it. A stationary closed gas system expands from .0100 to .0140 m$^3$ against constant external pressure 120 kPa. Its temperature rises by 8.00 K, and its supplied caloric model is $\Delta U=C\Delta T$ with $C=100\,\mathrm{J/K}$. A thermal source sends 1500 J into the system, while an unmeasured amount escapes to the surroundings. There is no other work.

1. Find boundary work, internal-energy increase and the escaped heat.

2. For the same imposed volume and temperature changes, find the minimum source heat compatible with an outward or zero leak. Interpret a claimed source input of only 1000 J.

Question 10 - Solution

A thermal energy balance determines an unknown heat leak

(a) Method and calculation

$$W=120000(.0140-.0100)=480\,\mathrm J,\quad\Delta U=100(8)=800\,\mathrm J.$$$$Q_{\mathrm{net}}=\Delta U+W=1280\,\mathrm J,\quad Q_{\mathrm{leak}}=1500-1280=220\,\mathrm J.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{source}}-Q_{\mathrm{leak}}=1280,\quad Q_{\mathrm{leak}}\ge0\ \Rightarrow\ Q_{\mathrm{source}}\ge1280\,\mathrm J.$$$$Q_{\mathrm{source}}=1000\,\mathrm J\ \Rightarrow\ Q_{\mathrm{leak}}=-280\,\mathrm J.$$That claim requires 280 J of additional inward heat, contrary to the outward-leak assumption; alternatively some measured data or transfer accounting is wrong.

Physical check

The temperature rise accounts for stored energy only; the source must also fund work delivered to the load.