Fixed Work Input Three-Row Scan - Thermodynamics and Statistical Mechanics Step by Step

A fixed-work-input scan holds the work input fixed and varies another factor across three rows, checking how the resulting heat or temperature change tracks the first-law budget row by row. The arithmetic is exact only for the stated work, heat, and rows in the ideal first-law model. Real measurements add temperature-scale error, pressure/volume calibration, thermal-equilibrium limits, gas nonideality, friction/losses, finite precision, and measurement uncertainty.

At fixed useful work, larger heat input lowers the efficiency fraction. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Same work from lower input is a larger fraction

Useful work is fixed at 12 joules. With input 24 joules, efficiency is one half.

\eta=12\ \text{J}/24\ \text{J}=\tfrac{1}{2}

More input lowers the fraction

The same 12 joules of work over 30 joules of input gives two fifths.

\eta=12\ \text{J}/30\ \text{J}=\tfrac{2}{5}

Largest input gives the smallest fraction

Input 40 joules with the same work gives three tenths. The fraction falls as the denominator grows.

\begin{array}{c|c|c|c}Q_{\text{in}}&W&Q_{\text{waste}}&\eta\\24\ \text{J}&12\ \text{J}&12\ \text{J}&\tfrac{1}{2}\\30\ \text{J}&12\ \text{J}&18\ \text{J}&\tfrac{2}{5}\\40\ \text{J}&12\ \text{J}&28\ \text{J}&\tfrac{3}{10}\\\end{array}