First-Law Split Three-Row Scan - Thermodynamics and Statistical Mechanics Step by Step

A first-law-split scan holds the heat input fixed and varies the work and internal-energy split across three rows, checking that Q_in = W + delta-U closes at the same total row by row. The arithmetic is exact only for the stated heat, work, internal-energy splits, 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.

The same heat input can split into work and internal energy in several exact ways. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Most energy can stay internal

A 20 joule input can split into 4 joules of work and 16 joules of internal-energy change.

Q_{\text{in}}=4\ \text{J}+16\ \text{J}=20\ \text{J}

The middle split still closes

The middle row uses 12 joules of work and 8 joules stored internally.

Q_{\text{in}}=12\ \text{J}+8\ \text{J}=20\ \text{J}

More work leaves less internal storage

The third row sends 16 joules to work and only 4 joules to internal energy. The total stays fixed.

\begin{array}{c|c|c}W&\Delta U&Q_{\text{in}}\\4\ \text{J}&16\ \text{J}&20\ \text{J}\\12\ \text{J}&8\ \text{J}&20\ \text{J}\\16\ \text{J}&4\ \text{J}&20\ \text{J}\\\end{array}