Combined Gas Three-State Scan - Thermodynamics and Statistical Mechanics Step by Step

A combined-gas three-state scan walks pressure, volume, and kelvin temperature through three states, checking that P * V / T stays constant across the rows. The arithmetic is exact only for the stated pressures, volumes, temperatures, and rows in the ideal-gas, fixed-amount model. Real measurements add temperature-scale error, pressure/volume calibration, thermal-equilibrium limits, gas nonideality, friction/losses, finite precision, and measurement uncertainty.

Three exact gas states keep pV over T fixed while different variables move. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Wide low-pressure state sets the ratio

The first row uses pressure 2 pascals, volume 6 cubic metres, and temperature 300 kelvin.

pV/T=2\ \text{Pa}\cdot6\ \text{m}^{3}/300\ \text{K}=\tfrac{1}{25}\ \text{J/K}

Pressure and volume can trade at same kelvin

At the same 300 kelvin, pressure 3 pascals and volume 4 cubic metres keep the same combined ratio.

pV/T=3\ \text{Pa}\cdot4\ \text{m}^{3}/300\ \text{K}=\tfrac{1}{25}\ \text{J/K}

Higher kelvin needs a larger pressure-volume product

The third row raises kelvin and the pressure-volume product together, so the ratio still matches.

\begin{array}{c|c|c|c}p&V&T&pV/T\\2\ \text{Pa}&6\ \text{m}^{3}&300\ \text{K}&\tfrac{1}{25}\ \text{J/K}\\3\ \text{Pa}&4\ \text{m}^{3}&300\ \text{K}&\tfrac{1}{25}\ \text{J/K}\\4\ \text{Pa}&4\ \text{m}^{3}&400\ \text{K}&\tfrac{1}{25}\ \text{J/K}\\\end{array}