Boyle's Law - Thermodynamics and Statistical Mechanics Step by Step

Boyle's law says that at constant temperature, pressure and volume vary inversely, P * V = constant, so squeezing the gas raises the pressure. It is the ideal-gas law with temperature held fixed. The exact trade-off holds only for a fixed amount of ideal gas in equilibrium; real gases show nonideal behavior at high density, and leaks, finite response time, and sensor calibration add uncertainty.

At fixed temperature, pressure times volume stays constant for this ideal gas model. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Start at fixed temperature

At the first state, pressure 2 pascals times volume 6 cubic meters gives 12 joules.

p_iV_i = 2\ \text{Pa}\cdot 6\ \text{m}^{3} = 12\ \text{J}

Increase pressure and reduce volume

At the second state, pressure is 4 pascals and volume is 3 cubic meters.

p_fV_f = 4\ \text{Pa}\cdot 3\ \text{m}^{3}

The product stays constant

The second product is also 12 joules, so higher pressure pairs with lower volume.

p_iV_i = p_fV_f = 12\ \text{J}

Several rows keep the same product

The diagram draws the first and third rows. At fixed temperature, larger pressure pairs with smaller volume so the product remains the same.

\begin{array}{c|c|c}p&V&pV\\2\ \text{Pa}&6\ \text{m}^{3}&12\ \text{J}\\3\ \text{Pa}&4\ \text{m}^{3}&12\ \text{J}\\4\ \text{Pa}&3\ \text{m}^{3}&12\ \text{J}\\6\ \text{Pa}&2\ \text{m}^{3}&12\ \text{J}\\\end{array}