Proportionality statement
According to boyle's law pressure and volume are what proportional? The relationship is inverse proportionality: pressure is inversely proportional to volume when temperature and the amount of gas remain constant.
Compact mathematical form
The constant \(k\) is fixed for a given sample of gas at a fixed temperature (isothermal condition) and fixed number of moles.
Physical conditions for Boyle’s law
Boyle’s law describes an isothermal change for a fixed amount of gas in which volume changes are accompanied by opposite changes in pressure. In typical general chemistry framing, the gas behaves approximately ideally over the range considered.
With \(n\) and \(T\) constant, the product \(nRT\) is constant, and the inverse \(1/V\) controls how \(P\) changes.
Meaning of “inversely proportional”
Inverse proportionality means that multiplying one variable by a factor divides the other variable by the same factor (within the validity of the law). The product remains constant:
| Change in volume | Consequence for pressure (constant \(T\), constant \(n\)) | Product check |
|---|---|---|
| Volume doubles: \(V_2 = 2V_1\) | Pressure halves: \(P_2 = \frac{1}{2}P_1\) | \(P_2V_2 = \left(\frac{1}{2}P_1\right)(2V_1) = P_1V_1\) |
| Volume triples: \(V_2 = 3V_1\) | Pressure becomes one-third: \(P_2 = \frac{1}{3}P_1\) | \(P_2V_2 = \left(\frac{1}{3}P_1\right)(3V_1) = P_1V_1\) |
| Volume halves: \(V_2 = \frac{1}{2}V_1\) | Pressure doubles: \(P_2 = 2P_1\) | \(P_2V_2 = (2P_1)\left(\frac{1}{2}V_1\right) = P_1V_1\) |
Common pitfalls
- Temperature changes: Boyle’s law requires constant \(T\); temperature variation introduces Charles’s law and the combined gas law.
- Amount of gas changes: leaks or reactions change \(n\), so \(PV\) is no longer constant for the original sample.
- Direct proportionality confusion: direct proportionality corresponds to a straight line through the origin for \(P\) versus \(V\), not the hyperbola expected from \(P \propto 1/V\).
- Unit handling: the proportionality statement is unit-free, while numerical calculations require consistent units across \(P\) and \(V\).