For a gas, which pair of variables is inversely proportional to each other (provided other conditions are constant), where P = pressure, T= temperature, V= ...
Question 1 Report
For a gas, which pair of variables is inversely proportional to each other (provided other conditions are constant), where P = pressure, T= temperature, V= volume, and n= number of molecules?
Answer Details
All the relationships follow from the ideal gas equation \[PV = nRT.\] To decide whether two quantities are directly or inversely proportional, hold the other two constant and see what the equation demands.
| Pair | Held constant | Relationship | Law |
|---|---|---|---|
| \(P\) and \(V\) | \(n, T\) | \(PV = \text{constant}\), so \(P \propto \dfrac{1}{V}\): inverse | Boyle |
| \(P\) and \(T\) | \(n, V\) | \(\dfrac{P}{T} = \text{constant}\): direct | Pressure law |
| \(V\) and \(T\) | \(n, P\) | \(\dfrac{V}{T} = \text{constant}\): direct | Charles |
| \(n\) and \(P\) | \(V, T\) | \(\dfrac{P}{n} = \text{constant}\): direct | Avogadro-type |
Only pressure and volume sit on the same side of the equation as a product, and a product held constant is the definition of inverse proportionality. So the inversely proportional pair is pressure and volume: squeeze a fixed mass of gas at constant temperature into half the space and the pressure doubles, because the molecules strike the walls twice as often.
A practical way to confirm the type of proportionality is the shape of the graph. Pressure against volume gives a curve (a hyperbola), while pressure against \(1/V\) gives a straight line through the origin. Pressure against absolute temperature and volume against absolute temperature both give straight lines through the origin directly. In an examination, always state which quantities are being held constant before quoting a gas law, since the same two variables can behave differently if a third is allowed to vary.
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