If we plot a graph between volume V and inverse of pressure P (i.e., 1/P) for an ideal gas at constant temperature T, the curve so obtained is
- (a)straight line
- (b)circle
- (c)parabola
- (d)hyperbola
Correct — A, a straight line. Start from the ideal gas equation PV = nRT and rearrange it as V = nRT × (1/P). The temperature is held constant and so are the amount of gas and the gas constant, so the whole bracket nRT is a fixed number; call it k. What is left is V = k × (1/P), which has the form y = mx. Plotting V on the vertical axis against 1/P on the horizontal axis therefore gives a straight line passing through the origin, with slope nRT. Choosing 1/P rather than P as the variable is exactly the trick that turns Boyle's curve into a straight line, which is why laboratory verifications of Boyle's law are plotted this way.
- (b)circle — A circle needs both variables squared and bounded, which no form of the gas equation produces. Neither V against P nor V against 1/P closes on itself; both run off to infinity in one direction.
- (c)parabola — A parabola would need one variable to depend on the square of the other, as in y = kx². The gas equation is first-order in every variable, so no squared term appears anywhere.
- (d)hyperbola — The trap, and the shape most candidates half-remember correctly but attach to the wrong axes. A rectangular hyperbola is what you get when you plot V against P itself at constant temperature, since PV = constant. The question deliberately asks for the plot against 1/P, and taking the reciprocal is precisely what straightens that hyperbola out.
Boyle's law states that at constant temperature the volume of a fixed mass of gas is inversely proportional to its pressure, PV = constant. The ideal gas equation PV = nRT combines Boyle's law with Charles's law and Avogadro's law in a single expression, where n is the number of moles and R the universal gas constant, 8.314 joules per mole per kelvin. A curve on which the temperature is held constant is called an isotherm.
Physics and chemistry papers repeatedly test whether a candidate can tell which pair of axes was chosen, because the same law looks completely different on different plots. At constant temperature, V against P is a rectangular hyperbola, V against 1/P is a straight line through the origin, and PV against P is a horizontal straight line. The trick to answering all of them is mechanical — put the given horizontal-axis quantity on the right-hand side of the rearranged equation, see whether what remains is a constant multiplier, and read off the shape.
- Boyle's law: at constant temperature, the pressure and volume of a fixed mass of gas obey PV = constant.
- A plot of V against P at constant temperature is a rectangular hyperbola; the same data plotted as V against 1/P is a straight line through the origin.
- The slope of the V against 1/P line equals nRT, so a hotter isotherm gives a steeper line.
- The universal gas constant R is 8.314 joules per mole per kelvin, and an isotherm is a curve of constant temperature.
Taking the reciprocal of pressure is what converts Boyle's hyperbola into a straight line.
- Answering 'hyperbola' from the memory of the standard Boyle's law diagram without noticing that the axis is 1/P, not P.
- Forgetting that the temperature is fixed here — if it were not, the slope would change and the plot would not be a single line.
Gas-law items in the GAT are usually about the shape of a graph or a short two-state numerical using P₁V₁ = P₂V₂, so be fluent in switching between the algebra and the picture.
A stainless steel chamber contains Ar gas at a temperature T and pressure P. The total number of Ar atoms in the chamber is n. Now Ar gas in the chamber is replaced by CO₂ gas and the total number of CO₂ molecules in the chamber is n/2 at the same temperature T. The pressure in the chamber now is P′. Which one of the following relations holds true? (Both the gases behave as ideal gases)
- (a) P′ = P
- (b) P′ = 2P
- (c) P′ = P/2
- (d) P′ = P/4
Answer(c) P′ = P/2
The same equation PV = nRT worked in the other direction — volume and temperature fixed, amount halved — which is the best way to check that you are reading the ideal gas relation rather than memorising one graph.
- practice — not a real PYQ
At constant temperature, a graph of the product PV against P for a fixed mass of an ideal gas is
- (a)a straight line through the origin
- (b)a straight line parallel to the pressure axis
- (c)a rectangular hyperbola
- (d)a parabola
Answer(b) a straight line parallel to the pressure axis — along an isotherm the product PV is a constant, so it does not change as P changes.
- practice — not a real PYQ
A gas at a pressure of 2 atmospheres occupies 3 litres. If the temperature is kept constant and the pressure is raised to 6 atmospheres, the new volume is
- (a)1 litre
- (b)2 litres
- (c)4.5 litres
- (d)9 litres
Answer(a) 1 litre — by Boyle's law P₁V₁ = P₂V₂, so V₂ = (2 × 3)/6 = 1 litre.