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

Q: What is the answer to NDA (II) 2016 GAT Q107?

A. 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.

(a)straight line
(b)circle
(c)parabola
(d)hyperbola

Answer

Why

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.

Why the others are wrong

(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.

Concept

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.

Key facts

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.

The same gas law on four different pairs of axes, all at constant temperature

Volume plotted against 1/P→Straight line through the origin, slope nRT: this is what the question asks for (the key's answer)

Volume plotted against P→Rectangular hyperbola, since PV is constant: the shape option (d) belongs to

The product PV plotted against P→Horizontal straight line, because the product does not change along an isotherm

Volume plotted against temperature at constant pressure→Straight line by Charles's law, which extrapolates back to −273.15 °C

Taking the reciprocal of pressure is what converts Boyle's hyperbola into a straight line.

Study next

Boyle's law, Charles's law, Gay-Lussac's law and how the ideal gas equation combines themAvogadro's law and the molar volume of a gas at standard temperature and pressureKinetic theory of gases and the pressure-from-molecular-collisions pictureReal gases, the van der Waals equation and why deviations grow at high pressure and low temperatureAbsolute zero and the Kelvin scale from the extrapolated Charles's law graph

Common traps

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.

Related PYQs

NDA_GAT_2018_II_Q692018Open & attempt →

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

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.