Ideal-gas equation and absolute temperature

What is the ideal-gas equation?

In plain words: all gases at low density behave in the same simple way. At fixed temperature, compression raises pressure; heating a fixed sample at constant pressure increases its volume, or at constant volume raises its pressure. These facts combine into one equation, which also points to the lowest possible temperature, absolute zero, and so to the kelvin scale.

Boyle’s law and Charles’ law

The state of a fixed quantity of gas is described by its pressure, volume and temperature. Experiments on low-density gases give two laws (NCERT, page 203):

• Boyle’s law (Robert Boyle, 1627–1691): at constant temperature, PV = constant. Halve the volume and the pressure doubles. At constant temperature, Boyle’s law makes a gas’s bulk modulus equal to its pressure, about 10^⁵ Pa for air at 1 atm, over 20 000 times smaller than water’s.
• Charles’ law (Jacques Charles, 1747–1823): at constant pressure, V/T = constant, with T the absolute temperature. Raise T by a quarter and the volume rises by a quarter. In general ΔV/V = ΔT/T, so at constant pressure an ideal gas has a coefficient of volume expansion α_V = 1/T (NCERT Eq. 10.6, page 206).

Together they give PV/T = constant for a given quantity of gas; the additional experimental fact that this constant is proportional to the number of moles gives the ideal gas law. Written for any quantity of any low-density gas, it is the ideal-gas equation:

Here μ is the number of moles. For a fixed quantity going from one state to another, P₁V₁/T₁ = P₂V₂/T₂.

Why “low density”? Real gases follow these laws closely when their molecules are far apart. At high densities or low temperatures they deviate, and near liquefaction they stop being gases at all (NCERT, pages 203–204).

The gas thermometer and absolute zero

At constant volume, the ideal-gas equation gives P ∝ T: the pressure of the gas measures its temperature.

Thus, with a constant-volume gas thermometer, temperature is read in terms of pressure.

NCERT Physics Part II (Class 11), Section 10.4, page 204

A graph of pressure against Celsius temperature for a gas at constant volume is a straight line. Extend it to lower temperatures, as if the gas stayed a gas, and it reaches zero pressure at −273.15 °C. Remarkably, every low-density gas, whatever its amount or kind, extrapolates to the same temperature. This temperature is absolute zero, the lowest temperature there can be (NCERT, page 204).

The kelvin scale

Absolute zero is the zero of the kelvin scale (absolute scale), named after Lord Kelvin: 0 K = −273.15 °C (NCERT, page 204).

The size of unit in Kelvin and Celsius temperature scales is the same.

NCERT Physics Part II (Class 11), Section 10.4, page 204

So the two scales differ only by a shift:

The ice point is about 273.15 K and the steam point about 373.15 K; on the Fahrenheit scale, absolute zero is −459.67 °F (NCERT Fig. 10.4 prints −459.69 °F; NCERT Eq. 10.1 gives −459.67 °F). NCERT’s Points to Ponder (page 223) adds that the scale is fixed by assigning exactly 273.16 K to the triple point of water, where ice, water and vapour coexist (since 2019 the kelvin has been defined through Boltzmann’s constant, and the triple-point thermodynamic temperature is approximately 273.16 K with measurement uncertainty, rather than an exact defining value); with this choice, water melts and boils at 1 atm very close to, but not exactly at, 0 °C and 100 °C. In every gas-law formula, T must be in kelvin: using °C would make V/T blow up at 0 °C, which has no physical meaning.

Solved examples

Take R = 8.31 J mol^⁻¹ K^⁻¹.

Common mistakes to avoid

Quick revision summary

• Boyle’s law: PV = constant at constant temperature. Charles’ law: V/T = constant at constant pressure.
• Ideal-gas equation for low-density gases: PV = μRT, with R = 8.31 J mol^⁻¹ K^⁻¹ and μ the number of moles.
• At constant volume P ∝ T: a gas thermometer reads temperature through pressure.
• Extrapolated to zero pressure, all gases point to −273.15 °C: absolute zero, 0 K.
• Kelvin and Celsius degrees are the same size: T = t_C + 273.15.
• In short: PV = μRT, with T in kelvin.

Questions students ask

What is the SI unit and dimensional formula of the gas constant R?

The universal gas constant R has the SI unit J mol^⁻¹ K^⁻¹ and the dimensional formula [M L^² T^⁻² K^⁻¹ mol^⁻¹], since R = PV/(μT) and PV is an energy (see the table of dimensional formulae). Its value is 8.314 J mol^⁻¹ K^⁻¹ (8.31 in NCERT), which is about 1.99 cal mol^⁻¹ K^⁻¹, so it rounds to R ≈ 2 cal mol^⁻¹ K^⁻¹. It is the same for every gas, and it equals Avogadro’s number times Boltzmann’s constant: R = N_Ak_B.

What are some real-life examples of Boyle’s law and Charles’ law?

Everyday examples of Boyle’s law (PV = constant for a fixed quantity of gas at constant temperature) include a syringe, which draws in liquid because pulling the plunger back lowers the pressure inside; the initial pressure reduction during breathing, before significant air enters; and a diver’s air bubbles, which grow as they rise into water at lower pressure. Once gas enters or leaves, the fixed-quantity law alone no longer describes the process. An approximate everyday example of Charles’ law (V/T = constant at constant pressure) is a balloon that shrinks in a refrigerator and swells again in a warm room. In Charles’ law T is the absolute temperature: a balloon cooled from 27 °C (300 K) to 0 °C (273 K) shrinks by only about 9%.

How do you find the density of a gas from the ideal-gas equation?

The density of an ideal gas is ρ = PM/(RT), where M is the molar mass in kg mol^⁻¹, because putting μ = m/M in PV = μRT gives P = (m/V)RT/M. So at fixed pressure a gas becomes less dense as it is heated, and at fixed temperature its density is proportional to its pressure. For example, air (M ≈ 0.029 kg mol^⁻¹) at 1.013 × 10^⁵ Pa and 300 K has ρ = 1.013 × 10^⁵ × 0.029/(8.31 × 300) ≈ 1.18 kg m^⁻³; the hotter, less dense air inside a hot-air balloon is what gives it lift.

Can a gas be cooled to absolute zero?

No, a gas, or anything else, cannot be cooled all the way to absolute zero, 0 K (−273.15 °C or −459.67 °F); it can only be approached, and experiments can approach extremely low temperatures. At ordinary pressures gases condense before reaching 0 K (helium boils at about 4.2 K at 1 atm), while dilute trapped gases can instead enter quantum-degenerate regimes, so the straight line of pressure against temperature that points to absolute zero is an extrapolation of how an ideal gas would behave, not a measurement down to it. Absolute zero is the lowest possible temperature, but not one at which all motion stops: quantum physics leaves particles a small zero-point energy even there.

What do the graphs of Boyle’s law look like?

For a fixed quantity of gas at constant temperature, Boyle’s law (PV = constant) gives three standard graphs: P against V is a rectangular hyperbola, P against 1/V is a straight line through the origin, and PV against P (or against V) is a horizontal straight line. On a P–V graph each such curve, an isotherm, lies farther from the axes at a higher temperature, since PV = μRT is then larger. Real gases follow these lines only at low pressure; at high pressure their PV against P graphs curve away from the horizontal.

Practice questions

Try each question first, then open the answer to check your working. Take R = 8.31 J mol^⁻¹ K^⁻¹.

1. Q1. The air in a tyre is at an absolute pressure of 300 kPa at 27 °C. After a long drive it warms to 57 °C. Treating the volume as fixed, find the new pressure.

   Show answerHide answer

   P ∝ T at constant volume: 300 × (57 + 273)/(27 + 273) = 300 × 330/300 = 330 kPa.

2. Q2. How many moles of gas are in a 10 L cylinder at 2.0 × 10^⁶ Pa and 300 K?

   Show answerHide answer

   μ = PV/(RT) = 2.0 × 10^⁶ × 0.010/(8.31 × 300), about 8.0 mol.

3. Q3. Why must temperatures be in kelvin, not degrees Celsius, in Charles’ law?

   Show answerHide answer

   Charles’ law says V is proportional to T, so V/T is constant. Since t_C = T − 273.15, V/t_C changes with temperature and would blow up at 0 °C, where the volume is not zero.