The relation between speed of sound (ν) and absolute temperature (T) in a gas is

The speed of sound in a gas is governed by the Newton–Laplace relation, v = √(γRT/M), where γ is the adiabatic index (ratio of specific heats), R is the universal gas constant, T is the absolute temperature, and M is the molar mass of the gas.

1. Write the Newton–Laplace relation for the speed of sound in a gas: v = √(γRT/M).

2. For a fixed gas, R and M are exact constants, and γ is treated as constant over the range of temperatures considered (the standard approximation at this level) — none of them change with temperature in that range.

3. Group the constants together: v = √(γR/M) × √T = k√T, where k = √(γR/M) is a fixed constant for that gas.

4. Since k is constant, v depends only on √T, i.e. v is directly proportional to T1/2.

Independent cross-check (kinetic theory): the root-mean-square molecular speed in a gas is vrms = √(3RT/M), which also varies as √T. Because the speed of sound and vrms differ only by a numerical factor (√(γ/3), likewise treated as constant over the same range) for a given gas, they must share the same temperature dependence — this independently confirms the √T scaling obtained above.

Hence, the speed of sound (ν) and absolute temperature (T) are related as ν ∝ T1/2.