Speed Of Sound In Gases

The speed of sound in a gas is examined in NCERT Class 11, Chapter 15 (Waves), where Newton first proposed $v=\sqrt{P/\rho }$ assuming isothermal compression. Laplace corrected this by noting that sound compressions and rarefactions happen so quickly that no heat is exchanged, making the process adiabatic rather than isothermal. The adiabatic bulk modulus is $\gamma P$, so the corrected speed is $v=\sqrt{\frac{\gamma P}{\rho }}$, where $\gamma$ is the ratio of specific heats. This factor of $\gamma$ raised the predicted speed to match experiment. The option using $\sqrt{P/\rho }$ without $\gamma$ is Newton's uncorrected and inaccurate formula. The option $\sqrt{\rho /P}$ inverts the ratio and gives wrong units. The option that speed is directly proportional to pressure is wrong because at constant temperature $P/\rho$ stays fixed, so speed does not simply rise with pressure. It also follows from this expression that, for a fixed gas at a given temperature, the speed of sound is independent of pressure changes at constant temperature, because raising the pressure raises the density in the same proportion so the ratio $P/\rho$ stays constant. However, since $P/\rho =RT/M$ from the ideal gas law, the speed does increase with the square root of absolute temperature, which is why sound travels faster on a hot day. A consistency check: including $\gamma \approx 1.4$ for air corrects Newton's underestimate of about 280 m/s up to the observed 331 m/s at standard conditions, and the factor $\sqrt{1.4}\approx 1.18$ accounts precisely for that gap, confirming Laplace's correction.