Physics · Waves · NEET
Newton assumed the compressions and rarefactions in air happen slowly, so the temperature stays constant (isothermal). For an isothermal change PV = constant, which makes the bulk modulus B = P. So he got v = sqrt(P/rho). Putting P = 1.01 x 10^5 Pa and rho = 1.29 kg/m^3 gives about 280 m/s. The measured value is about 332 m/s, so his answer was roughly 15 percent too low. The mistake was the isothermal assumption, not the math.
Sound is a fast process. The compressions and rarefactions happen thousands of times per second, so heat produced in a compression has no time to flow out to the neighbouring rarefaction. Air is also a poor conductor of heat. Because no heat enters or leaves, the process is adiabatic, not isothermal. This is the key idea behind the Laplace correction.
For an adiabatic change PV^gamma = constant, so the bulk modulus becomes B = gamma P instead of just P. Substituting into v = sqrt(B/rho) gives Laplace's formula v = sqrt(gamma P/rho). Here gamma is the ratio of specific heats (Cp/Cv). For air, which is mostly diatomic, gamma = 1.4. So the corrected speed is sqrt(1.4) times Newton's value, which turns 280 m/s into about 332 m/s.
No, not directly. Although P appears in v = sqrt(gamma P/rho), when you raise pressure at constant temperature the density rho rises in the same proportion (from PV = nRT). So the ratio P/rho stays constant and v does not change. This is why NEET says speed of sound is independent of pressure at constant temperature. Speed does depend on temperature, because v is proportional to sqrt(T).
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Newton's formula is v = sqrt(P/rho), where P is the atmospheric pressure and rho is the density of air. It assumes sound travels as an isothermal process and predicts about 280 m/s at 0 degrees C.
Laplace corrected Newton by treating sound propagation as adiabatic, giving v = sqrt(gamma P/rho), where gamma = Cp/Cv = 1.4 for air. This raises the predicted speed to about 332 m/s, matching experiment.
For air, gamma (the ratio of specific heats Cp/Cv) is 1.4, because air is mostly made of diatomic gases like nitrogen and oxygen.
Newton predicted about 280 m/s while the true speed is about 332 m/s. That is an error of roughly 52 m/s, or about 15 to 16 percent below the correct value.
Because sound is adiabatic, the bulk modulus of air becomes B = gamma P instead of B = P. Since v = sqrt(B/rho), the factor gamma enters and gives v = sqrt(gamma P/rho).
