Discuss Newton’s formula for velocity of sound in air medium and apply Laplace’s correction.

Explanation

Newton's assumed that when sound propagates in ari, the formation of compression and rarefaction takes place in a very slow manner so that the process is isothermal in nature. It is found that, the heat produced during compression (pressure increases, volume decreases), and heat lost during rarefaction (Pressure decreases, volume increases) occure over a period of time such in a wavy that the temperature of the medium remains constant. Hene, by treating the air molecules to form an ideal gas, the changes in pressure and volume obey Boyle's law, Mathematically
`PV="chonstant" " "..(1)`
Differentiating equation (1) we get
`PdV+VdP=0`
(or) `P-V(dP)/(dV)=B_(T)" "...(2)`
Where `B_(T)` is an isothermal bulk modulus of air.
`v= sqrt((B)/(rho))" "...(3)`
Substituting equatio (2) in equation (3), the speed of sound in air is
`v_(T)= sqrt((B_(T))/(rho))= sqrt((P)/(rho))`
Since P is the pressure of air whose value at NTP (Norma Temperature and Pressure) is 76 cm of mercury, we have
`P=(0.76xx13.6xx10^(3)xx9.8)Nm^(-2)`
`rho=1.239 kg m^(-3)`. Here `rho` is density of ari. Then the speed of sound in air at normal temperature and pressure (NTP) is
`v_(T) = sqrt(((0.76xx13.6xx10^(3)xx9.8))/(1.293))`
`=270.80 ms^(-1)`
`=280 ms^(-1)` (theoretical value) But the speed of sound in air at `0^(@)C` is experimentally observed as `332 ms^(-1)` that is close upto 16 % more than theoretical value.
Lapace correction : Laplace satisfactorily corrected this discrepancy by assuming that when the sound propagates through a medium, the particles oscillate very rapidly such that the compression and rarefaction occure very fast. Hene the exchange of heat produed due to compression and cooling effect due to rarefaction do not take place, because, air (medium) is a poor conductopr of heat. Since, temperature is no longer considered as a constant here, propagation of sound is an adiabatic process. By adiabatic considerations, the gas obeys Poisson's law (as Newton assumed), that is
`PV^(gamma)= "Constant" " "....(4)`
where `gamma=(C_(P))/(C_(v))`, that is the ration between specific heat at constant pressure and specific heat at constant volume.
Differentiating equation (4) on both the sides, we get
`V^(gamma)dP+P(gammaV^(gamma-1)+dV)=0`
Or , `gamma P=-V(dp)/(dV)=-B_(A) " "...(5)`
where `B_(A)` is the adiabatic bulk modulus of air.
`v= sqrt((B)/(rho)) " "...(6)`
Now, substituting equation (5) in equation (6), the speed of sound in air is
`v_(A)= sqrt((B_(A))/(rho))= sqrt((gammaP)/(rho))`
`= sqrt(gamma)v_(T)`
Since air contains mainly, nitrogen, oxygen, hydrogen etc. ( diatomic gas ) , we take ` gamma = 1.47`. Hence, speed of sound in air is `v_(A)=(sqrt(1.4))(280 ms^(-1)=331.30 ms^(-1)` , which is very much closer to experimental data .