Derive an expression for escape speed.

Explanation

Laplace.s correction: In 1816, 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 occur very fast. Hence the exchange of heat produced due to compression and cooling effect due to rarefaction do not take place, because, air (medium) is a bad conductor of heat. Since, temperature is no longer considered as a constant here, sound propagation is an adiabatic process. By adiabatic considerations, the gas obeys Poisson.s law (not Boyle.s law as Newton assumed), which is
`PV^(gamma)` = Constant ...(4)
where, `gamma = (C_(p))/(C_(v))` which is the ratio 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 `gammaP = - V (dp)/(dV) = B_(A)` ...(5)
where, `B_(A)` is the adiabatic bulk modulus of air. Now, substituting equation (5) in equation `V=sqrt(B/rho)` the speed of sound in air is `V_(A) = sqrt((B_(A))/(rho)) = sqrt((gammaP)/(rho)) = sqrt(gammav_(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 m `s^(-1)`) = 331.30 m `s^(-1)`, which is very much closer to experimental data.