Question

Newton’s formula for velocity of sound in gas is
A). $v=\sqrt{\dfrac{P}{\rho }}$
B). $v=\dfrac{2}{3}\sqrt{\dfrac{P}{\rho }}$
C). $v=\sqrt{\dfrac{\rho }{P}}$
D). $v=\sqrt{\dfrac{2P}{\rho }}$

Answer 1

Hint: After his theoretical considerations Newton gave the formula for velocity of sound through a gaseous medium as:
$v=\sqrt{\dfrac{B}{\rho }}$
Here B is the bulk modulus of elasticity and $\rho$ is the density of the gas. He also said that sound travels under isothermal conditions.

Complete step by step answer:
Newton gave an empirical formula to calculate the velocity of sound in gasses:
$v=\sqrt{\dfrac{B}{\rho }}$
Sounds travel in the form of compressions and rarefactions. Newton assumed that the changes in pressure and volume of the sound wave travelling in gas are isothermal. At the place of compression temperature increases and at rarefaction it decreases. So, heat transfers from a place of compression to a place of rarefaction and the temperature remains the same.
We will now find the relation between Bulk modulus under isothermal conditions and Pressure P.
Consider a certain gas with initial pressure P and initial volume V
Under isothermal conditions we know:
$\text{PV=constant}$
On differentiating both sides we get
$\text{PdV+VdP=0}$
$\text{P=-}\dfrac{dP}{{dV}/{V}\;}$
And by definition of bulk modulus we get,
$\Rightarrow P=B$
On substituting P in place of B in the first equation we get:
$v=\sqrt{\dfrac{P}{\rho }}$
Therefore, the correct answer is Option A. $v=\sqrt{\dfrac{P}{\rho }}$.

Note: Students must remember that the formula given by newton had an error of $16$ when compared with experimental calculations. This formula was later corrected by Laplace as he assumed that the changes are adiabatic instead of isothermal and he gave the formula $v=\sqrt{\dfrac{\gamma P}{\rho }}$. Here, $\gamma =\dfrac{{{C}_{p}}}{{{C}_{v}}}$is the ratio of two principal specific heats.