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

The Sound can travel in different mediums like solids, liquids and gasses. In every medium it travels as vibrations or through compressions and rarefactions. Newton's formula for speed in a gas is related to the pressure of the gas and its density.
Formulas used:
$v=\sqrt{\dfrac{RT}{M}}$
$PV=nRT$
$n=\dfrac{m}{M}$
$\rho =\dfrac{m}{V}$

Complete answer:
The speed of sound is highest in solids, then liquids and lowest in gases. Sound is a mechanical wave and it travels by forming compressions and rarefactions in the medium. The Newton’s formula for velocity of gas is given by-
$v=\sqrt{\dfrac{RT}{M}}$ - (1)
Here, $v$ is the velocity of sound in air
$R$ is the gas constant
$T$ is temperature
$M$ is the molecular mass of the gas
According to the ideal gas equation, we have,
$PV=nRT$ - (2)
Here, $P$ is the pressure
$V$ is the volume of the gas
$n$ is the number of moles
From eq (2), we have,
$\begin{aligned} & PV=nRT \\\ & \Rightarrow RT=\dfrac{PV}{n} \\\ \end{aligned}$
Substituting in eq (1), we get,
$v=\sqrt{\dfrac{RT}{M}}$
$\Rightarrow v=\sqrt{\dfrac{PV}{nM}}$ - (3)
We know that,
$n=\dfrac{m}{M}$
Here, $m$ is the given mass of the gas
Substituting above relation in eq (3), we get,
$\begin{aligned} & v=\sqrt{\dfrac{PV}{nM}} \\\ & \Rightarrow v=\sqrt{\dfrac{PV}{\dfrac{m}{M}\times M}} \\\ \end{aligned}$
$\Rightarrow v=\sqrt{\dfrac{PV}{m}}$ - (4)
We know that density is mass per unit volume. Its SI unit is $kg\,{{m}^{-3}}$.
$\rho =\dfrac{m}{V}$
Here, $\rho$ is the density of gas
$m$ is the mass of the gas
$V$ is the volume of the gas
We substitute density in eq (4), to get,
$\therefore v=\sqrt{\dfrac{P}{\rho }}$
Therefore, Newton's formula for speed in sound is $v=\sqrt{\dfrac{P}{\rho }}$.

Hence, the correct option is (A).

Note:
Sound cannot travel with a medium. Newton’s formula was later declared as wrong and was corrected by Laplace. For speed of sound in air, the mean of values for all gases is taken. In air, the speed of sound is independent of pressure. The corrected formula for speed of sound in air is
$v=\sqrt{\dfrac{\gamma RT}{M}}$ ($\gamma$ is the adiabatic index).