Question

State Newton's formula for the velocity of sound in a gas. What is the Laplace's correction? $\left[8\text{marks}\right]$

Answer 1

According to Newton, velocity of sound in any medium is given by
$v=\sqrt{\frac{E}{\rho }}$
where, $E$ is the modulus of elasticity and $\rho$ is the density of the medium. For gases $E=B$, bulk modulus
$\therefore v=\sqrt{\frac{B}{\rho }}....\left(1\right)$
$\left[2\text{marks}\right]$
When sound waves travel through a gas alternate compressions and rarefactions are produced. At the compression region pressure increases and volume decreases and at the rerefaction region pressure decreases and volume increases. Newton assumed that these changes take place under isothermal conditions i.e., at a constant temperature.
Under isothermal condition, $B=P$, pressure of the gas.
$\therefore$ In $\left(1\right)v=\sqrt{\frac{P}{\rho }}....\left(2\right)$
This is Newton's formula for velocity of sound in gas.
$\left[2\text{marks}\right]$
For air at NTP, $P=101.3\text{kPa}$ and $\rho =1.293{\text{kg/m}}^3$
Substituting the values of $P$ and $\rho$ in equation $\left(2\right)$ we get $v=280\text{m/s}$. This is much lower than the experimental value of $332\text{m/s}$. Thus Newton's formula is discarded.
$\left[1\text{mark}\right]$
Laplace's correction:
According to Laplace, in a compressed region temperature increases and in a rarefied region it decreases and these changes take place rapidly. Since air is an insulator, there is no conduction of heat. Thus changes are not isothermal but adiabatic.
$\left[1\text{mark}\right]$
Under adiabatic condition, $B=\gamma P$, where $\gamma$ is the ratio of specific heats of the gas.
Substituting in equation $\left(1\right)v=\sqrt{\frac{\gamma P}{\rho }}$
The above equation is called Newton - Laplace's equation Substituting the values of $P$, $\rho$ and $\gamma$ in the above equation, give the velocity of sound in air at NTP to be about $331\text{m/s}$. This is in close agreement with the experimental value.
$\left[2\text{marks}\right]$