Question

As per Newton’s formula, what is the velocity of sound at N.T.P. ?

Answer 1

Hint: To calculate the velocity of sound at Normal temperature and pressure (NTP), we need to first find the condition of NTP. The parameters at Normal temperature and pressure (NTP) are ${{20}^{\circ }}C$ and 1 atm pressure. We will use this value of pressure and temperature to calculate the speed of sound using Newton’s formula:

Complete answer:
Let us assume that the speed of sound at Normal temperature and pressure (NTP) is denoted by ‘v’. Then, the Newton’s formula for speed of sound is given by:
$\Rightarrow v=\sqrt{\dfrac{P}{\rho }}$
Where,
P is the pressure of the medium in which the sound wave is travelling. And,
$\rho $ is the density of the medium (in this case, air)
Now, to calculate the density of air, we know that at NTP, 1 mole of any gas occupies 24.04Ltr of volume.
Thus, the density of air can be calculated as:
$\begin{align}
  & \Rightarrow \rho =\dfrac{29\times {{10}^{-3}}kg}{24.04\times {{10}^{-3}}{{m}^{3}}} \\
 & \Rightarrow \rho =1.206kg{{m}^{-3}} \\
\end{align}$
Also, 1 atm pressure when converted into standard unit comes out to be:
$\begin{align}
  & \Rightarrow P=1.01325\times {{10}^{5}}Pa \\
 & \Rightarrow P=101325Pa \\
\end{align}$
Now, putting the value of pressure and density in Newton's equation for speed of sound. We get:
$\begin{align}
  & \Rightarrow v=\sqrt{\dfrac{101325Pa}{1.206kg{{m}^{-3}}}} \\
 & \Rightarrow v\approx 290m{{s}^{-1}} \\
\end{align}$
Hence, as per Newton's formula the velocity of sound at N.T.P. comes out to be $290m{{s}^{-1}}$ .

Note:
While deriving the above speed of sound formula, Newton assumed that compression and rare-fraction in the air is an isothermal process. But, this assumption was later on corrected by Laplace who said that the compression and rare-fraction of air is an adiabatic process. This is known as Laplace correction, and the new formula is: $v=\sqrt{\dfrac{\gamma P}{\rho }}$ , where ‘$\gamma $’ is the adiabatic index.