Question

At what temperature is the velocity of sound in oxygen equal to half its velocity in chlorine at 42${}^{\circ }C$? The ratio of molecular weights of oxygen and chlorine is 16 : 35.

Answer 1

Use the formula for the speed of sound waves travelling in a given gas. Consider the speed of sound in oxygen be v and that of chlorine be $\dfrac{v}{2}$. Then write the equations for both the speeds. Then divide the two equations to find the temperature of oxygen.
Formula used:
$v=\sqrt{\dfrac{\gamma RT}{M}}$

Complete answer:
We know that the speed of sound waves in air is approximately equal to 340$m{{s}^{-1}}$. However, it is not constant in all the gases. The speed of the sound waves changes with change of the gas in which it travels.
The speed of sound traveling in a gas is given as $v=\sqrt{\dfrac{\gamma RT}{M}}$ …. (i).
Here, $\gamma$ is the adiabatic index, R is the universal gas constant, T is the temperature of the gas and M is the molecular mass of the gas.
It is given that velocity of sound in oxygen is equal to half its velocity in chlorine at a temperature 42${}^{\circ }C$.
Let the speed of sound in oxygen be v. Then the speed of sound in chlorine is $\dfrac{v}{2}$.
Let the temperature of oxygen at which we can get this relation be T. And the temperature of chlorine is $42{}^{\circ }C=315K$
Let the molecular weights of oxygen and chlorine be ${{M}_{o}}$ and ${{M}_{c}}$.
The value of $\gamma$ for both will be the same.
Therefore, from (i) we get,
$v=\sqrt{\dfrac{\gamma RT}{{{M}_{o}}}}$ …. (ii).
And $\dfrac{v}{2}=\sqrt{\dfrac{\gamma R(315)}{{{M}_{c}}}}$ …. (iii).
Divide (ii) and (iii).
$\Rightarrow \dfrac{v}{\dfrac{v}{2}}=\dfrac{\sqrt{\dfrac{\gamma RT}{{{M}_{o}}}}}{\sqrt{\dfrac{\gamma R}{{{M}_{c}}}}}$
$\Rightarrow 2=\sqrt{\dfrac{{{M}_{c}}T}{315{{M}_{o}}}}$.
It is given that $\dfrac{{{M}_{c}}}{{{M}_{o}}}=\dfrac{35}{16}$.
$\Rightarrow 2=\sqrt{\dfrac{35T}{315(16)}}$
$\Rightarrow 4=\dfrac{35T}{315(16)}$
$\Rightarrow T=\dfrac{4\times 315\times 16}{35}=576K$.
Therefore, the temperature of oxygen must be 576K.

Note:
Some students may make a mistake by substituting the values of the temperature of chlorine in degree Celsius. The temperature in the formula for speed of sound is in Kelvin.
Therefore, always remember to substitute the temperature in Kelvin in any problem that requires temperature. Otherwise, the answer may go wrong.