Question

The velocity of sound in air at $27{}^\circ C$ is 340m/s. Calculate the velocity of sound in air at $127{}^\circ C$.

Answer 1

Hint: As a first step, you could recall the expression of velocity in terms of Temperature. Assuming all the other quantities in that expression are constant, we could find the proportionality relation between velocity and temperature. Thus we could find the answer easily from it.
Formula used:
Velocity,
$v=\sqrt{\dfrac{\gamma RT}{M}}$

Complete answer:
In the question we are given the velocity of sound in air at room temperature that is $27{}^\circ C$ to be 340m/s and we are supposed to find the velocity of sound in air at a greater temperature, that is, $127{}^\circ C$.
All we have to do here is to find the relation between the velocity of sound in air with temperature. We know that this relation could be given by,
$v=\sqrt{\dfrac{\gamma RT}{M}}$
Where, $\gamma $ is the adiabatic index, R is the universal gas constant, T is the absolute temperature and M is the molecular mass.
Since the medium remains the same (air) in both the given cases we could say that $\gamma $ and M remains constant and also, R is already a constant. So, we could make this conclusion that,
$v\alpha \sqrt{T}$
$\Rightarrow \dfrac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\dfrac{{{T}_{1}}}{{{T}_{2}}}}$ …………………………………………. (1)
Before substituting, let us convert the temperatures into their SI units.
${{T}_{1}}=27{}^\circ C=300K$
${{T}_{2}}=127{}^\circ C=400K$
Now, we could substitute these in (1) to get,
${{v}_{2}}={{v}_{1}}\sqrt{\dfrac{{{T}_{2}}}{{{T}_{1}}}}=340\sqrt{\dfrac{400}{300}}$
$\therefore {{v}_{2}}=392.6m/s$
Therefore, we found the velocity of sound in air at $127{}^\circ C$ to be 392.6m/s.

Note:
In problems involving the calculation with temperatures, make sure that you convert the given temperatures into their respective SI units. If you forget to do this step, you will get the answer wrong. We have taken molecular mass M and adiabatic index to be constant as they are dependent on the medium and the medium remains the same in both cases.