Question

The speed of sound in air is ___________ temperature?
A. independent of
B. directly proportional to
C. indirectly proportional to
D. None

Answer 1

Hint: Concept of speed of sound in gases is used and the factors on which it depends.

Formula used:
$v = \sqrt {\dfrac{{rP}}{q}} $

Complete step by step solution:
We know that Laplace formula for the speed of sound in a gas is
$v = \sqrt {\dfrac{{rP}}{q}} $ ……………………………… (i)
Where v = is the speed of sound in a gas.
r $ = $ ratio of specific heats.
P $ = $ pressure
q $ = $ density
Now, for one male of a gas, ideal gas equation is given by
$PV = RT$ (as n $ = 1$)
Where R is gas constant and T is absolute temperature
It M is the molecular mass of the gas and f is its density, then
Density = $\dfrac{{Mass}}{{Volume}}$
$ \Rightarrow q = \dfrac{M}{V}$
$ \Rightarrow V = \dfrac{M}{q}$
Therefore, $\dfrac{{PM}}{q} = RT$
$ \Rightarrow \dfrac{P}{q} = \dfrac{{RT}}{M}$ ……………………………… (ii)
 Put (ii) is (i), we get
$v = \sqrt {\dfrac{{rRT}}{M}} $
Clearly, $V \times \sqrt T $
Hence the speed of sound in a gas is directly proportional to square root of its absolute temperature.
So, for air, speed of sound is directly proportional to temperature. Hence,

Option (B) is the correct option.

Additional Information: The factor on which the speed of sound in a gas depends are:-
1. Pressure has a number effect on speed of sound in a gas.
2. At constant pressure, the speed of sound in a gas is inversely proportional to the square root of its density.
3. Speed of sound is more in moist air than in dry air.
4. Speed of sound in a gas is directly proportional to square root of its absolute temperature . The velocity of sound in air increases by $61\,cm\, {s^{ - 1}}$ for every ${1^o}C$ rise in temperature.
5. Velocity of sound in gases increases in the direction of wind and decreases in the direction opposite to wind.

Note: Although from Laplace formula,
$v = \dfrac{{\sqrt {rP} }}{q}$
From this, one can think that velocity of sound in a gas is independent of temperature but one should remember that pressure but one should remember that pressure is a function of temperature. So, velocity of sound in gas also depends on temperature.