Question

At what temperature will the speed of sound in air be $1.75$ times its speed at N.T.P?

Answer 1

N.T.P is the normal temperature and pressure. According to IUPAC full form is the international union of pure and applied chemistry. Here, we will use the kinetic theory of gases formula where, RMS velocity of a gas molecule is directly proportional to square root of absolute temperature.

Complete step-by-step answer:
The NTP value of temperature and pressure is$293.15K$ and $1atm$.The speed of sound is the distance travelled per unit time it propagates through medium .Speed of sound is also called an ideal gas that depends only on temperature and composition. Sounds speed vary from medium to medium such as sound travels slow in gas, fast in liquid and fastest in solid. Sound wave in solid is composed of compression waves and there is a shear wave which is very different kind of sound wave which occurs only in solids. Wave propagates in two types in solid first is longitudinal wave and second is transverse wave .These wave have different speed at the same frequency.
A longitudinal wave is associated with compression and decompression in the direction sound travels but the transverse wave which is also called shear wave occurs only in solids because only solid supports elastic deformation. In a non-dispersive medium speed of sound is independent of sound frequency. In a dispersive medium, speed of sound is a function of sound frequency, through the dispersion relation each frequency component propagates at its own speed also called phase velocity.
From kinetic theory of gases RMS velocity of a gas molecule is directly proportional to square root of absolute temperature.
${{V}_{r.m.s}}\alpha \sqrt{Temperature}$
$\dfrac{{{V}_{1}}}{{{V}_{2}}}=\sqrt{\dfrac{{{T}_{1}}}{{{T}_{2}}}}$
As, we know that at NTP
${{T}_{1}}=293.15K$
And from the question we know,
${{V}_{2}}=1.75{{V}_{1}}$
Using the values in the above equation we get:
$\dfrac{{{V}_{1}}}{{{V}_{2}}}=\dfrac{1}{1.75}=\sqrt{\dfrac{293}{{{T}_{2}}}}$
${{T}_{2}}=293\times {{\left( 1.75 \right)}^{2}}$
On simplifying above equation we get:

& {{T}_{2}}=897.31k \\\ & {{T}_{2}}=624{}^\circ C \\\ \end{aligned}$$ Hence at $$624{}^\circ C$$ speed of sound in air will be 1.75 times its speed at N.T.P. **Note:** Air contains a small amount of carbon dioxide, which is a dispersive medium and causes dispersion to air at ultrasonic frequency. The loudest natural sound on earth is caused by an erupting volcano. Flies cannot hear any kind of sound. Horror films like to use infrasound.