Question

Explain why the speed of sound depends on the temperature of the medium? Why this temperature dependence is more noticeable in a gas rather than in a solid or a liquid?

Answer 1

The speed of sound is given by:

$v=\sqrt{\gamma }\times \sqrt{P}/\sqrt{\rho }$
where P is the pressure and ρ is the density of the gas. γ

is a constant called the $adiabatic$ $index$.

The equation should make intuitive sense. The density is a measure of how heavy the gas is, and heavy things oscillate slower. The pressure is a measure of how stiff the gas is, and stiff things oscillate faster.

Now let's consider the effect of temperature. When you're heating the gas you need to decide if you're going to keep the volume constant and let the pressure rise, or keep the pressure constant and let the volume rise, or something in between. Let's consider the possibilities.

Suppose we keep the volume constant, in which case the pressure will rise as we heat the gas. That means in equation (1) $P$increases while $\rho$ stays constant, so the speed of the sound goes up. The speed of sound is increasing because we're effectively making the gas stiffer.

Now suppose we keep the pressure constant and let the gas expand as it's heated. That means in equation (1) ρ$\rho$decreases while $P$ stays constant and again the speed of sound increases. The speed of sound is increasing because we're making the gas lighter so it oscillates faster.

And if we take a middle course and let the pressure and the volume increase then $P$increases and $\rho$ decreases and again the speed of sound goes up.

So whatever we do, increasing the temperature increases the speed of sound, but it does it in different ways depending on how we let the gas expand as it's heated.

Just as a footnote, an ideal gas obeys the equation of state:

$PV=\eta RT$

(2)

where $\eta$

is the number of moles of the gas. The (molar) density ρ is just the number of moles per unit volume, ρ=n/V, which means n=ρV. If we substitute for n

in equation (2) we get:

$PV=\rho VRT$

which rearranges to:

$P/\rho =RT$

Substitute this into equation (1) and we get:

$v=\sqrt{\gamma RT}$

so:

$v\propto \sqrt{T}$

which is where we came in.

Since molecular vibration is very less in case of liquids and solids therefore temperature dependence is more noticeable in a gas rather than in a solid or a liquid