I was reading a textbook. I found that it was mentioned the speed of sound increases with increase in temperature. But sound is a mechanical wave, and it travels faster when molecules are closer.
But an increase in temperature will draw molecules away from each other, and then accordingly the speed of sound should be slower. How is it possible that the speed of sound increases if temperature increases? What is the relation of speed of sound and temperature?
Best Answer
The speed of sound is given by:
$$v = \sqrt{\gamma\frac{P}{\rho}} \tag{1} $$
where $P$ is the pressure and $\rho$ is the density of the gas. $\gamma$ is a constant called the adiabatic index. This equation was first devised by Newton and then modified by Laplace by introducing $\gamma$.
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) $\rho$ 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:
$$ PV = nRT \tag{2} $$
where $n$ is the number of moles of the gas. The (molar) density $\rho$ is just the number of moles per unit volume, $\rho = n/V$, which means $n = \rho V$. If we substitute for $n$ in equation (2) we get:
$$ PV = \rho VRT $$
which rearranges to:
$$ \frac{P}{\rho} = RT $$
Substitute this into equation (1) and we get:
$$ v = \sqrt{\gamma RT} $$
so:
$$ v \propto \sqrt{T} $$
which is where we came in. However in this form the equation conceals what is really going on, hence your confusion.
Experimentally, the constant of proportionality for the above equation is approx. 20.