I was reading a textbook. I discovered that the was pointed out the speed of sound increases with increase in temperature. However sound is a mechanically wave, and also it travels quicker when molecules are closer.

But rise in temperature will attract molecules away from every other, and also then accordingly the rate of sound have to be slower. Exactly how is it possible that the speed of sound boosts if temperature increases? What is the relation of speed of sound and also temperature?  The speed of sound is given by:

$$v = \sqrt\gamma\fracP\rho \tag1$$

where $P$ is the pressure and $\rho$ is the density of the gas. $\gamma$ is a continuous called the adiabatic index. This equation was an initial devised by Newton and also then modification by Laplace by introducing $\gamma$.

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

Now let"s take into consideration the impact of temperature. As soon as you"re heating the gas you have to decide if you"re walk to keep the volume constant and allow the push rise, or store the pressure continuous and allow the volume rise, or miscellaneous in between. Let"s take into consideration the possibilities.

Suppose we save the volume constant, in which instance the press will climb as we warmth the gas. That means in equation (1) $P$ rises while $\rho$ continues to be constant, so the speed of the sound goes up. The speed of sound is increasing because we"re successfully making the gas stiffer.

Now suppose we save the pressure constant and permit the gas expand as it"s heated. That method in equation (1) $\rho$ decreases when $P$ stays consistent and again the speed of sound increases. The speed of sound is increasing because we"re make the gas lighter so the oscillates faster.

And if us take a center course and also let the pressure and the volume increase then $P$ increases and also $\rho$ decreases and again the speed of sound walk up.

So every little thing we do, raising the temperature boosts the rate of sound, yet it does that in various ways depending on how we let the gas broaden as it"s heated.

Just together a footnote, suitable gas obeys the equation that state:

$$PV = nRT \tag2$$

where $n$ is the variety of moles of the gas. The (molar) thickness $\rho$ is just the variety of moles every unit volume, $\rho = n/V$, which way $n = \rho V$. If us substitute for $n$ in equation (2) us get:

$$PV = \rho VRT$$

which rearranges to:

$$\fracP\rho = RT$$

Substitute this into equation (1) and we get:

$$v = \sqrt\gamma RT$$

so:

$$v \propto \sqrtT$$

which is whereby we came in. But in this form the equation conceals what is really going on, hence your confusion.

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Experimentally, the consistent of proportionality because that the over equation is approx. 20.