PROBLEM $66$

In class we calculated the root$-$mean$-$square speed of the water molecules at room temperature. Following the same line of thinking as in Question $65,$ we realize that the root$-$mean$-$square speed of molecules in air $($mostly $N_2)$ should be comparable to the speed of sound in air $($or in an ideal gas$).$ This should not be too surprising to you with the knowledge now you have. $($a$)$ Using the equation of state of an ideal gas, calculate the bulk modulus $($at temperature $T),$ which is defined as

$B=\frac{\text{v}\text{o}\text{l}\text{u}\text{m}\text{e}\text{s}\text{t}\text{r}\text{e}\text{s}\text{s}}{\text{v}\text{o}\text{l}\text{u}\text{m}\text{e}\text{s}\text{t}\text{r}\text{a}\text{i}\text{n}}=\frac{\frac{F}{A}}{\frac{V}{V}}=-\frac{P}{\frac{V}{V}}$

$($b$)$ Recall that the speed of sound in a fluid $v=\sqrt[\phantom{2}]{\frac{B}{}}$ depends on the elastic and inertial properties of the fluid, where $B$ is the bulk modulus and $$ is the density of air. Express the speed of sound waves in terms of molecular mass $m,$ temperature $T,$ as well as the Boltzmann's constant $k_B$

$($c$)$ Compute the result in $($b$)$ at room temperature. The result was first obtained by Isaac Newton, but it is lower than the measured value due to the failure to include the effect of fluctuating temperature.

$($d$)$ Pierre$-$Simon Laplace later pointed out that as a sound wave passes through a gas, the compressions are so rapid or so far apart that energy flow by heat is prevented by lack of time or by insulation. The compressions and rarefactions are adiabatic. As a result, the speed of sound has an additional factor of $\sqrt[\phantom{2}]{},$ where $$ is the adiabatic index $(=\frac{7}{5}=1\mathrm{.}400$ for diatomic molecules at room temperature$).$ Evaluate Laplace's result for the speed of sound and compare it to the numerical value that you know or you can find online.

$($e$)$ Compare your result with the root$-$mean$-$square molecular speed.

$(20$ points$)$