In class we calculated the root$-$mean$-$square speed of the water molecules at room tem$-$perature. Following the same line of thinking as in Question $65,$ we realize that theroot$-$mean$-$square speed of molecules in air $($mostly N$2)$ should be comparable to thespeed of sound in air $($or in an ideal gas$).$ This should not be too surprising to you withthe knowledge now you have. $($a$)$ Using the equation of state of an ideal gas, calculatethe bulk modulus $($at temperature T$),$ which is defined asB$=$volume stressvolume strain$=$F$/$AV$/$V$($V$)/$V$($b$)$ Recall that the speed of sound in a fluid v $=$ B$/$ depends on the elastic andinertial 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 wellas the Boltzmanns constant kB$($c$)$ Compute the result in $($b$)$ at room temperature. The result was first obtained byIsaac Newton, but it is lower than the measured value due to the failure to include theeffect 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 bylack of time or by insulation. The compressions and rarefactions are adiabatic. As aresult, the speed of sound has an additional factor of $,$ where is the adiabatic index$(=7/5=1\mathrm{.}400$ for diatomic molecules at room temperature$).$ Evaluate Laplacesresult for the speed of sound and compare it to the numerical value that you know oryou can find online.$($e$)$ Compare your result with the root$-$mean$-$square molecular speed.