Prove that `V= sqrt( gamma CT)` from the ideal gas equation for speed of sound is gases.

Explanation

Let us consider an ideal gas whose equation of state is
`PV= nRT " " ...(1)`
where P-pressure, V- volume, T-temperature, n-number of mole and R-universal gas contant.
For a given mass of molecules equation (1) can be written as
`(PV)/(T)` constant `" "...(2)`
for a fixed mass m, density of the gas inversely varies with volume i.e.,
`rho prop (1)/(V),V=(m)/(rho) " "....(3)`
Substituting equation (3) in equation (2), we get
`(P)/(rho)=cT " "...(3)`
where c proportionality is constant.
`:. v=sqrt((gamma P)/(rho))= sqrt(gamma cT)`