For the first part refer to this question : Laplace's correction in Newton's formula for velocity of sound in air.
Second Part: The velocity of sound in an elastic medium is given by \(v=\sqrt{\frac E\rho}\), where E is elasticity and rho is the density.
For air medium, E = B = γP, where γ is constant called the ratio of molar heat capacity of gas constant pressure to that at constant volume and P is the pressure.
So, \(v=\sqrt{\frac{\gamma P}\rho}\)
- The effect of pressure: The velocity of sound in air is \(v=\sqrt{\frac{\gamma P}\rho}\). For one mole of gas, equation of state is PV = RT. At constant temperature PV = Constant. Since \(v=\sqrt{\frac{\gamma P}\rho}\).
∴ \(P\frac M\rho\) = constant ⇒ \(\frac P\rho\) = \(\frac{const}\rho\) = constant
Hence, \(v=\sqrt{\frac{\gamma P}\rho}\) = \(\sqrt{\gamma\times constant}\) = constant
Thus, the velocity of sound in a gas is independent of the pressure of the gas provided temperature remains constant.
- The Effects of temperature: For mole of gas, the equation of state is PV = RT. If M and ρ be the molecular weight and density of the gas, then, \(V=\frac M\rho\)
∴ \(\frac{PM}\rho=RT\) ⇒ \(\frac P\rho=\frac{RT}M\)
Thus, \(v=\sqrt{\frac{\gamma P}\rho}=\sqrt{\frac{\gamma RT}M}\).
For given gas, γ, R, M are constants.
⇒ \(v\propto\sqrt T\)
Hence, the velocity of sound in a gas is directly proportional to the square root of its absolute temperature.
- The effect of density: Consider two gases having same value of γ at the same pressure P but having densities ρ1 and ρ2 respectively. Then, the velocity of sound in them is
\(v_1=\sqrt{\frac{\gamma P}{\rho_1}}\)
\(v_2=\sqrt{\frac{\gamma P}{\rho_2}}\)
∴ \(\frac{v_1}{v_2}=\sqrt{\frac{\rho_2}{\rho_1}}\) ⇒ \(v\propto\frac1{\sqrt\rho}\)
Hence, velocity of sound in a gas is inversely proportional to the square root of the density of the gas.
- The Effect of mass: Let us consider M1 and M2 be molecular weight of the gas of the gases, then at constant temperature, the velocity of sound in them is
\(v_1=\sqrt{\frac{\gamma RT}{M_1}}\) [Here, we consider on mole of each gas such that PV = RT]
\(v_2=\sqrt{\frac{\gamma RT}{M_2}}\)
Here, for a given gas at constant temperature, R, T and γ are constants.
∴ \(\frac{v_1}{v_2}=\sqrt{\frac{M_2}{M_1}}\Rightarrow v\propto\frac1{\sqrt M}\)
Hence, velocity of sound in a gas is inversely proportional to the square root of the molecular mass of the gas.