Show that the adiabatic bulk modulus of an ideal gas is equal to the product of the pressure of the gas and the ratio of its two specific heat `(gamma)`.

Explanation

In case of adiabatic change of some fixed mass of ideal gas, `pV^(gamma) = "constant"`,
Where,
`gamma = ("specific heat of the gas at constant pressure")/("specific heat of the gas at constant volume")`
Differentiating the above relation we get,
`V^(gamma) dp + pgamma V^(gamma-1) dV = 0`
or, `V^(gamma)dp + gamma p (V^(gamma))/(V) dV = 0`
or, `(dp)/(p) = -gamma (dV)/(V)`
`:. K_(s) = (dp)/(-(dV)/V) = gamma p`, [where `K_(s)` is the adiabatic bulk modulus of the gas]
So, the adiabatic bulk modulus of an ideal gas is equal to the product of the pressure of the gas and the ratio of its two specific heats `(gamma)`.