Question

A given Quantity of a gas is at a pressure p and Absolute Temperature T . The isothermal bulk modulus of the gas is.
2p/3
P
3p/2
2p

Answer 1

Isothermal bulk modulus is defined as volume times negative partial derivative of pressure with respect to volume at constant temperature:
K = - V ∙ (∂P/∂V)_T

An ideal gas satisfies the following equation of state:
P∙V = n∙R∙T

So the pressure for an ideal gas is given by:
P = n∙R∙T/V
The partial derivative of pressure w.r.t. volume at constant temperature is:
(∂P/∂V)_T = (∂(n∙R∙T/V)/∂P)_T
= n∙R∙T ∙ d(1/V)/dV
= - n∙R∙T/V²
= (- n∙R∙T/V) ∙ (1/V)
= - P/V

Hence,
K = - V ∙ ( - P/V ) = P