# Why bulk modulus of air is initial pressure?

So in the definition of bulk modulus the strain part is defined as (∆v/v) where ∆v is change in volume and v is the initial volume and while driving it for air we use the ideal gas law PV= constant and differentiate to to show that bulk modulus is equal to the initial pressure.

But in the gas law PV=k the "P" and "V" are instantaneous states, then why after differentiating do we set it equal to initial values and not the instantaneous ones? I understand that if ∆v is small then initial p is bulk modulus but what if ∆v large enough that the states change appreciably? Should the initial pressure still be considered the bulk modulus?

Note: I am assuming constant temperature.

• Who says the bulk modulus is the initial pressure? It's the instantaneous pressure, as you note. The initial pressure can be used as an approximation if the changes are suitably small. – Chemomechanics Apr 1 at 20:22
• It should more precisely be $-dP/d\ln{V}$ which is equal to the instantaneous pressure P. – Chet Miller Apr 1 at 21:50
• Yes this is what I was thinking exactly while going through the derivation – Lucifer Apr 2 at 3:56