What is the relationship between bulk modulus and pressure in an ideal gas?

[adrian116]

The questions is that

The equation of state (the equation relating pressure, volume, and temperature) for an ideal gas is pV=nRT, where n and R are constants.
a.) Show that if the gas is compressed while the temperature T is held constant, the bulk modulus is equal to the pressure.
b.) When an ideal gas is compressed without the transfer of any heat into or out of it, the pressure and volume are related by pV^(x)=constant, where x is a constant having different values for different gases. Show that, in this case, the bulk modulus is given by B=xp.

I have no idea what i should do first...

 

[Physics Monkey]

You should probably start by writing down the definition of the bulk modulus. As you will see when you write the definition down, you need to know how pressure varies with volume for a) an isothermal process and b) an adiabatic process. Now simply calculate using the definition for each kind of process.

 

[krome]

Almost a year later

Isothermal: d(PV)=PdV+VdP= NRdT= 0 since dT= 0. Thus, -dV/V= dP/P and plugging this into the definition of the bulk modulus B=dp/(-dV/V)= P.

Adiabatic: d(PV^x)= xPV^(x-1) dV+ V^x dP= 0 since PV^x is constant. Thus, -dV/V= dP/xP and plugging this in as above gives B= xP.