ideal gas - How does the bulk modulus of air change with rising pressure?

I can't seem to find the answer to what should be a trivial question:

I have a rigid air-tight container of fixed volume and I am pumping air inside. The pressure is increasing (very slowly) from ~100kPa to ~50MPa - is the bulk modulus of air constant throughout the process or does it increase/decrease with increasing pressure?

I am assuming that the bulk modulus of gas should increase with increasing pressure as there is more force acting inside the gas (more gas molecules interactions) and the fluid itself is increasing in density.

Can you please offer any advice or reference me to some link.

Answer

If the temperature of the gas is kept constant during the compression then the bulk modulus of an ideal gas is just equal to the pressure.

The definition of the bulk modulus is:

$$ K = -V\frac{dP}{dV} \tag{1} $$

For an ideal gas $PV = RT$, so $P = RT/V$. If the temperature is constant this gives:

$$ \frac{dP}{dV} = -\frac{RT}{V^2} \tag{2} $$

and substituting into (1) we get:

$$ K = V \frac{RT}{V^2} = \frac{RT}{V} $$

and $RT/V$ is just $P$ so we get:

$$ K = P $$

Note that if the compression is not isothermal, or the gas is not ideal, equation (2) will not apply and the bulk modulus will not be equal to the pressure.