# 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.

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.

• Thank you John for your answer, it completely clarified the problem :) – user2820052 May 15 '15 at 13:56
• I was just wandering - how do I correct for the fact that air is not an ideal gas? I was thinking that Van der Waals equation would provide me with better estimate of how will the pressure change, but how do I correct bulk modulus for the fact that air isn't ideal gas? Any ideas would be greatly appreciated... – user2820052 Jun 24 '15 at 13:53
• @user2820052 looks like John didn't get back to you; did you figure this out by other means? It appears thermodynamic properties have more to do with predicting bulk modulus rather than material properties (molecular weight, etc.). So tables of specific heat ratio of various gases may be useful. – docscience Feb 28 '17 at 18:47

As we know that density $D=\frac{M}{V}$ here $V$ is constant, so $dD=dM$ for unit volume, Now bulk modulus is given as

$$K = D\frac{dp}{dD}= M \frac{dp}{dM}$$ i.e $K$ is proportional to $\frac{dp}{dM}$

But the change in Mass is very less as compare to change in pressure therefore $k$ is increases with pressure.

• Hi, welcome to Physics SE! Please don't post formulae as pictures or plain text, but use MathJax instead. MathJax is easy for people on all devices to read, and can show up clearer on different screen sizes and resolutions. I've edited it here as an example. Look at this Math SE meta post for a quick tutorial. – user191954 Aug 5 '18 at 12:40