# Isothermal bulk modulus of an ideal gas

To calculate isothermal bulk modulus, I have two methods.

Method 1:

$P_1V_1=P_2V_2=K$

$B=-\dfrac{(P_2-P_1)V_1}{V_2-V_1}$

$B=-\dfrac{(\dfrac{K}{V_2}-\dfrac{K}{V_1})V_1}{V_2-V_1}$

$B=-\dfrac{\dfrac{K(V_1-V_2)}{V_2V_1}V_1}{V_2-V_1}$

$B=\dfrac{KV_1}{V_2V_1}$

$B=\dfrac{K}{V_2}$

$B=P_2$

Method 2:

I love calculus

$PV=K$

On differentiating,

$dPV+PdV=0$

$\dfrac{dP}{dV}=-\dfrac{P}{V}$ .......(1)

$B=-\dfrac{dP}{dV}V$

Using equation (1)

$B=-(-\dfrac{P}{V})V$

$B=P$

Using method (1) I got $P_2$ as the answer whereas in method (2) I got $P_1$ as the answer. I know $P_1\neq P_2$. Which one is correct then?

(2) gives the answer $P$, the pressure at any instant, and not $P_1$. Method(1) is wrong. $B=-\frac{dP}{dV}V$ includes $\frac{dP}{dV}$, which is the instantaneous rate of change of pressure with respect to volume and B varies with P(or V) in this process. What you are doing is calculating the average change over a significant change in pressure and volume. Instantaneous change can be approximated by average change only if the change is very small.

• If I tell you that the ideal gas was initially at pressure $P_1$ and ask you what is the isothermal bulk modulus, what will you answer – user203191 Aug 27 '18 at 9:58
• Method (1) is not "wrong" if $P_2$ is very close to $P_1$ - it is just explicitly writing down the limit, rather than using calculus. If the change is very small, then in the limit $P_1$ = $P_2$ = $P$. The mistake in the OP's first method is assuming that $K$ is constant when $P$ changes. Of course $K$ is independent of pressure (to a good approximation) for solids and liquids, but not for gases. – alephzero Aug 27 '18 at 12:18
• @LoopBack No matter what the initial pressure is the bulk modulus of an ideal gas for an isothermal process at a particular instant is given by the pressure at that instant. If the change is very small then method (1) is not completely wrong and bulk modulus is quite close to $P_1$, so you can safely say that it's $P_1$. – Asit Srivastava Aug 27 '18 at 13:07

Isothermal bulk modulus is defined as volume times negative partial derivative of pressure with respect to volume at constant temperature:

$K = -V \dfrac{\partial P}{\partial V}$

An ideal gas satisfies the following equation of state:

$PV = nRT$

So the pressure for an ideal gas is given by:

$P = \dfrac{nRT}{V}$

The partial derivative of pressure w.r.t. volume at constant temperature is:

$\dfrac{\partial P}{\partial V} = \dfrac{∂\dfrac{nRT}{V}}{∂P}$

$= nRT \dfrac{d\dfrac{1}{V}}{dV}$

$= - nRT/V²$

$= - \dfrac{\dfrac{nRT}{V}} {V}$

$= - \dfrac{P}{V}$

Hence,

$K = P$