Why should bulk modulus always be positive?

The minus sign that appears in Equation 12.39 is for consistency, to ensure that $$B$$ is a positive quantity. Note that the minus sign ($$–$$) is necessary because an increase $$\Delta p$$ in pressure (a positive quantity) always causes a decrease $$\Delta V$$ in volume, and decrease in volume is a negative quantity

This is the reasoning which the source I am reading from gives but I don’t get why bulk modulus would have to be positive. I also can’t understand why bulk modulus would always be positive for an object in an equilibrium system

• Because if it was negative the system would collapse? Commented Feb 9, 2023 at 13:49
• Could you explain it a bit more Commented Feb 9, 2023 at 13:52
• @Nitish hint to understand the collapse/instability: consider the harmonic oscillator differential equation with a spring that has a negative elasticity constant: physics.stackexchange.com/q/380173/226902 . In fact, the modulus is the "elastic constant" for waves (i.e. oscillations) in the material: physics.stackexchange.com/q/392681/226902 Commented Feb 9, 2023 at 17:51
• are you asking why it's defined to be always positive instead of changing the minus to a plus and saying it's always negative, or are you asking why it always has to be one or the other? Commented Feb 10, 2023 at 11:58
• I was asking why it always has to be one or the other Commented Feb 28, 2023 at 4:54

If the bulk modulus were negative, the system would be in an unstable configuration. (Thermodynamically, this means that it is not a minimum of the free energy. However, the instability is really just purely mechanical; no references to $$S$$ and $$T$$ are requried.) Consider a system (for simplicity and definiteness, make it a fluid) with negative bulk modulus $$B$$ at a given pressure $$P$$ and volume $$V$$. What happens if we compress (by applying an external force) the system, so that $$V\rightarrow V+\Delta V$$, with $$\Delta V<0$$?

If $$B$$ is negative, then the pressure drops by a positive amount, $$|\Delta P|$$, where the actual $$\Delta P<0$$. In a realistic system, the pressure would increase as the system is compressed, so that it would eventually come back to equilibrium, when the increased pressure balances the applied external force. However, here the volume decreases and the pressure drops, which means that as the system shrinks, it becomes less and less able to resist the external force, meaning that it will continue to shrink even more! The reverse happens if we start out by allowing the system to expand a little and increase its pressure, which would cause it to expand even further. Either way, a miniscule change in volume leads to a runaway effect, with volume either increasing or decreasing until the equation of state changes enough to make $$B$$ positive again.

It is possible, in principle, to have a nonequilibrium system with a negative $$B$$, but the configuration is unstable. For example, direct application of the van der Waals equation of state, $$\left(P+\frac{an^{2}}{V^{2}}\right)(V-nb)=nRT,$$ predicts than when $$T$$ is small enough (less than the critical point temperature $$T_{c}$$) there a region where the bulk modulus is negative (as shown in this image from Wikipedia).

If a fluid is quickly supercooled into the region with negative $$B$$, the anomalous behavior can be directly observed, but only briefly. What happens after a very short period of time is that the instabilities take over. Some regions of the fluid (where the local pressure happened to be lower) rapidly expand and other regions (where the pressure was randomly a little higher) rapidly contract. These become regions of gas and liquid, respectively! The instability is resolved by a phase separation; a system with negative $$B$$ has greater free energy than the same system split into separate liquid and gas components.

• This sounds quite complicated for me as I haven’t done thermal properties of solids yet. I do understand that bulk modulus being negative has something to do with the system being unstable but it would be really helpful if you could put it in simple terms. Commented Feb 9, 2023 at 14:30
• @Nitish As I said, it doesn't really have anything to do with thermodynamics, just forces (as discussed in the answer by Chemomechanics).
– Buzz
Commented Feb 9, 2023 at 19:17
• It get it now. Thanks mate! Commented Feb 10, 2023 at 1:24

As requested, a simpler version of the excellent answer by Buzz: If you pulled on this hypothetical material, it would shrink. We just don’t see that behavior in passive materials.

Alternatively, if you squeezed it in a vise, it would expand, which would pressurize it even more, which would cause further expansion. This is called an explosion and again is not seen in passive materials.

And a nonsimple, math-intensive supplement: The ultimate reason for positive bulk moduli in passive materials arises from the Second Law; materials at equilibrium tend to minimize, not maximize, their stored strain energy.