# Negative specific heat

I just read a paper containing this term, "negative specific heat". I have only come across specific heat which is positive not a negative one. How is a negative specific heat be defined or possible?

SpaceTiger
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touqra said:
I just read a paper containing this term, "negative specific heat". I have only come across specific heat which is positive not a negative one. How is a negative specific heat be defined or possible?

It means that when energy is put into the system, the temperature of the system drops. How could this be? Well, consider the following.

I have a group of stars that are bound together by their own gravity (it happens all the time and is called a "cluster"). The only thing that prevents the stars from collapsing into one another is their motion, which can be related to their temperature. The faster the stars move, the higher their temperature, and the more they can resist the pull of gravity. (Note: I'm not talking about the temperature of the individual stars, but the temperature of the cluster, as defined by the motion of the stars. It's directly analogous to the motions of molecules in a gas.)

Ok, now imagine that we give each of those stars a kick; that is, we put energy into the system. What happens? Well, the cluster expands, becoming more loosely bound and requiring less motion to keep itself from collapsing. The system will reach equilbrium when its motion is just enough to prevent collapse, and since the cluster is now more spread out, this will be when the stars are moving more slowly. Thus, they have a lower temperature. What happened to the energy? It went into the gravitational potential!

If you're uncomfortable with the idea of giving clusters a temperature, then know that the same thing happens in the sun, which is composed of gas and has a temperature in the traditional sense. I think the above example is easier to visualize, however.

In more technical terms, this arises from the virial theorem:

$$E_{tot} = U+K = -K$$

$$K \propto T$$

$$\frac{dE}{dT} \propto \frac{dE}{dK} < 0$$

where U is the potential energy (gravitational and electric will both work), K is the kinetic energy, and T is the temperature.

touqra said:
I just read a paper containing this term, "negative specific heat". I have only come across specific heat which is positive not a negative one. How is a negative specific heat be defined or possible?

SpaceTiger explained very well for gravitational bodies.

also in small matter arise negative heat capacities. For example in hot nucleus. Therein the negative heat capacities arise of existence of surface terms

$$dS = dS _{volume} + dS _{surface}$$

and

$$dU = dU _{volume} + dU _{surface}$$

The surface terms vanish for macroscopic matter but are important in mesoscopic and microscopic regimes. They are characterized also by negative heat capacities responsible of several exotic phenomena of nuclear matter, including phase transitions of nuclei.

I knew there was something fishy here , and there it is
$$K \propto T$$
That is not correct, the temperature is not proprtional to the kinetic energy. Rather, it should be determined from the statistical sum, which considered energy states (not only kinetic energy values).
On the other hand, micro- and mesoscopic systems and, may be , nonequilibrium systems may have "negative" heat capacity, because the value is strictly defined only for macroscopic systems in thermodynamic equilibrium.

SpaceTiger
Staff Emeritus