# Entropy and vacuums

If I have a vacuum (completely devoid of any particles; Q.M. effects aside), then turn a nozzle and let a gas freely expand into the vacuum, will I have increased the entropy of the vacuum?

I get confused a lot when we talk about a system. Most of the time, we seem to mean a fluid within a container (thermodynamics), but what about the actual container itself? Can I make a claim about the entropy of an empty container before/after I fill it with something?

• You don't have increased the entropy of the vacuum. You transfer entropy to vacuum. – lucas Apr 19 '16 at 2:24
• Aren't you implying that I have increased the entropy from zero to some value?? – whatwhatwhat Apr 19 '16 at 2:25
• No. When you don't have a system, you cannot define system's properties like entropy, energy, etc. – lucas Apr 19 '16 at 2:28
• Ah ok, so the vacuum is not part of the "system" whenever we talk about a gas within a vacuum. – whatwhatwhat Apr 19 '16 at 2:40
• Whenever you talk about a gas within a vacuum, is there a vacuum? – lucas Apr 19 '16 at 2:51

There are many ways to think about the entropy of a vacuum (assuming there is no radiation and thus T=0), but all give the same result, the entropy is zero. One easy way is to notice that the walls are made of something (it doesn't matter what) that cannot change its state, so the number of microstates, $\Omega$, is equal to 1. Then $S=k\ln\Omega=0$.

The thermodynamic definition of entropy of a system is

where T is the absolute temperature of the system, dividing an incremental reversible transfer of heat into that system (δQrev). (If heat is transferred out the sign would be reversed giving a decrease in entropy of the system.)

You may consider a vacuum as a system under study, but in this definition of entropy since there is zero over zero the change in entropy is undefined , so there is no meaning to "entropy of vacuum". Heat and temperature characterize large ensembles of particles, and also entropy.

you say :

I get confused a lot when we talk about a system. Most of the time, we seem to mean a fluid within a container (thermodynamics),

A system is what we have under consideration for an evaluation of thermodynamic quantities. Systems can be open, or closed/isolated; if energy can be exchanged with other systems they are open. The law of increasing entropy holds for closed or isolated systems.

In practice, when considering a problem, we can have approximately closed systems if one makes sure that the energy exchanges with the environment are minimal.

but what about the actual container itself? Can I make a claim about the entropy of an empty container before/after I fill it with something?

The container if it has a complete vacuum has no definable entropy. Once you introduce material you can have entropy.

I find conceptually more useful the definition of entropy in statistical mechanics, , where it is connected with the number of microstates in the system and can be shown to be identical to the thermodynamic entropy definition.

The macroscopic state of a system is defined by a distribution on the microstates. The entropy of this distribution is given by the Gibbs entropy formula, named after J. Willard Gibbs. For a classical system (i.e., a collection of classical particles) with a discrete set of microstates, if E_i is the energy of microstate i, and p_i is the probability that it occurs during the system's fluctuations, then the entropy of the system is

The quantity k_B is a physical constant known as Boltzmann's constant, which, like the entropy, has units of heat capacity. The logarithm is dimensionless.

In this definition it is again obvious that a complete vacuum has no definable entropy.