## 12 Volume Entropy

A gas particle is somewhere in a volume *V _{1}*. We turn a dial doubling the volume where the particle can be. According to the ideas of the last section, we are also doubling the number of micro-states available to the particle. It doesn't matter what elementary unit volume

*V*jwe assign to a micro-state, the entropy increases, and we have

_{0}In physics the *natural logarithm* is used to measure the entropy. But this differs only by a constant factor from the base 2 logarithm. Much more important is the fact, that in the context of physics the logarithm is multiplied with the Boltzmann constant *k* . Thus our above change of volume for 1 particle has the measure

By multiplying by *k*j, the entropy artificially, so to speak, obtains the units of *k*j, i.e. J/K. In the next section we will see the advantages of this definition. This new scaling changes nothing at all for the present. However, now the entropy is invariant if and only if *k* is !

If one has *N* jparticles instead of one, each residing independently somewhere in the volume, then a change in volume from *V _{1}* jto

*V*jcauses a corresponding change in the volume entropy of

_{2}By whatever factor the volume transforms for a moving observer the volume entropy must transform the same as the Boltzmann constant *k* !

We would obtain very similar results with the* concentration entropy S _{K}* , the

*entropy of mixing S*,

_{M}*free entropy S*nor

_{F}*spin entropy S*.

_{S}All these changes in entropy equal the product of *k* jand the logarithm of a ratio of micro-states, and the number of the micro-states do not change due to a relative movement of the observer. They are themselves calculated as a ratio of two 'volumes' (maybe also volumes in phase space). It is irrelevant how exactly the numerator and denominator transform, because the corresponding factor cancels out.

However, the calculation of the total entropy of a given amount of an ideal gas is by no means trivial, since the various micro-states, for example of kinetic energy of a gas particle, may assume very different probabilities. It is not just about counting the number of micro-states, since these are not evenly distributed. The corresponding calculations can be found in the literature. For secondary school, the book "Entropie und Information - naturwissenschaftliche Schlüsselbegriffe" by Wolfgang Salm (Aulis Verlag 2002) is recommended. The sections **11** and **12** of this work are based largely on that publication.