(7) "If a system of ideal gases has changed to another state (possibly in an entirely unknown way) without changes remaining in other bodies, then the final entropy can certainly not be smaller, it can only be greater than or equal to the initial condition. In the former case this process is an irreversible one, in the latter case a reversible one.
"Equality of entropy in the two states therefore constitutes a sufficient and at the same time a necessary condition for the complete reversibility of the passage from one state to the other, provided no changes are to remain behind in other bodies."
(8) "This proposition has a very considerable range of validity; for there was expressly no limiting assumption made concerning the way in which the gas system reached its final condition; the proposition is therefore valid not only for slowly and simply changing processes but also for any physical and chemical processes provided at the end no changes remained in any body outside of the gas system. Nor need we believe that entropy of a gas has significance only for states of equilibrium, provided we can suppose the gas mass (moving in any way) to consist of sufficiently small parts each so homogeneous that it possesses entropy."[26]
Then the summation must extend over all these gas parts. "The velocity has no influence on the entropy, just as little as the height of the heavy gas parts above a particular horizontal plane."
(9) "The laws thus far deduced for ideal gases can in the same way be transferred to any other bodies, the main difference in general being that the expression for the entropy of any body cannot be written in finite magnitudes because the equation of condition is not generally known. But it can always be shown—and this is the decisive point—that for any other body there really exists a function possessing the characteristic properties of entropy."
Now let us assume any physically "homogeneous body, by which is meant that the smallest visible space parts of the system are completely alike. Here it does not matter whether or no the substance is chemically homogeneous, i.e., whether it consists of entirely like molecules, and consequently it also does not here matter whether in the course of the prospective changes of state it experiences chemical transformation.... When the substance is stationary the whole energy of the system will consist of the so-called 'inner' energy
, which depends only on the mass and inner constitution of the substance, which constitution is conditioned by the temperature and density."
(10) Let us suppose that with such a homogeneous body there is conducted a certain reversible or irreversible cycle process which therefore brings the body exactly back again to its initial condition. Let the external influences on the body consist in the performance of work and in heat supply or withdrawal, which heat exchange is to be effected by any number of suitable heat reservoirs. At the end of the process no changes remain in the body itself, only the heat reservoirs have altered their state. Now let us suppose the heat carriers in the reservoirs to be composed of purely ideal gases, which may be kept at constant volume or under constant pressure, at any rate only be subject to reversible changes of volume. According to the last proved proposition, the sum of the entropies of all the gases cannot have become smaller, for at the end of the process no changes remain in any other body, not even in the body which completed the cycle process.
(11) Let