# Origin and justification for stating the second law of thermodynamics as $dS=d_{\rm{e}}S+d_{\rm{i}}S$?

The second law of thermodynamics is sometimes stated as:

$$dS=d_{\rm{e}}S+d_{\rm{i}}S$$

with :

• $$dS$$, the variation of entropy of the system;
• $$d_{\rm{e}}S$$, the entropy exchanged with or supplied to the system from its surroundings ($$d_{\rm{e}}S$$ can be positive, null or negative; and for a closed system that can only exchange heat with its surroundings: $$d_{\rm{e}}S=\delta Q/T$$);
• $$d_{\rm{i}}S$$, the entropy produced inside the system ($$d_{\rm{i}}S≥0$$; and null only for reversible processes).

However, up to now, I have not seen a reference given along with this equation to justify it with regards to previous statements of the second law. For example, in Non-Equilibrium Thermodynamics, from S. R. DE Groot and P. Mazur (1961), Chapter 3, §1, it just says that this statement is "a well known form of the second law of thermodynamics".

When and where was this statement originally formulated and its equivalence with previous alternative statements first demonstrated?

Clausius in his "The mechanical theory of heat" discusses what he calls "uncompensated transformation". First he derives that $$\int \frac{dQ}{T}=0$$ for a reversible cyclic transformation, then that $$\int \frac{dQ}{T} \le 0$$ for an irreversible transformation, finally on page 214 of the Macmillan 1879 edition he introduces the quantity $$N = -\int \frac{dQ}{T}\ge 0$$ and calls it the "uncompensated transformation" (uncompensirten Verwandlung). The infinitesimal increment of this $$N$$ is $$d_iS$$.
§2, Magnitude of the Uncompensated Transformation. In many cases the magnitude of the Uncompensated Transformation is obtained directly from the equivalence value of the transformations, as determined by the method of Chapter IV. If for example a quantity of heat $$Q$$ is generated by any process such as friction, and this is finally imparted to a body of temperature $$T$$, the uncompensated transformation thus produced has the value $$\frac{Q}{T}$$. Again, if a quantity of heat Q has passed by conduction from a body of temperature $$T_1$$ to another of temperature $$T_2$$ then the uncompensated transformation is $$Q\left(\frac{1}{T_1}-\frac{1}{T_2}\right).$$ If a body has passed through a non-reversible cyclical process, and we wish to determine the resulting uncompensated transformation, which we may call $$N$$, we have, by the principles explained in Chapter IV., the equation $$N = -\int \frac{dQ}{T} \tag{1}\label{1}.$$ As however a cyclical process may be made up of several individual changes of condition in a given body, some of which may be reversible, others non-reversible, it is in many cases interesting to know how much any particular one of the latter has contributed towards making up the whole sum of uncompensated transformations. For this purpose we may suppose that after the change of condition which we wish to enquire into, the variable body is brought by any reversible process into its former condition. By this means we form a smaller cyclical process, in which equation $$\eqref{1}$$ may be applied just as well as in the whole process. Thus if we know the quantities of heat which the body has taken in during this process, and the temperatures which appertain to them, the negative integral $$-\int \frac{dQ}{T}$$ gives the uncompensated transformations which have taken place. But as the return to the original condition, which has taken place in a reversible manner, can have contributed nothing to increase this sum, the expression above gives the uncompensated transformation which was sought, and which was caused by the given change in condition.