SOME NEW ASPECTS OF THERMODYNAMICS*

There have been several attempts to present the second law of thermodynamics—the principle of increasing entropy—not in the form of an inequality. This, evidently, is not difficult to do if one introduces into consideration the quantity of the increment of entropy occurring owing to irreversible processes taking place within the system, \(\Delta S_{irr}\). In the paper under review, the methods of introducing this concept into thermodynamic equations are set forth in detail, and the practical convenience of the new relations that arise is shown.

Considering a system in a state of thermal and mechanical exchange and exchange of matter with the medium, the authors write the fundamental laws of thermodynamics in the form:

\[ \Delta E = \sum_m E_m + \sum_n Q_n - W, \]

\[ \Delta S = \sum_m S_m + \sum_n \frac{Q_n}{T_n} + \Delta S_{irr}. \]

Here \(\Delta E\) and \(\Delta S\) are the changes in energy and entropy occurring in the system over the chosen time interval; \(E_m\) and \(S_m\) are the energy and entropy brought into the system by a flow of matter; the remaining notation is conventional.

* R. Tolman and P. Fine, On the irreversible production of entropy, Rev. Mod. Phys., 20, 51, 1948.

Applying these equations to cyclic and stationary processes, one may, by setting \(\Delta E\) and \(\Delta S\) equal to zero, combine the two laws of thermodynamics into a single equation, called by the authors the “efficiency equation.” It is obtained by multiplying the entropy equation by the temperature of the medium \(T_0\) and subtracting it from the energy equation:

\[ W = \sum_m (E_m - T_0 S_m) + \sum_n \frac{T_n - T_0}{T_n} Q_n - T_0 \Delta S_{irr}. \]

Since \(\Delta S_{irr}\)—the amount of entropy produced by irreversible processes taking place within the system—is a positive quantity, the equation shows quite clearly the effect of irreversible processes on the “loss” of possible work. From this equation follows at once the possibility of estimating the inefficiency of a given process that is an element in the operation of some machine. It is important to emphasize that the “loss” of possible work \(T_0 \Delta S_{irr}\) is an additive quantity and is composed of analogous contributions for all irreversible processes occurring in the system. This enables the engineer to assess the reduction or increase in the efficiency of the operation of the whole system due to some particular cause, without analyzing the operation of the entire system as a whole.

In order for the constructed equality to be of use in considering concrete phenomena, one must be able to calculate the change of entropy occurring owing to irreversible processes. As is known, the definition of entropy as the sum of reduced heats pertains to reversible processes. On the other hand, it should be remembered that entropy is a function of state. Consequently, if the initial and final states of two processes, of which one is reversible and the other is not, coincide, then the entropy changes will be the same. In this way one may calculate the quantity \(\Delta S_{irr}\), but this method will be limited by the circumstance that the initial and final states of the system must be equilibrium states; otherwise we cannot bring the system from the first state into the second by a reversible path.

In the paper under review, calculations are made of the irreversible change of entropy for a number of cases. Where possible, the authors give formulas for the rate of irreversible change of entropy. These calculations are carried out for the degradation of energy in friction and in the passage of an electric current through a resistance, for irreversible heat flow, for the case of free expansion, for diffusion in solution, and for the case of passage of impact waves in gases, and for chemical reactions.

In the following part of this work the authors return to the efficiency equation cited above and show how this equation may now be applied (knowing the expressions for the irreversible changes of entropy for various processes) to the calculation of the efficiency of the operation of various systems. Here cyclic processes are considered, occurring in a simple heat engine and in a heat engine in which the working fluid returns to the heater, and two stationary processes: the first—a stationary flow of a homogeneous fluid through an apparatus supplied with a turbine producing work, and the second—the stationary admission into an apparatus of certain chemical products and the stationary discharge of other substances.

Using two examples of viscous flow and the thermoelectric effect, the authors show new possibilities of thermodynamics when the second law is written as an equality rather than an inequality. In conclusion, several remarks are made about the concept of temperature. It is emphasized that this concept has a strict meaning only for an equilibrium state.

A. K.