Allotropy 3

The first big practical advance that was made in our knowledge of the relations between thermal changes and chemical affinity was due to van't Hoff's demonstration that the law of mass action was a necessary consequence of the second law of thermodynamics and of the quantitative connection between the value of the "equilibrium constant" and the change in total energy due to the reaction. The second law of thermodynamics, like the first, is a statement of experience, perhaps one might say a collection of inter-related statements of experience. The first law states that it is impossible to create energy; the second law states that it is im possible to convert the heat energy of our surroundings continu ously into useful work. The second law deals with a question which the first law does not answer, namely, under what condi tions can heat energy be converted into useful work (that is to say, mechanical energy). If heat energy passes spontaneously from one body to another, the first body is said to be at a higher temperature. It is a matter of universal experience that the re verse change, i.e., the passage of heat from one body to another of higher temperature never takes place spontaneously. It is also a matter of experience that any spontaneous process can be made to yield a definite amount of useful work. For instance, the cool ing of a furnace is a spontaneous process. By the use of steam and suitably designed engines, it is possible to obtain useful work from this process.

Carnot's Definition.—The question how much useful work can be obtained by the spontaneous passage of heat from one temper ature to another is of fundamental importance not only in mechan ical engineering but in the science of chemistry. Carnot, a French engineer, actually solved this problem in 1824 before the law of conservation of energy was universally accepted. He showed that the maximum amount of work which can possibly be obtained from the "spontaneous" transference of a quantity of heat Q from a reservoir at temperature to another of the lower temperature T2 was Q(TI — T2)/Ti. It will be noted that Carnot's result involves a quantitative definition of "absolute" temperature. The so-called "thermodynamic scale of temperatures" which is implicit in his result, is based on the behaviour of a perfect gas, the energy content of which is independent of its volume and dependent only on the temperature. The absolute temperature of a perfect gas is defined by the fundamental gas law T=PV/R, where P is the gas pressure, V its volume, and R a constant. The value of R depends on the units in which the pressure and volume are expressed. Many gases approach very nearly the properties of the imaginary "perfect" gas, particularly at low densities.

The next point of interest is that Carnot in deriving his result imagined a perfectly frictionless engine, with a perfect gas as a working fluid, taking heat from the high temperature reservoir (as steam takes heat from a furnace), converting part of the heat into work, and rejecting the rest to the low-temperature reservoir. The argument is that such an engine will always yield the maxi mum amount of work, because if it is reversed, the expenditure of work, only infinitesimally greater in amount than the engine yields on the direct operation, will suffice to restore the heat from the low temperature to the high temperature reservoir. For suppose it was possible to make another reversible and frictionless engine which was capable of converting a greater proportion of the trans ferred heat into work. Then it would be possible to use part of

the work yielded by this change to reverse the first engine, and to restore the whole system to its original condition, leaving a definite amount of work over which has been created out of nothing. This is contrary to the first law. Hence we say that the work that can be done by a perfectly efficient reversible engine or process is independent of the nature of the process.

The Gibbs-Helmholtz Equation.

This conclusion leads to a mathematical expression of the first and second laws of thermo dynamics in a form which can be applied to all chemical as well as other phenomena. For if, instead of having a perfect engine to transfer heat from the reservoir at T-I-dT to another at T, we allow a chemical process to take place in the first reservoir, yielding a maximum amount of work A and absorbing heat from the reservoir equal to Q, and we then reverse this chemical proc ess in the second reservoir by the expenditure of A — dA of work, we have gained an amount of work = dA, and if the process is conducted reversibly this work must be equal to that gained by any other perfect process; in other words, dA=Q.dT /T. But Q, the heat absorbed, is equal (by the law of conservation of energy) to the maximum work A, less the diminution in total energy U, i.e., of work that can be obtained from a (reversible) chemical process taking place spontaneously and at constant volume at the temper ature T; Ad-dA is the work that would be obtained by allowing it to take place at the same volume at the higher temperature Td-dT; and U is the diminution of total energy which corre sponds to the heat evolved when the reaction takes place without the performance of any work, e.g., in a closed vessel. A is of ten known as the decrease in free energy, to distinguish it from U, the decrease in total energy.

The most direct use of this equation is found in its application to the electric cell or battery. For the electric cell is a contrivance for converting chemical energy into useful work. If the cell is reversible, the work will be the maximum obtainable from the chemical reaction which is the source of the electric current. What we mean by a reversible cell is that, if a potential infinitesimally greater than that yielded by the cell is applied to the electrodes, the current will travel in the reverse direction in the cell and the chemical reaction will be reversed. The lead accumulator is nearly reversible in this sense ; so is the Daniell cell. The electro motive force of the Daniell cell is found to be i•i volts. For every gram-equivalent of copper deposited and zinc dissolved in the cell, an amount of work is done =EF, where F is the quantity of electricity (96,540 coulombs) associated with one gram-equivalent. These units can be expressed in calories ; calcu lation shows that if E is the potential, the work done by the reaction is 23,050E calories. Hence the work done by the Daniell cell is 25,26o calories per gram-equivalent of copper deposited. The heat of the reaction, when it takes place without performance of work, is found to be 25,060 calories. Hence in this case A = U nearly. We should find therefore that dA/dT, or dE/dT, which is the change of electromotive force with the temperature, is very small. Measurement shows that it is only 0.000034 volt per degree.