Nernst’s Heat Theorem

Nernst s heat theorem (1906) As T —> 0, the entropy change in any reversible process tends to zero. 

In September of 1920, Polanyi took a position in Berlin at the Kaiser Wilhelm Institute for Fiber Chemistry, which was housed in the buildings of the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry directed by Fritz Haber. [8] By this time Polanyi had published his doctoral thesis and papers in several areas of thermodynamics, including papers on Nernst s heat theorem and Einstein s quantum theory for specific heats. [9] In the next 13 years, before he was forced to leave Germany in 1933, Polanyi worked in several areas of physical chemistry in Berlin, afterwards heading Manchester s physical chemistry laboratory for fifteen years. In 1948 Polanyi exchanged his professorship in chemistry at Manchester for a chair in social studies, thus formally becoming a philosopher. 

The hydration entropy can also be deduced experimentally (Latimer 18) as the difference between the standard entropy of the hydrated ions (deduced from measurements of the specific heat on the basis of Nernst s Heat Theorem or the Third Law of Thermodynamics) and the theoretically calculated absolute entropy of the gaseous ion, both reckoned per unit volume at constant concentration. This entropy can also be calculated (Eley and Evans18). 

Knowing the values of the entropy constants S and also the specific heats and their temperature coefiicients (i.e. /3, y,. ..) for all gases, we should be able to calculate the equilibrium constant K from the heat of the reaction for all gas reactions at all temperatures. The constants /3, y,. .. can be determined for each gas by direct measurements of the specific heats at different temperatures. The two laws of thermodynamics alone, however, do not enable us to express the entropy constants S in terms of experimental data. This has only recently been made possible by the discovery of Nernst s heat theorem (see Chapter XIV.). 

The work done by the condensed system during the reversible reaction can be calculated by Nernst s heat theorem from thermal quantities alone. It is... 

Thus Nernst s heat theorem enables us to calculate the entropy constant of a gas from its specific heat and from the vapour pressure curve of the condensed gas, and vice versa. Equation (166), however, involves the assumption that the specific heat of the gas can be expressed as a series of powers of T. 

This discrepancy is, however, not to be attributed to the invalidity of Nernst s heat theorem, but to the obvious inaccuracy of the assumptions regarding the specific heats. Two solids and only one gas take part in the reaction, and the error made in the calculation of C is apparently not nearly so completely eliminated as in pure gas reactions. 

We have thus shown that Nernst s heat theorem can be deduced from two hypotheses of a very general and plausible character, namely, first that all physical properties which remain finite as the temperature is lowered converge gradually to their limiting value, and, second, that the entropy does not become infinite at the absolute zero, but converges likewise to a finite limiting value. 

It is very important to realise that the affinity of the condensed substances is measured by an expression which involves the concentrations of the saturated vapours together with the equilibrium constant characteristic of the same reaction in the gaseous state This conclusion depends upon the assumption that the pressure or concentration of a saturated vapour is a true measure of the reactivity of the condensed substance The above expression is of great impoitance foi it allows us to calculate the affinity of condensed reactions from measurements made upon the substances m the gaseous state This point will be referred to again in discussing the application of Nernst s Heat Theorem to gaseous reactions... 

Systems not m equilibrium (continued)—Relation between the affinity and the heat of a reaction—Nernst s heat theorem and some of its applications... 

The third law of thermodynamics was first stated by Nernst For certain isothermal chemical reactions between solids, the entropy changes approach zero as the thermodynamic temperature approaches zero. Nernst based this statement on his analysis of experimental data obtained by T. W. Richards, who studied the entropy changes of chemical reactions between solids as the temperature was made lower and lower. The statement of Nernst was sometimes called Nernst s heat theorem, although it is a statement of experimental fact and not a mathematical theorem. 

The mathematical structure of thermodynamics is based on two laws. The third law, also called Nernst s heat theorem, properly belongs to statistical theory. Its main use in thermodynamics is in establishing an entropy scale. In 1931 Fowler raised the postulate regarding the existence of thermal equilibrium to the status of the zeroth law of thermodynamics. We need discuss neither the zeroth or the third laws here. 

When we come to deal with gases or vapours, we pass at once out of the region of direct applicability of Nernst s theorem. If we assume, approximately, that the specific heat of the gas is constant over a small range near the absolute zero, we have ( 79) ... 

The calculation of AG from the caloric data is straightforward, independent of the path, that is, whether the reaction takes place in a single step or through a series of steps by using Hess s law and Nernst heat theorem [15-17]. Furthermore, we can calculate AG for the reaction of interest from the combination of other reactions involved for which the thermodynamic data are known. However, both the initial and final states in many cases are hypothetical. Even in the case of measurements executed very carefully and accurately, there might be problems in defining the states of the compounds, or even metals ( ) that take part in the reaction. 

The first theoretician of the vitrification process was Simon (1930), who pointed out that it can be interpreted as a "freezing-in" process. Simon measured specific heats and entropies of glycerol in the liquid, crystalline and glassy state below Tg the entropy of the supercooled liquid could, as a matter of fact, only be estimated. Linear extrapolation would lead to a negative entropy at zero temperature (paradox of Kauzmann, 1948) which would be in contradiction with Nernst s theorem. So one has to assume a sharp change in the slope of the entropy, which suggested a second order transition as defined by Ehrenfest. 

The supposed differences between chemical constants determined from vapour pressures (7/) and from chemical equilibrium data from the Nernst Heat Theorem Jn found by Eucken, Karwat, and Fried, are illusory, and due to inaccuracies in the experiments made in Eucken s laboratory, The supposed constant differences between the calorimetric and optical values (from moments of inertia found from band spectra. or otherwise) listed, with a theoretical explanation, by Eucken, Karwat, and Fried, are also illusory for the same reason. The relation between the spectroscopic and calorimetric entropies, as found from experimental results meriting confidence, is fully discussed in 70.11 and 26-29.IV. Apart from the incorrect data used by Eucken, Karwat, and Fried, 11 their method of extrapolation of specific heats of gases to 0°K. is much cruder than they suspected. 12... 

The substances which are formed with evolution of heat (e.g. the solid phase) mrlst therefore always have the smaller specific heal. This regularity was first discovered by van t HofF, who confirmed it by the following table. His proof, however, was based on an erroneous assumption as to the course of the A and Q curves, and must be replaced by Nernst s theorem. 

Even in its original form the theorem deals with chemical reactions and changes of state, that is to say, with the most important natural phenomena accompanied by evolution or absorption of heat. It is therefore natural to suspect that the heat theorem, like the two law s of thermodynamics, has its origin in the nature of heat itself. The laws of thermodynamics, as was shown in Chapters III. and V., could be traced back to the results of our everyday experience (impossibility of perpetual motion of the first and second kinds). This simple method of derivation fails in the case of the new theorem because temperatures in the neighbourhood of the absolute zero can never be the immediate objects of experience. They can only be reached by refined experimental methods. For this reason Nernst s theorem can never be susceptible of direct experimental proof, and can only be tested by its consequences. We can deduce the theorem, however, from a more general principle regarding the nature of heat and the properties of the thermodynamic functions. 

In other words, every chemical reaction takes place without change in entropy at the absolute zero. From this it follows that the entropy of a compound is equal to the sum of the atomic entropies. The assumption made by Planck in addition to Nernst s theorem, viz. that the entropy of all substances vanish like the specific heats at the absolute zero, is sufficient but not necessary for the derivation of the heat theorem. 

Now if the condensed phase is a perfect crystal at absolute zero we may equate s 0, 0) to zero (Nernst s theorem, chap. IX). Furthermore as T -> 0 the specific heat tends to zero sufficiently rapidly for the integral CT c... 

Equation (14.12) enables us to make a direct test of the Nernst heat theorem, for if this is valid then s 0, 0) is zero and... 

Since the last two terms in Eq. (11.74) can be calculated from heat capacities and heats of reaction, the only unknown quantity is ASS, the change in entropy of the reaction at 0 K. In 1906, Nernst suggested that for all chemical reactions involving pure crystalline solids, ASS is zero at the absolute zero the Nernst heat theorem. In 1913, Planck suggested that the reason that ASS is zero is that the entropy of each individual substance taking part in such a reaction is zero. It is clear that Planck s statement includes the Nernst theorem. 

Historically, the third law of thermodynamics emerges from the heat theorem, by Nernst which states A chemical reaction between pure crystalline phases that occurs at absolute zero produces no entropy change. This means that adiabatic and isothermal processes approach each other at very low temperatures. The importance of Nernst s theorem is that it gives a solid base for the calculation of thermodynamic equilibria. 

After much discussion of its range of validity, the Nernst heat theorem has been stated in the form that the entropy of all factors within a system which are in internal thermodynamic equilibrium disappears at absolute zero temperature. Nernst s other work is considered later (see p. 633). 

Here we should note that from (3.57) we cannot determine the absolute value of s, which is specified by virtue of the Nernst Heat Theorem (i.e., the Third Law of... 

This expression correlates the mechanical property B to the specific heat Cy or the internal vibration energy, H T/0o)- Equation (27.3) satisfies the Nernst s theorem, where the temperature derivatives of the elastic constants must vanish at 0 K. The B correlates to Y by Y/B = 3 x (1 — 2v), where v denoting the Poisson ratio is negligibly small, and therefore Y 3B. This relation could reproduce the measured Y(7) reasonably well at low temperatures by taking the ySROo/iBoVo) as an adjusting parameter. 

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