Maxwell Relations for Substances with Negative Thermal Expansion and Negative Compressibility

It is shown that taking into account the negative compressibility of substances changes Maxwell relations. The earlier results of the author indicating that these relations differ for substances with negative thermal expansion have received additional confirmation. Universal Maxwell relations have been derived. The results obtained have been confirmed experimentally by a number of authors.


THEORY
The first Maxwell relation is: where S is entropy. This relation stems from the first law of thermodynamics. This law for heat exchange can be written as: where δQ is the heat introduced into the system and U is internal energy. One introduces a quantity of heat into the system and it turns into the change in internal energy and work produced by the system. On the left-hand side, the motive force of the process is written, and its effect is written on the right-hand side. In [10−16] the general form of the first law of thermodynamics for the heat exchange was obtained: However, for the heating of substance by compression, the first law of thermodynamics cannot be derived from Eq. (2) and must be derived independently [10,11,13,17]: (without the heat losses). Again, the motive force of the process is written on the left-hand side, and its effect is written on the right-hand side. One can prove this result very easily. Equation (2) cannot describe the compression of the substances with negative compressibility. The equation for that must be the following one: One can adduce another good argument. According to tables of thermodynamic derivatives [18], This is the derivative for the heat exchange process. However, for mechanical compression this derivative must be obtained from Eq. (4), and it is: Therefore, the thermodynamics of compression differs from the thermodynamics of the heat exchange.
From Eq. (3) it follows that the first general Maxwell relation will be as follows: In the Appendix it is shown that this has been confirmed by many experiments and that Eq. (1) contradicts them.
The second Maxwell relation is: Its traditional derivation is the following [19]. One introduces dU from Eq. (2) into the differential of enthalpy: and obtains: From this equation, Eq. (9) results.
One can notice a mistake in this derivation. Let us prove that Eq. (2) is valid only for a constant pressure. Let us assume that the pressure is not constant in it. One can notice that δQ = TdS = dH P in Eq. (2) is a full differential, where dH P is the enthalpy change at a constant pressure.
Therefore, the derivatives ( ) One can see that: is the heat exchange at a constant pressure, and is the heat exchange at a constant volume [17,20,21]. From Eq. (15), omitting the subscripts, the second Maxwell relation can be derived: This equation has a different sign compared with Eq. (9).
From the well-known thermodynamic identity [22], it follows that: This means that Eq. (15) will look like: From this equation the second general Maxwell relation follows: The third Maxwell relation is: Consider its traditional derivation [19]. One introduces dU from Eq. (2) into the differential of Helmholtz energy: which results in: From this, Eq. (20) is obtained. However, this derivation is non-strict: an equation which describes the heat exchange at a constant pressure with varying volume is introduced into the equation which describes a process with a constant volume and varying pressure. A more strict derivation should be one such as: corresponds to the quantity of heat introduced into the system at a constant volume (we introduce into Eq. (24) dU from Eq. (2) with dV = 0), and and the third general Maxwell relation becomes: In the Appendix it is shown that this equation has been confirmed by many experiments and that Eq. (20) contradicts them.
The fourth Maxwell relation is: Consider its traditional derivation [19]. One introduces dU from Eq. (2) into the differential of Gibbs energy: and obtains: From this, Eq. (28) results.
One can see that this derivation is non-strict: dU from Eq. (2) is at a constant pressure and does not equal dU from Eq. (29), which is at a varying pressure. Let us try to derive Eq. (28) more strictly. One introduces a quantity of heat (TdS) into the system at varying volumes and pressures and the Gibbs energy of the system changes: where dG 1 is the change in the Gibbs energy at a constant pressure: and dG 2 is the change in it at a constant volume:

CONCLUSION
It has been shown that the negative compressibility of substances effects the Maxwell relations. The earlier results of the author indicating that negative thermal expansion also effects these relations have been strongly confirmed. General Maxwell relations have been obtained which take into account the sign of compressibility and thermal expansion: Eqs. (8) The sign of the left part must be negative because when one increases the temperature of a system, its entropy increases and for the entropy to remain constant the volume must decrease.
The left part of it is positive. Its right part describes the following process: one introduces a quantity of heat into the system (dS > 0) and its volume decreases. If one wants to keep the temperature constant, one has to increase the volume, and hence this derivative is greater than zero. Again, the traditional Maxwell relation, Eq. (20), contradicts the experiment.
Let us introduce a quantity of heat into a substance (Eq. (2)) and let us suppose that it expands. According to the definition of work in thermodynamics, P in Eq. (2) is the internal pressure (produced by the substance) and is positive in our case because it expands the substance [25]. Its absolute value equals the sum of the pressure caused by surface tension and atmospheric pressure, with the latter being negligibly small compared to the former. If the substance possesses negative thermal expansion, then the pressure produced by the substance is negative and Eq. (2) can be rewritten as: which coincides with Eq. (3). _________________________________________________________________________________