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Research Papers

# What is the Third Law?*

[+] Author and Article Information
Gian Paolo Beretta

Universitá di Brescia,
Brescia 15 - 25121, Italy
Massachusetts Institute of Technology,
Cambridge, MA 02139

Elias P. Gyftopoulos

Massachusetts Institute of Technology,
Cambridge, MA 02139

The concept of compatibility of a state with a given set of values of amounts of constituents and parameters plays a special role in the statement of the second law. It is defined as follows. A state A1 with values (n)1, (β)1, (P)1, where (P)1 denotes the set of values of all the properties of the system, is compatible with a given set of values n of the amounts of constituents and β of the parameters if the two sets (n)1, (β)1 and n, β are compatible. Two sets of values of amounts of constituents and parameters (n)1, (β)1 and (n)2, (β)2 are compatible if the change from one set to the other can occur as a result of the allowed internal mechanisms of the system, such as chemical reactions, interconnections, and internal forces. For example, if a system has two compartments of volumes V′ and V″, respectively, interconnected so as to satisfy the constraint V′ = V″= constant, then the two sets of values $V1'$,$V1"$ and $V2',V2"$ are compatible if $V1'+V1"+V2'+V2"$ because then the internal interconnection between the two volumes allows the change from one set of values, say, $V1'$ = 3 m3 and $V1"$ = 5 m3, to the other set of values, say, $V2'=2 m3$ and $V2"=6 m3$. The same two sets of values would not be compatible if the two compartments were not interconnected.

In some expositions, the third law is referred to as Nernst's principle.

Indeed,

$CV=(∂E∂T)n,β=1kT2[∑j=0∞εj2Djexp[-εj/kT]∑j=0∞Djexp[-εj/kT]-(∑j=0∞εjDjexp[-εj/kT∑j=0∞Djexp[-εj/kT)2]=D0∑j=1∞(εj-ε0)2Djexp[-(εj-ε0/kT]/kT2(D0+∑j=1∞Djexp[-(εj-ε0)/kT])2 +∑i=1∞∑j=1∞(εi-εj)2Diexp[-(εi-ε0)/kT]Djexp[-(εj-ε0)/kT]/kT22(D0+∑j=1∞Djexp[(εj-ε0)/kT)2$

Because $εj-ε0$> 0 for every j$≥$1, and exp(−a/T)/T2 → 0 for a > 0 and $T→0$, we readily verify Relation 7.

4As we discuss in Section 7, we are especially concerned with systems with few particules. Accordingly, the discussions in Refs. [8] to [10] about taking the limit as the number of particules goes to infinity are not germaine to our purposes.

*Proceedings of the Florence World Energy Research Symposium FLOWERS'90, Firenze, Italy, 28 May–1 June 1990 in A Future for Energy, edited by S. Stecco and M. J. Moran, Pergamon Press, pp. 434–444 (1990). Reprinted with permission.

J. Energy Resour. Technol 137(2), 021004 (Mar 01, 2015) (5 pages) Paper No: 04-BerettaGyftopoulos-FL; doi: 10.1115/1.4026380 History: Online November 18, 2014

## Abstract

We discuss entropies of systems at very low temperatures or, equivalently, the third law of thermodynamics. We conclude that definitive values of such entropies can be established only by experiments on systems with very few degrees of freedom, such as one-particle systems.

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## Figures

Fig. 1

Schematic representation of the projection of the states of a system with given values of the amounts of constituents and the parameters on the energy versus entropy plane

Fig. 2

Projections of the states of a system with given values of amounts of constituents, and two different values of the volume. V1 and V2 (a) Projection on the same E versus S plane; and (b) projections on two different planes in E-S-V space.

Fig. 3

Schematic representation of the energy versus entropy relation for stable equilibrium states with given values of n and β at low temperatures

Fig. 4

Shape of the energy versus entropy diagram consistent with the statement of the second law modified to account for the possibility that for a given set of values of n and β a system admits more that a single ground-energy state.

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