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THERMODYNAMIC CIRCLE

The thermodynamic circle is a mnemonic model developed for easily deducing or recalling Maxwell relations associated with the various thermodynamic potentials such as internal energy (U), enthalpy (H), Helmholtz free energy (A) and Gibbs free energy (G) and other thermodynamic relations of closed and open thermodynamic systems.

The ‘Thermodynamic circle’ concept was first developed by Prof. K. Natarajan of Vel Tech (Deemed to be University), Chennai, India. The first literature publication of the thermodynamic circle method appeared in the 2014 article in the ‘International Journal of Mechanical Engineering Education’ (IJMEE)[1]. wherein he describes two mnemonic models named as ‘Thermodynamic circle-1’ (for the closed system) and ‘Thermodynamic circle-2’ (for the open system) for easily recalling Maxwell relations of simple compressible closed system and the Maxwell relations of a single component, simple compressible, open system with independently varying mole number of species. He followed this with the second 2016 article in IJMEE [2] wherein he describes the use of these two thermodynamic circles for easily recalling other thermodynamic relations and the Maxwell relations of a single component, simple, compressible, open system with independently varying chemical potential of the species.

The systematic ways of obtaining the various thermodynamics relations, from these models, have been described in detail. After acquainting with these systematic procedures, the students can easily organize the network of these relations in their minds without any ambiguity. In addition, using these models, the students can easily realize how the various thermodynamic properties and the potentials are closely related geometrically The ‘Thermodynamic circle-1’ devised for the closed system is shown as Figure 1 in Reference [1]. It has the properties P,V and T,S placed at the ends of its horizontal and vertical diameters and the thermodynamic potentials A, G, H and U placed in its four quadrants in alphabetical order in the anticlockwise direction starting from the top right quadrant.

The Maxwell relations associated with the four thermodynamic potentials: Enthalpy (H), internal energy (U), Gibbs free energy (G) and Helmholtz free energy (A) of a simple compressible closed system can be deduced from the ‘Thermodynamic circle-1’ as given below: Any three properties taken successively in clockwise or anticlockwise direction along the circumference of the ‘Thermodynamic circle-1’ will respectively form the numerator, the denominator and the constant in one of the partial derivative of the Maxwell relation of the thermodynamic potential whose natural variables are the last two properties chosen along the circumference of the circle. The fourth property on the circumference of the circle followed by two more properties taken successively in the opposite direction along the circumference circle will similarly form the numerator, the denominator and the constant in the second partial derivative of the same Maxwell relation. A negative sign will precede a partial derivative if any one of the properties P or S appears in the numerator of that partial derivative.

This procedure is graphically illustrated in Reference [1] for the associated Maxwell relations of the four Thermodynamic potentials A,G,H and U. The beginning and the end of the path that passes along the properties which form one of the partial derivatives of the Maxwell relation are marked with number 1. Similarly the beginning and the end of the path that passes along the properties which form the second partial derivative of the same Maxwell relation are marked with number 2. The path directions are clearly indicated by the arrows.

If the Maxwell relation of a particular thermodynamic potential is desired, one should begin with the property other than the natural variable of that potential, while choosing the three successive properties along the circumference of the circle. For example, if the Maxwell relation for the potential U is desired, beginning should be made with the property P or the property T on the circumference of the circle, which are not the natural variables of the potential U. Similarly, if the Maxwell relation of the potential H is desired, the path should begin with the property V or the property T. If the Maxwell relation of the potential G is desired, the path should begin with the property V or the property S. If the Maxwell relation potential A is desired, the path should begin with the property P or the property S.

The details of the ‘Thermodynamics circle-2’ and its use in deducing the Maxwell relations associated with the open thermodynamic systems are given in References 1 and 2. REFERENCES

1.	K. Natarajan, ‘Graphical summaries for Maxwell relations of closed and open thermodynamic systems’, Int. J. Mech. Engg. Edu. 2014, 42, 1-17

2.	K. Natarajan, ‘Graphical summaries for Gibbs and other thermodynamic relations of closed and open thermodynamic systems’, Int. J. Mech. Engg. Edu. 2016, 44(3):198-219

3.	H. B. Callen, Thermodynamics and An Introduction to Thermo Statistics,, second edition (John Wiley, New York), 1985, 181-185.

4.	J. Rodriguez and AJ. Bralnard. ‘An improved mnemonic diagram for thermodynamic relationships’, J. Chem Educ 1989; 66: 495-496.

5.	L. Pogliani and C. La Mesa. ‘The mnemonic diagram for thermodynamic relationships’. J Chem Educ 1992; 69: 808-809.

6.	JE. Fieberg and CA. Girard. ‘Mnemonic device for relating the eight thermodynamic state variables: the energy pie’. J. Chem Educ 2011; 88: 1544-1546.