A summary of the "Euler reciprocity relation", a synonym for what Rudolf Clausius calls the "condition for an exact differential", which would seem to indicate that this logic was originated by Swiss mathematician Leonhard Euler. [5] |

du = P dx + Q dy

the expression on the right side of the equation

meaning that the partial derivative of the function

Examples

The expression

Proof

To prove or derive this condition, we first start with the function of two variables:

du = P dx + Q dy

and note that the definition of a total differential

hence, by comparison of these two expressions, we find that:

and

Differentiating the first of these with respect to

and

Whence, by the calculus principle that the order of differentiation is immaterial, we have:

whereby it is proved that:

and that this is called the "condition for an exact differential".

Maxwell's equations

The application of the condition for an exact differential for functions of two variables that are already known to exist or be actual functions, such as internal energySee main: Maxwell's relations

Potential functionRelationdU = T dS – P dV

hencedH = T dS + V dP

hencedA = –P dV – S dT

hencedG = dH – S dT

hence

These relations are sometimes also called Maxwell relations. [3] It is not necessary, according to American physicist Mark Zemansky, to memorize these relations since they are so easily derived. The Maxwell equations do not refer to a process but merely express relations that hold at any equilibrium state of a chemical system. [2] The great interest of Maxwell's equations according French thermodynamicst Pierre Perrot, is that they lead to the partial derivatives of entropy as a function of physical quantities directly available by experiment. [4]

References

1. Osborne, George A. (1891).

2. Zemansky, Mark W. (1943).

3. Maxwell relations - Wikipedia.

4. Perrot, Pierre. (1998).

5. The Terrible Beauty of Thermodynamics (2010) – ScribD.com.