In thermodynamics, the combined law of thermodynamics, also called the Gibbs fundamental equation, is a mathematical summation of the first law of thermodynamics and the second law of thermodynamics subsumed into a single concise mathematical statement as shown below: [1]

dU – TdS + PdV ≤ 0

where dU is a variation in internal energy, T is temperature, dS is variation entropy, P is pressure, and dV is variation in volume of a simple working body in which there are no flows of particles or out of the body nor external forces, other than gravity, acting on the body. In theoretical structure, in addition to the obvious inclusion of the first two laws, the combined law incorporates the implications of the zeroth law, via temperature T, and the third law, through its use of free energy as related to the calculation of chemical affinities near absolute zero. [2]

Shortened form
In concise form, knowing that the expression to the left of the inequality sign is the Gibbs free energy dG, the combined law
for cyclical heat driven processes, in closed reactive systems, at constant pressure and constant temperature:

dG ≤ 0

states that a "spontaneous process" will ensue when their is a decrease in the free energy of the system. This can be restated in terms of the difference between the measure of Gibbs free energy between two states of a system

Gfinal – Ginitial ≤ 0
∆G ≤ 0

In simple terms, according to geo-thermodynamicists Raymond Kern and Alain Weisbrod, for earth systems, the combined law of thermodynamics tells us that: “when a system evolves naturally, i.e. undergoes a natural process, in an isothermal manner, at constant volume or pressure, its Gibbs free energy G (or Helmholtz free energy F) always decreases.” They define this as the conditional statement for all types of evolution in isothermal earth-bound systems in which thermo-mechanical and thermo-chemical changes take place at constant volume or constant pressure. For more complex systems, in which generalized forces act or in which species migrate across the system boundary, then the generalized combined law of thermodynamics takes the form: [3]

Combined law of thermodynamics (complex systems)
Here, T denotes temperature, S the entropy, U the internal energy, p pressure, V volume, Xk any generalized force except pressure, xk any generalized coordinate except volume, μk chemical potential, mk the mass of the k-th substance, which can be replaced by the number of moles. [4]

1. (a) Thims, Libb. (2007). Human Chemistry (Volume One), (preview), (ch. 4: "Combined Law of Thermodynamics", pgs. 73-119). Morrisville, NC: LuLu.
(b) Alberty, S. (2001). Physical Chemistry, 3rd Ed. (textbook). New York: John Wiley & Sons, Inc.; combined law of thermodynamics (definition): dU = TdS - PdV
(c) Combined Law of Thermodynamics - Wolfram's World of Science.
2. (a) Quote: “the following important relation, which is the central equation of thermodynamics, relates the zeroth law through the concept of temperature, the first law through the internal energy function, and the second law through entropy and the absolute temperature: dU = T∆S – pdV.
(b) Dugdale, J.S. (1998). Entropy and its Physical Meaning, (pg. 49). Philadelphia: Taylor & Francis.
3. Gladyshev, G.P. (1997). Thermodynamic Theory of the Evolution of Living Beings. (pg. 117). New York, Commack: Nova Science Publishers, Inc.
4. Gladyshev, G. P. (2000). Differential equations of macrothermodynamics. The systems and the processes .
International Academy of Creative Endeavors [Moscow, Russia].
5. Gladyshev, Georgi. (2010). “On the Thermodynamics of the Evolution and Aging of Biological Matter.” Journal of Human Thermodynamics, 6: 26-38.
6. Kern, R. & Weisbrod, A. (1967). Thermodynamics for Geologists, (pgs. 64-65). San Francisco: Freeman, Cooper and Company.

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