In thermodynamics, the ten fundamental equations of thermodynamics are the ten core equations of the mechanical theory of heat, signified by roman numerals (I-X), as found in German physicist Rudolf Clausius’ 1875 textbook The Mechanical Theory of Heat, distinguished from the other 448 equations, in the same book, signified numerically, per chapter. [1] In founding chemical thermodynamics, in 1876, American engineer Willard Gibbs formulated 700 equations, based on Clausius' original 458 equations. In total, this amounts to 1,158 equations that one must master in order to become fluent in extended applications, such as in human chemical thermodynamics or in study of other types of earth-bound natural systems.

Fundamentals
These ten fundamental equations, which represent the foundation of thermodynamics, are listed below.

Equation
Description#
Pg.

dQ = dH + dL \, Development of the first main principle
(Heat dQ imparted to any body whatsoever acts to increase
the quantity of heat dH and the quantity of work dL done) 		I(27)
dQ = dH + dJ + dW \, Forces against which work is done
(the quantity of work dL may be divided into two classes: (a) internal work dJ, those which the molecules of the body exert among themselves, and which depend on the nature of the body itself; (b) external work dW, those which which arise from external influences, to which the body is subjected)		II(27)
dQ = dU + dW \, First main principle
(first law of thermodynamics)		III(31)
dQ = dU + pdv \, Pressure-volume work
(Case in which the only external force is a uniform pressure normal to the surface)		IV(38)
\int \frac{dQ}{T} = 0 \, Convenient expression for the second main principle
(if in a reversible cyclical process every element of heat taken in (positive or negative) be divided by the absolute temperature at which it is taken in, and the difference so formed be integrated for the whole course of the process, the integral so obtained is equal to zero) 		V(89)
dQ = TdS \, Second main principle
(second law of thermodynamics)		VI(90)
\int \frac{dQ}{\tau} \,
Discussion on the function τ as being the absolute temperature T
VII(105)
dQ = \tau dS \, VIII(107)
\int \frac{dQ}{T} \le 0 \,
Non-reversible processes
(uncompensated transformations must always be positive)		IX(213)
dQ \le TdS \, X(214)

Of these, equations III and VI represent the “two main principles” (first main principle and second main principle) of the mechanical theory of heat, as discussed in Clausius' chapter five “Formation of the Two fundamental Equations.”

Gibbs fundamental equation
See main: Gibbs fundamental equation
In 1876, American chemical engineer Willard Gibbs expanded on the two main principles, by adding together equations III, IV, and VI, to form what is known as the combined law of thermodynamics:

dU = TdS - PdV \,

but then went a step further by assuming that other factors, or rather 'forces', could act on the body, thus having an effect on the change of the bodies energy dU during any type of process. In short, according to Gibbs, one fully quantify the internal energy U of a system as the sum of all of its influencing conjugate variables pairs, which can be formed mathematically whereby with any extensity xi (extensive variable, e.g. volume v) it is always possible to associate a tension variable Xi (intensive variable, e.g. pressure P), as a partial derivative of the internal energy of the system with respect to the partial of the extensive variable:

X_i = \frac{\partial U}{\partial x_i}

which is called the "conjugate", with the product of the two intensive extensive variables Xdx being called the "conjugate variable pair". Thus, according to the first law, as explained by Gibbs, the change in internal energy dU of a system is given can be defined rigorously as the summation of the product of all conjugate pairs acting on the system:

dU = \sum_{i=1}^k X_i dx_i

whereby the first main principle can be re-written as:

dU = TdS - PdV + \sum_{i=1}^k X_i x_i \,

whereby the condition for a process progressing irreversibly is that the "uncompensated transformations will be positive, expressed by saying that the variation of the entropy of the body will increase in each cycle, dS > 0, as defined by equations nine and ten, whereas the 'condition for equilibrium' is that the variation of the entropy dS of the body will be zero at the equilibrium state. The main nine types of conjugate pairs are listed below:

Intensive Variable
Extensive Variable
Energy
 Function ProductPerson
Pressure P Volume dV pressure-volume work δW pdVClapeyron (1834)
Temperature T Entropy dS internal work δQ TdSClausius (1865)
Chemical potential μParticle number dnspecies transfer work
μdnGibbs (1876)
Force F Length dl elongation/contraction work
Fdl

Electromotive force ε Charge de electrical work
εde

Surface tension γ Surface area dA surface work
γdA

Gravitational potential ψ Mass dm gravitation work
ψdm

Electric field E Electric dipole moment dp electric polarization
Edp

Magnetic field B Magnetic moment dm magnetic polarization
Bdm


What this table says is that when a system does work or has work done on it the system internal energy is effected, and this effect can be quantified by conjugate variable pairs. To go through one example, when the system has a volume dV change due the action of a pressure P, i.e. pressure-volume work (dW = pdV), an expansion of a material section or elongation dl in response to the application of a force F, i.e. elongation work (dW = Fdl), does work (μdn) by transporting a certain number of atoms or molecules dn against a concentration gradient, where μ is the chemical potential, i.e. transport work (dW = μdn), or, among many other possible examples, does work in the action of charge transport in which an amount of charge dq is transported against an electric potential ψ, i.e. electrical work (dW = ψdq), then the Gibbs fundamental equation becomes: [3]

Gibbs fundamental equation (longer version)

In short, what Gibbs did was to add on the 'other conditions' to Clausius' first main principle (III) by which the equilibrium of the system could be effected, as described in his now-famous treatise On the Equilibrium of Heterogeneous Substances, containing exactly 700 numbered equations, which has called the principia of thermodynamics. [2] In modern shorthand, the cliff-note version for primary results of Gibbs work, as applied in chemistry, for simple isothermal-isobaric processes, wherein reactants are put in a reaction vessel, is the following spontaneity criterion:

Condition
Description

dG lt 0Reaction or process is spontaneous in the forward direction.
dG > 0 \, Reaction or process is nonspontaneous (reaction is favored in the opposite direction).
dG = 0 \, System is at equilibrium (there is no net change).



which are simplified rules which quantify and predict the "spontaneity" of the said reaction or process under study.

See also
● Quantity
● Characteristic function
● Characteristic function notation table

References
1. Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (2nd ed). London: Macmillan & Co.
2. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy, III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.

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