In thermodynamics, the Gibbs-Clausius equations are the 1,158 equations that constitute the combined works of German physicist Rudolf Clausius (448 numbered equations and 10 fundamental equations) and American engineer Willard Gibbs (700 numbered equations), as found in the The Mechanical Theory of Heat (1875) and On the Equilibrium of Heterogeneous Substances (1876), which together define modern thermodynamics, and in particular modern chemical thermodynamics.

Clausius fundamental equations
The following are the ten fundamental equations of thermodynamics, according to German physicist Rudolf Clausius, who assigned these with Roman numerals, as contrasted with the other 448-numbered equations found in his 1875 second edition:

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

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

Gibbs fundamental equation
These were advanced further in 1876 by American engineer Willard Gibbs, in his On the Equilibrium of Heterogeneous Substances, in the form of the combined law of thermodynamics and the Gibbs fundamental equation, a treatise containing exactly 700-numbered equations. Others to build on the ten equations of Clausius include: Boltzmann entropy, Prigogine entropy, Gibbs entropy, among others.