Gibbs' fundamental relations are important to thermodynamics. Their focus isn't on heat transfer processes, instead they look at the states of matter in a system. In brief Gibbs' fundamental relations describe the entropy of a system as a function of it's extensive variables.

The first law of thermodynamics can be expressed mathematically.

dU = dQ + dW
Where U represents energy, Q represents heat, and W represents work. Gibbs had also expressed the law for fluids in terms of temperature (T), pressure (P), volume (V), and entropy (S).
dU = T•dS - P•dV
The solution to the above equation is U = U(S,V). The limitation is that the amount of matter interint a system must be fixed for this relation. It can be avoided if the matter entering the system is treated as a thermodynamic variable as well. Gibbs' fundamental relations for simple systems which can be derived from the above are as follows:

Gibbs' fundamental relation (entropy representation)

S = S(U, V, N1, N2, ... Nr)
Gibbs' fundamental relation (energy representation)
U = U(S, V, N1, N2, ... Nr)

Those two equations are considered the fundamental relations of thermodynamics. Where Ni, from i = 1 to r, are the chemical components of the system in moles.

The first law can be rewritten to be:

dU = (∂U/∂S)|V,N1... dS + (∂U/∂V)|S,N1... dV
The second law, and zeroth law can be described by the principles of least energy, and maximum entropy. The third law states that as temperature approaches zero, the entropy of the system approaches zero. This puts a boundary condition on any function of state regarding the function for entropy.

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