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Concept
Fundamental thermodynamic relation
Summary
In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like G (Gibbs free energy) or H (enthalpy). The relation is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way.
Here, U is internal energy, T is absolute temperature, S is entropy, P is pressure, and V is volume.
This is only one expression of the fundamental thermodynamic relation. It may be expressed in other ways, using different variables (e.g. using thermodynamic potentials). For example, the fundamental relation may be expressed in terms of the enthalpy H as
in terms of the Helmholtz free energy F as
and in terms of the Gibbs free energy G as
The first law of thermodynamics states that:
where and are infinitesimal amounts of heat supplied to the system by its surroundings and work done by the system on its surroundings, respectively.
According to the second law of thermodynamics we have for a reversible process:
Hence:
By substituting this into the first law, we have:
Letting be reversible pressure-volume work done by the system on its surroundings,
we have:
This equation has been derived in the case of reversible changes. However, since U, S, and V are thermodynamic state functions that depends on only the initial and final states of a thermodynamic process, the above relation holds also for non-reversible changes. If the composition, i.e. the amounts of the chemical components, in a system of uniform temperature and pressure can also change, e.g. due to a chemical reaction, the fundamental thermodynamic relation generalizes to:
The are the chemical potentials corresponding to particles of type .
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