Relation of Gibbs-Helmholtz

The relation of Gibbs-Helmholtz (name given in the honor of the physicists Gibbs and Helmholtz) is given by:

$\backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; left\; (\backslash \; frac\; \{G\}\; \{T\}\; \backslash \; right)\}\{\backslash \; partial\; T\}\; \backslash \; right)\; \_p\; =\; -\; \backslash \; frac\; \{H\}\; \{T^2\}$

With:

• H the Enthalpy
• G the free Enthalpy
• T the Temperature (absolute)

One finds also other formulations equivalent of this relation:

$\backslash \; qquad\; H\; =\; G\; -\; T\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$

$\backslash \; qquad\; H\; =\; G\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; ln\; T\}\; \backslash \; right)\; \_p$

$H=\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; left\; (\backslash \; frac\; \{G\}\; \{T\}\; \backslash \; right)\}\{\backslash \; partial\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{T\}\; \backslash \; right)\}\backslash \; right)\; \_p$

$-H=\; T^2\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; left\; (\backslash \; frac\; \{G\}\; \{T\}\; \backslash \; right)\}\{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$

Demonstration

This relation is shown simply on the basis of the Relations of Maxwell, and in particular that relating to S:

$S=\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$

• By replacing in the expression of definition of the free Enthalpy: br/>

$G=H-TS~$

• with what precedes:
  

$G=H+T\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$

• from where: $H=G-T\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$

• while multiplying by $-\; \backslash \; frac\; \{1\}\; \{T^2\}$ the preceding relation:
  
  

$-\; \backslash \; frac\; \{H\}\; \{T^2\}\; =\; \backslash \; frac\; \{G\}\; \{T^2\}\; +\; \backslash \; frac\; \{1\}\; \{T\}.\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$

• one recognizes with the 2nd term the derivative partial of $\backslash \; frac\; \{G\}\; \{T\}$ compared to T, with p constant:

$\backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; left\; (\backslash \; frac\; \{G\}\; \{T\}\; \backslash \; right)\}\{\backslash \; partial\; T\}\; \backslash \; right)\; \_p\; =\; G.\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{T\}\; \backslash \; right)\}\{\backslash \; partial\; T\}\; \backslash \; right)\; \_p\; +\; \backslash \; frac\; \{1\}\; \{T\}.\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p\; =\; -\; \backslash \; frac\; \{G\}\; \{T^2\}\; +\; \backslash \; frac\; \{1\}\; \{T\}.\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$

• one from of deduced the relation from relation of Gibbs-Helmholtz :
  
  

Other formulations

One meets also the relations of Gibbs-Helmholtz in the following equivalent forms:

$\backslash \; qquad\; \backslash \; triangle\; U\; =\; \backslash \; triangle\; F\; -\; T\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; triangle\; F\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_\; \{V,\; N\}$

$\backslash \; qquad\; \backslash \; triangle\; H\; =\; \backslash \; triangle\; G\; -\; T\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; triangle\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_\; \{V,\; N\}$

Notations used in this article

• V Volume.

• p Pressure.
• T Temperature.
• U Énergie interns.
• H Enthalpy
• F Free energy.
• G Enthalpy free.
• S Entropy
• N number of mole S

Interest

This relation gives access easily the free enthalpy when one knows the variations of the enthalpy compared to the temperature with constant pressure, and vice versa. It belongs to the relations extrèmement useful in Thermodynamique to pass from a function of state to another.

Notice

A similar relation exists between F (the free energy), U (the energy interns), and T , even if this one is much used than the preceding one:

$\backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \backslash \; left\; (\backslash \; frac\; \{F\}\; \{T\}\; \backslash \; right)\}\{\backslash \; partial\; T\}\; \backslash \; right)\; \_V\; =\; -\; \backslash \; frac\; \{U\}\; \{T^2\}$

This one shows same manner, on the basis of the definition of F (F=U-TS) and of the relation of Maxwell:

$S=\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_V$

These formulas make apparaitrent the difference between the variations of Enthalpie and free Enthalpie for the first formula and the difference between the variations of internal energy and free energy for the second formula

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