Chemical affinity

Definition

Affinity is defined by the formula:

$A\; =\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; \backslash \; xi\}\; \backslash \; right)\; \_\; \{p,\; T\}\; ~$

Differential of the free Enthalpy of a chemical system

Chemical affinity comes from the definition of the Différentielle of the free enthlapie, G.

The free function enthalpy being a Fonction of state, its differential is total exact, i.e. that it is equal to the sum of the partial differentials compared to each independent variable:

$G\; =\; F\; (T,\; p,\; n\_i)\; ~$

$dG\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_\; \{p,\; n\_i\}\; dT\; +\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; p\}\; \backslash \; right)\; \_\; \{T,\; n\_i\}\; dp\; +\; \backslash \; sum\_\; \{I\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; n\_i\}\; \backslash \; right)\; \_\; \{T,\; p,\; n\_j\}\; dn\_i~$

maybe by posing $\backslash \; mu\_i=\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; n\_i\}\; \backslash \; right)\; \_\; \{T,\; p,\; n\_j\}\; ~$, the chemical Potential , one obtains:

$dG\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_\; \{p,\; n\_i\}\; dT\; +\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; p\}\; \backslash \; right)\; \_\; \{T,\; n\_i\}\; dp\; +\; \backslash \; sum\_\; \{I\}\; \backslash \; mu\_idn\_i~$

that is to say:

$dG\; =\; -\; SdT\; +\; Vdp\; +\; \backslash \; sum\_\; \{I\}\; \backslash \; mu\_idn\_i~$

with

$dn\_i=\; \backslash \; nu\_id\; \backslash \; xi\; ~$ (see chemical Balance).

thus dG is also expressed:

$dG\; =\; -\; SdT\; +\; Vdp\; +\; \backslash \; sum\_\; \{I\}\; \backslash \; mu\_i\; \backslash \; nu\_id\; \backslash \; xi~$

from where: $\backslash \; sum\_\; \{I\}\; \backslash \; mu\_i\; \backslash \; nu\_i\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; \backslash \; xi\}\; \backslash \; right)\; \_\; \{T,\; p\}\; ~$

One thus defines chemical affinity by the formula:

$has\; =\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; \backslash \; xi\}\; \backslash \; right)\; \_\; \{p,\; T\}\; =\; -\; \backslash \; sum\_\; \{I\}\; \backslash \; mu\_i\; \backslash \; nu\_i\; ~$

It is thus introduced naturally into the differential:

$dG\; =\; -\; SdT\; +\; Vdp\; -\; AD\; \backslash \; xi\; ~$

in the same way:

$dF\; =\; -\; SdT\; -\; pdV\; -\; AD\; \backslash \; xi\; ~$

$dH\; =\; TdS\; +\; Vdp\; -\; AD\; \backslash \; xi\; ~$

$of\; =\; TdS\; -\; pdV\; -\; AD\; \backslash \; xi\; ~$

from where other possible definitions of chemical affinity:

$has\; =\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; H\}\; \{\backslash \; partial\; \backslash \; xi\}\; \backslash \; right)\; \_\; \{p,\; S\}\; =\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; \backslash \; xi\}\; \backslash \; right)\; \_\; \{V,\; S\}\; =\; -\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; \backslash \; xi\}\; \backslash \; right)\; \_\; \{V,\; T\}\; ~$

(although in practice, only the first is useful)

• $G\; ~$ Enthalpy free.
• $F\; ~$ Free energy.
• $H\; ~$ Enthalpy.
• $U\; ~$ Énergie interns.
• $S\; ~$ Entropy.
• $V\; ~$ Volume.
• $p\; ~$ Pressure.
• $\backslash \; xi\; ~$ Advance of reaction
• $n\_i\; ~$ quantity of mantière of component I
• $\backslash \; nu\_i\; ~$ coefficient of component I

The chemical affinity, which is an extensive variable, is the combined variable of the intensive variable Avancement of definite reaction at the time of a chemical balance.

Other Properties

One notices inter alia $has\; =\; -\; \backslash \; Delta\_rG\_\; \{T,\; p\}\; ~$ from where $A\; (\backslash \; xi)\; =R.T.\; \backslash \; ln\; \backslash \; frac\; \{K\_\; \{(T)\}\}\; \{Q\_R\; (\backslash \; xi)\}~$

Another definition

In 1922, the Belgian chemist Theophilus de Donder (1872-1957) gave another definition of chemical affinity.

Statement

where $\backslash \; delta\_i\; S\; ~$ is the entropy creates by irreversibility of the reaction and $T~$ the absolute temperature (in Kelvin).

Demonstration

This time it is indeed a demonstation to prove that the 2 writings of chemical affinity are concordant.

By definition, the free enthalpy is:

$G=H-TS=U+pV-TS~$

from where the expression of its differential:

$dG=dU+pdV+Vdp-TdS-SdT~$

According to the First principle of thermodynamics:

$\backslash \; Delta\; U\; =\; W\; +\; W\text{'}\; +\; Q\; ~$

By considering $W\text{'}~$, work of the forces other than the compressive forces, no one: $\backslash \; Delta\; U\; =\; W\; +\; Q\; ~$.

that is to say $dU\; =\; \backslash \; delta\; W\; +\; \backslash \; delta\; Q\; ~$ for an infinitesimal variation.

with $\backslash \; delta\; W\; =\; -\; p\_edV~$, the work of the forces of external pressure, $p\_e~$, is with $p=p\_e~$: $\backslash \; delta\; W\; =\; -\; pdV~$.

According to the Second principle of thermodynamics:

$dS\; =\; \backslash \; delta\_e\; S\; +\; \backslash \; delta\_i\; S\; ~$

where $\backslash \; delta\_e\; S=\; \backslash \; frac\; \{\backslash \; delta\; Q\}\; \{T\}\; ~$ is the entropy exchanged with the medium. and $\backslash \; delta\_i\; S~$ is the entropy created by the irreversibility of the reaction.

thus $dG=-pdV+\; \backslash \; delta\; Q+pdV+Vdp-T\; (\backslash \; frac\; \{\backslash \; delta\; Q\}\; \{T\}\; +\; \backslash \; delta\_i\; S)\; -\; SdT~$

Maybe while simplifying:

$dG=Vdp-SdT\; -\; T\; \backslash \; delta\_i\; S~$ that one compares with the expression using the first défintion of $A~$: $dG=Vdp-SdT\; -\; AD\; \backslash \; xi~$

from where $\backslash \; delta\_i\; S\; =\; \backslash \; frac\; \{AD\; \backslash \; xi\}\; \{T\}\; ~$

This second definition of $A~$ according to $\backslash \; xi\; ~$, $\backslash \; delta\_i\; S\; ~$ and $T\; ~$ is used to define the Condition of Natural Evolution (CEN), also called Condition of Spontaneous Evolution (THESE).

Category: Thermochemistry

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