Entropy Production in Chemical Reactions

Consider a chemical reaction occurring at constant temperature and constant pressure. The reaction may be represented in the general form

\[
\sum_i \nu_i A_i=0
\]

Here, \(A_i\) denotes the \(i\)-th chemical species and \(\nu_i\) is its stoichiometric coefficient. The stoichiometric coefficient is negative for a reactant and positive for a product. For example, the reaction

\[
\mathrm{N_2+3H_2\rightleftharpoons2NH_3}
\]

can be written as

\[
-\mathrm{N_2}-3\mathrm{H_2}+2\mathrm{NH_3}=0
\]

Thus,

\[
\nu_{\mathrm{N_2}}=-1,\qquad
\nu_{\mathrm{H_2}}=-3,\qquad
\nu_{\mathrm{NH_3}}=2
\]

Let \(d\xi\) be the infinitesimal extent of reaction. The change in the amount of the \(i\)-th component is related to the extent of reaction by

\[
\boxed{dn_i=\nu_i\,d\xi}
\]

For a multicomponent system, the Gibbs free energy is given by

\[
dG=-S\,dT+V\,dP+\sum_i\mu_i\,dn_i
\]

Since the reaction is considered at constant temperature and pressure,

\[
dT=0
\]

\[
dP=0
\]

Therefore,

\[
dG=\sum_i\mu_i\,dn_i
\]

Substituting

\[
dn_i=\nu_i\,d\xi
\]

gives

\[
dG=\sum_i\mu_i\nu_i\,d\xi
\]

or

\[
\boxed{
dG=
\left(\sum_i\nu_i\mu_i\right)d\xi
}
\]

Chemical Affinity

The chemical affinity of the reaction is defined as

\[
\boxed{
A=\sum_i\nu_i\mu_i
}
\]

Therefore, the change in Gibbs free energy can be written as

\[
\boxed{dG=A\,d\xi}
\]

Thus, the affinity is directly related to the change in Gibbs free energy with the extent of reaction:

\[
\boxed{
A=\left(\frac{\partial G}{\partial\xi}\right)_{T,P}
}
\]

The affinity therefore provides a thermodynamic measure of the tendency of the chemical reaction to proceed in the chosen direction.

Entropy Balance for the Reaction

For an irreversible process, the entropy balance is

\[
dS=d_eS+d_iS
\]

where \(d_eS\) is the entropy exchanged with the surroundings and \(d_iS\) is the entropy produced inside the system.

At constant temperature, if the surroundings receive an infinitesimal quantity of enthalpy \(dH\), the entropy transferred to the surroundings is

\[
d_eS=\frac{dH}{T}
\]

Hence,

\[
dS=\frac{dH}{T}+d_iS
\]

Multiplying by \(T\),

\[
TdS=dH+Td_iS
\]

Rearranging,

\[
Td_iS=TdS-dH
\]

or

\[
-Td_iS=dH-TdS
\]

At constant temperature and pressure, the Gibbs free-energy relation is

\[
dG=dH-TdS
\]

Therefore,

\[
-Td_iS=dG
\]

and hence

\[
\boxed{
d_iS=-\frac{dG}{T}
}
\]

Entropy Production in Terms of Affinity

From the definition of chemical affinity,

\[
dG=A\,d\xi
\]

Substitution into the entropy-production equation gives

\[
d_iS=-\frac{A\,d\xi}{T}
\]

Therefore,

\[
\boxed{
d_iS=-\frac{A}{T}\,d\xi
}
\]

Dividing by \(dt\),

\[
\frac{d_iS}{dt}
=
-\frac{A}{T}\frac{d\xi}{dt}
\]

If the reaction rate is defined as

\[
v=\frac{d\xi}{dt}
\]

then the rate of entropy production is

\[
\boxed{
\sigma=-\frac{Av}{T}
}
\]

where \(\sigma\) is the rate of entropy production.

For a spontaneous reaction proceeding in the forward direction, \(d\xi>0\) and the Gibbs free energy decreases. Thus,

\[
dG<0 \]

and from

\[
dG=A\,d\xi
\]

we have

\[
A<0 \]

Consequently,

\[
-\frac{Av}{T}>0
\]

which satisfies the second law of thermodynamics:

\[
\boxed{\sigma\geq0}
\]

Condition for Chemical Equilibrium

At chemical equilibrium, there is no net thermodynamic tendency for the reaction to proceed. Therefore, the affinity becomes zero:

\[
\boxed{A=0}
\]

Since

\[
A=\sum_i\nu_i\mu_i
\]

the condition for chemical equilibrium is

\[
\boxed{
\sum_i\nu_i\mu_i=0
}
\]

At this condition,

\[
dG=A\,d\xi=0
\]

and consequently

\[
d_iS=0
\]

Thus, at constant temperature and pressure, the entropy production associated with a chemical reaction is related to the Gibbs free-energy change and can be expressed in terms of the chemical affinity and the reaction rate.

The important relation obtained for the reaction is

\[
\boxed{
\frac{d_iS}{dt}=-\frac{Av}{T}
}
\]

where \(A\) is the chemical affinity and \(v\) is the rate of reaction.