The chemical potential of a species is the change in Gibbs energy with respect to the change in the number of moles of the species, with temperature, pressure and the amount of other substances held constant. It is a measure of the potential of a species for change.
In the previous article, we derived eq143 and eq148 for a closed system that does only pV work. These equations are not applicable to open or closed systems with composition changes or open systems that interchange matter with their surroundings. We now derive equations that apply to such systems.
Consider a one-phase system containing a single chemical species (i.e. a pure substance) in a container that is in thermal equilibrium. The container may be open or closed. The Gibbs energy of the system is represented by the function:
The total differential of for this system is:
where is defined as the chemical potential of the pure substance.
Substituting the definition of molar Gibbs energy in
:
Since is an intensive quantity,
Hence, for a pure substance, its chemical potential is equal to its molar Gibbs energy and we can write eq149 as
Comparing eq150 with the total differential of for a closed system,
is the chemical potential due to the exchange of material between the system and its surroundings for an open system. It cannot be due to the change in system composition (including the change in state) because we did not assume any reaction occurring.
If the one-phase system contains different chemical species, its Gibbs energy is represented by the function:
where is the number of moles of the
-th species in the system.
To derive an expression for the change in Gibbs energy for a multi-component system, we again start from the total differential of :
where refers to maintaining the composition of all species constant, while
means keeping the composition of all species constant except .
Comparing eq151 with eq148,
where is the chemical potential of the
-th species in the system.
We now have a Gibbs energy equation that is applicable to a ‘one-phase, multi-species’ system that is either open or closed. If the system is closed and involves a reaction, in eq152 refers to composition changes within the system. For an open system,
may be attributed to either composition changes within the system or material exchange with the surroundings or both, depending on the specified conditions. Eq152 reduces to eq149 if there is only one component in the summation. Note that
in a ‘one-phase, multi-species’ system may not be equal to
of the pure species
.
Consider a one-phase closed system containing two species that react reversibly (note that reversibility here refers to the reaction having a forward component and backward component, and not in terms of a reversible thermodynamic process) as follows:
If the reaction is occurring at constant pressure and constant temperature, eq152 becomes
Suppose the forward reaction is spontaneous. From eq145, and hence,
, i.e.
Let and
,
As , we have
or
Similarly, in the event that the reverse reaction is spontaneous, and
, and
, which gives
The spontaneity of a reaction due to the difference in the chemical potentials of the reaction species is analogous to the tendency of the gravitational potential energy of a body to change from a higher value to a lower value. In other words, the term ‘chemical potential’ relates to the potential energy of the system to effect a change.
With reference to eq154 and eq155, the remaining scenario is
When both substances have the same chemical potential, they have equal tendency to change. This means that the reaction is at equilibrium.
We can also express the change in chemical potential of the reaction in a closed system is a different way. Let’s rewrite the equation as and define
where is the stoichiometric number of the
-th species in the reversible reaction and
is the amount of substance that is being changed in the reaction, i.e. a proportionality constant that measures how much reaction has occurred.
is dimensionless, while
is called the extent of a reaction and has units of amount in moles. This proportionality constant is the same for all species. By convention, the stoichiometric number for reactants and products in a reversible reaction are negative and positive respectively with respect to the forward reaction.
From eq157,
Since the Gibbs energy of a reaction system decreases for a spontaneous reaction until equilibrium is achieved,
For example, for the reaction with no
initially,
mole, with
moles and
mole, assuming all
is consumed. If the reaction is at equilibrium when
mole, we have
mole and
mole. At this point, there are 1 mole of
and 0.5 mole of
in the system. When the extent of the reaction reaches
mole (point R), there are 0.8 mole of
and 1.2 moles of
in the system. Let’s now analyse the reverse reaction from point R to the equilibrium point.
, with
and
. Therefore,
in a reversible reaction is always positive for the forward reaction and negative for the reverse reaction.
Substituting eq157 in eq153, we have
For a spontaneous forward reaction, and
, which makes
. When the reaction attains equilibrium,
and hence
. For a spontaneous reverse reaction,
and
, which makes
. By denoting
, we have a new indicator of the nature of a reversible chemical reaction, where
We call the indicator , the reaction Gibbs energy. A reaction where
is known as an exergonic reaction (Greek for work producing), while a reaction where
is an endergonic reaction (Greek for work consuming). The change in Gibbs energy with respect to the change in extent of a reaction for a reaction at constant
and
is shown in the diagram below.
Finally, we can also derive the first law of thermodynamics for a one-phase open system that involves pV work only, by substituting the definition of enthalpy in eq143 to give , whose differential form is:
Substituting eq152 in eq159
Similarly, the change in enthalpy for an open system is derived as follows:
Substitute eq159 and eq160 in the above equation,