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:
_{T,n}dp+\left(\frac{\partial G}{\partial T}\right)_{p,n}dT+\mu dn\;\;\;\;\;\;\;\;(149))
where _{p,T})
is defined as the chemical potential of the pure substance.
Substituting the definition of molar Gibbs energy 
in :
}{\partial n}\right]_{p,T})
Since 
is an intensive quantity for a pure substance,
_{p,T}=G_m\;\;\;\;\;\;\;\;(149a))
Hence, for a pure substance, its chemical potential is equal to its molar Gibbs energy and we can write eq149 as
_{T,n}dp+\left(\frac{\partial G}{\partial T}\right)_{p,n}dT+G_m dn\;\;\;\;\;\;\;\;(150))
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.
Question
Does eq150 only apply to a single-component (pure substance), single-phase system?
Answer
Yes. If the pure substance undergoes molecular rearrangment (e.g. cis-trans isomerisation), dissociation (e.g. 
) or melting, eq150 does not apply. Systems involving cis-trans isomerisation and dissociation reactions are considered multi-component, whereas a pure substance undergoing melting is treated as a single-component, two-phase system.
If the one-phase system contains different chemical species (e.g. cis-trans isomerisation or dissociation reaction), its Gibbs energy is represented by the function:
)
where 
is the number of moles of the 
-th species in the system.
The corresponding total differential of for a multi-component, single-phase system is:
_{T,n}dp+\left(\frac{\partial G}{\partial T}\right)_{p,n}dT+\left(\frac{\partial G}{\partial n_1}\right)_{p,T,n_{j\neq1}}dn_1)
_{p,T,n_{n\neq 2}}dn_2+\cdots +\left(\frac{\partial G}{\partial n_i}\right)_{p,T,n_{j\neq i}}dn_i\;\;\;\;\;\;\;\;(151))
where the subscripts 
refers to maintaining the composition of all species constant, while 
means keeping the composition of all species constant except .
Substituting eq148a and eq148b into eq151 yields
)
where _{p,T,n_{j\neq i}})
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 
,
dn<0)
As 
, we have 
or
)
Similarly, in the event that the reverse reaction is spontaneous, 
and 
, and dn<0)
, 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
_{p,T}=\sum_{i=1}\mu_iv_i\;\;\;\;\;\;(158))
For a spontaneous forward reaction, 
and 
, which makes _{p,T}<0)
. When the reaction attains equilibrium, 
and hence _{p,T}=0)
. For a spontaneous reverse reaction, and 
, which makes _{p,T}>0)
. By denoting _{p,T}=\Delta_rG)
, 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.
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,
)
Finally, the total differentials of for a single-component, multi-phase system (e.g. melting of ice), and a multi-component, multi-phase system (e.g. seawater in contact with air) are
_{T,n^{\alpha}}dp+\biggr\(\frac{\partial G^{\alpha}}{\partial T}\biggr\)_{p,n^{\alpha}}dT+\mu^{\alpha}dn^{\alpha}\biggr\]\;\;\;\;\;\;(162))
and
_{T,n}dp+\biggr\(\frac{\partial G^{\alpha}}{\partial T}\biggr\)_{p,n}dT+\sum_i\mu^{\alpha}_idn^{\alpha}_i\biggr\]\;\;\;\;\;\;(163))
respectively, where 
Eq162 and eq163 can also be expressed, by combining with eq148, as:
)
and
)
respectively.
Consider a single component, two-phase system at constant pressure and temperature. At equilibrium, eq162 becomes 
. Since the net change of each component is zero at equilibrium, 
. This implies that 
, i.e. the chemical potentials a component in two coexisting phases are equal at equilibrium. It follows that the chemical potentials a component in three coexisting phases are also equal at equilibrium, because and 
, so 
and therefore 
. This has important applications in phase equilibria.
