Modern Engineering Thermodynamics

15.12 Chemical Equilibrium and Dissociation 627
Irreversible reaction equations are written as n A A + n B B ! n C C + n D D with the implication that A and B are completely
and irreversibly consumed in the reaction as C and D are produced. In an irreversible reaction equation,
the stoichiometric coefficients represent the actual number of moles of each element present in the reaction vessel.
In an equilibrium reaction equation, on the other hand, while the stoichiometric coefficients still represent
the number of moles that enter into the equilibrium reaction, they do not necessarily represent the number of
moles present in the reaction vessel. Therefore, the mole fraction concentrations present cannot be determined
from the equilibrium reaction equation alone. Consequently, we continue to use the symbol n i to represent the
total number of moles of species i present in the reaction vessel, but we must now introduce the symbol v i ≤ n i
to represent the number of moles of species i that actually enter into the equilibrium reaction.
For example, at high temperature, the reaction of A and B to form C and D may partially reverse and reform A
and B from C and D, and at equilibrium, both the forward and the reverse reactions take place simultaneously,
resulting in a reversible “equilibrium” composition containing all four substances. To denote a reversible chemical
equilibrium reaction that implies the coexistence of all four substances A, B, C, andD, weuseadouble arrow
between the reactants and the products and we use v i for the stoichiometric coefficients of the reaction, as follows:
v A A + v B B ⇆ v C C + v D D
Van’t Hoff argued that chemical equilibrium occurs only when a reversible system is in a state of constant uniform
pressure and temperature. For a closed system whose only work mode is p − V, the combined first and second
laws in differential form (neglecting any changes in kinetic or potential energy) are
or
dQ − dW = dU = TdS− TdðS P Þ − pd V
TdS= dU + pdV+ TdðS P Þ = dH − V dp + TdðS P Þ (15.30)
where we use the definition of total enthalpy H = U + pV, anddðS P Þ is the differential entropy production rate,
which is required to be greater than or equal to zero by the second law of thermodynamics. In Chapter 11, we
introduce the Gibbs function G as (see Eq. (11.7)) G = H − TS, and on differentiation, it becomes
Rearranging gives
and combining this result with Eq. (15.30) gives
dG = dH − TdS− SdT
TdS= dH − dG − SdT
dG = V dp − SdT− TdðS P Þ (15.31)
For chemical equilibrium, we require that, T, p, andS P all be constants; then dG = 0 and consequently G = constant.
Otherwise, for nonequilibrium chemical reactions that take place at constant temperature and pressure,
the second law of thermodynamics requires that
or
dG = −TdðS P Þ < 0
Z
G 2 − G 1 = − TdðS P Þ < 0
Consequently, we have the following three results for the Gibbs function of a chemical reaction that occurs at
constant temperature and pressure:
a. dG < 0 ðor G 2 − G 1 < 0Þ implies that the chemical reaction has the potential to occur (but this does not imply
that it will spontaneously occur).
b. dG = 0 ðor G 2 = G 1 = constantÞ implies that chemical equilibrium exists and no further reactions can occur
beyond the equilibrium reactions.
c. dG > 0 ðor G 2 − G 1 > 0Þ implies that the reaction cannot occur at all because to do so would violate the second
law of thermodynamics.
For a system that has undergone a chemical reaction at constant temperature and pressure, item b requires that a
final state of chemical equilibrium occurs only when G P′ = G R′ = constant, where P′ and R′ denote the products
and the reactants in the equilibrium reaction. Then,
G R′ = ∑
R′
v i g i = ∑ v i g i = G P′ (15.32)
P′