Chapter 9
Chemical Equilibrium
This chapter is concerned with chemical equilibrium. The canonical ensemble is
employed to obtain a statistical thermodynamic expression for the Gibbs energy of
a pure polyatomic gas. This expression is then employed to obtain the change in
the Gibbs energy for a reacting gas mixture and an expression for the equilibrium
constant for a chemical reaction at constant pressure and temperature, thereby
giving a statistical mechanical working relation for it. Four extended examples are
examined in order to illustrate the use of the statistical expression. The relation
between the constant-pressure and constant-concentration equilibrium constants is
established. The chapter finishes with an application to the chemical kinetic rate
constant for a chemical reaction that proceeds via a model two-step mechanism that
employs an ‘activated complex’ or ‘transition state’ to facilitate completion of the
reaction.
9.1 Setting the Stage
We shall begin by looking carefully at the form of the Gibbs energy for a pure gas.
In order to do this, let us recall that we have seen in Eqs. (3.2.22) and (6.2.7) that
the canonical partition function Z N may be well approximated by
Z N =
1
N!
(z)
N
=
1
N !
z
N
tr z
N
rot z
N
vib z
N
el .
(9.1.1)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_9
481
Chemical Equilibrium
This chapter is concerned with chemical equilibrium. The canonical ensemble is
employed to obtain a statistical thermodynamic expression for the Gibbs energy of
a pure polyatomic gas. This expression is then employed to obtain the change in
the Gibbs energy for a reacting gas mixture and an expression for the equilibrium
constant for a chemical reaction at constant pressure and temperature, thereby
giving a statistical mechanical working relation for it. Four extended examples are
examined in order to illustrate the use of the statistical expression. The relation
between the constant-pressure and constant-concentration equilibrium constants is
established. The chapter finishes with an application to the chemical kinetic rate
constant for a chemical reaction that proceeds via a model two-step mechanism that
employs an ‘activated complex’ or ‘transition state’ to facilitate completion of the
reaction.
9.1 Setting the Stage
We shall begin by looking carefully at the form of the Gibbs energy for a pure gas.
In order to do this, let us recall that we have seen in Eqs. (3.2.22) and (6.2.7) that
the canonical partition function Z N may be well approximated by
Z N =
1
N!
(z)
N
=
1
N !
z
N
tr z
N
rot z
N
vib z
N
el .
(9.1.1)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_9
481