Page 152 - Physical Chemistry

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where n is the number of moles of substance i present at the start of the reaction. j Section 4.8
i,0
measures how much reaction has occurred. Since n is dimensionless and n has units Reaction Equilibrium
i i
of moles, j has units of moles. j is positive if the reaction has proceeded left to right,
and negative if it has proceeded right to left.

EXAMPLE 4.7 Extent of reaction

Suppose 0.6 mol of O reacts according to 3O → 2O . Find j.
2 2 3
The change in number of moles of species i during a reaction is proportional
to its stoichiometric number n , where the proportionality constant is the extent of
i
reaction j; n n j. Since n 3 and n 0.6 mol, we have 0.6 mol
i i O 2 O 2
3j and j 0.2 mol.
Exercise
In the reaction 2NH → N 3H , suppose that initially 0.80 mol of NH ,
3 2 2 3
0.70 mol of H , and 0.40 mol of N are present. At a later time t, 0.55 mol of H is
2 2 2
present. Find j and find the moles of NH and N present at t. (Answer: 0.05 mol,
3 2
0.90 mol, 0.35 mol.)

a
a
The material equilibrium condition is m dn 0 [Eq. (4.83)]. Section 4.7
i a i i
showed that at equilibrium the chemical potential of species i is the same in every
a
phase that contains i, so we can drop the phase superscript a from m and write the
i
material equilibrium condition as
a
a a m dn a m a dn b a m dn 0 (4.96)
a
a
i a
i
i
i
i
i
i a i a i
where dn is the change in the total number of moles of i in the closed system and m is
i i
the chemical potential of i in any phase that contains i.
For a finite extent of reaction j, we have n n j [Eq. (4.95)]. For an infinitesi-
i i
mal extent of reaction dj, we have
dn n dj (4.97)
i
i
Substitution of dn n dj into the equilibrium condition m dn 0 [Eq. (4.96)]
i i i i i
gives ( n m ) dj 0. This equation must hold for arbitrary infinitesimal values of
i i i
dj. Hence
The condition for chemical-reaction equilibrium in a closed system is that
N M 0.
i i i
When the reaction 0 → n A has reached equilibrium, then
i i i
a n m 0 reaction equilib. in closed syst., P-V work only (4.98)*
i
i
i
where n and m are the stoichiometric number and chemical potential of species A .
i i i
The relation of (4.98) to the more familiar concept of the equilibrium constant will be-
come clear in later chapters. Note that (4.98) is valid no matter how the closed system
reaches equilibrium. For example, it holds for equilibrium reached in a system held at
constant T and P, or at constant T and V, or in an isolated system.
The equilibrium condition (4.98) is easily remembered by noting that it is obtained
by simply replacing each substance in the reaction equation (4.92) by its chemical
potential.