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Because T and P are constant, K is still 8.33. After the N is added, but before any Section 6.6
2
x
shift in equilibrium occurs, we have Shifts in Ideal-Gas Reaction Equilibria
11>5.12 2
Q 8.39
x
13.1>5.1211>5.12 3
Q now exceeds K , and the equilibrium must therefore shift to the left in order to re-
x
x
duce Q to 8.33; this shift produces more N . Addition of N under these conditions
2
x
2
shifts the equilibrium to produce more N . Although the addition of N increases
2
2
x(N ), it decreases x(H ) [and x(NH )], and the fact that x(H ) is cubed in the denom-
2
3
2
2
inator outweighs the increase in x(N ) and the decrease in x(NH ). Therefore, in this
2
3
case, Q increases on addition of N . For the general conditions under which addition
x
2
of a reagent at constant T and P shifts the equilibrium to produce more of the added
species, see Prob. 6.50. In this discussion, we assumed that Q always decreases when
x
the reaction shifts to the left and increases when the reaction shifts to the right. For a
proof of this, see L. Katz, J. Chem. Educ., 38, 375 (1961).
Le Châtelier’s principle is often stated as follows: In a system at equilibrium, a
change in one of the variables that determines the equilibrium will shift the equilib-
rium in the direction counteracting the change in that variable. The example just given
shows this statement is false. A change in a variable may or may not shift the equilib-
rium in a direction that counteracts the change.
Some advocates of the “counteracting change” formulation of Le Châtelier’s prin-
ciple claim that if the principle is restricted to intensive variables (such as temperature,
pressure, and mole fraction), it becomes valid. In the NH example just given, al-
3
though the equilibrium shifts to produce more N when N is added at constant T and
2
2
P, this shift does decrease the N mole fraction. [After the N is added but before the
2
2
shift occurs, we have n(N ) 3.1, n(H ) 1, n(NH ) 1, n tot 5.1, and x(N )
2
3
2
2
0.607843. When the equilibrium shifts, one finds (Prob. 6.51) j 0.0005438, n(N )
2
3.1 j 3.1005438, n tot 5.1 2j 5.1010876, and x(N ) 0.607820
2
0.607843.] However, consider a system at equilibrium with n(N ) 2 mol, n(H )
2
2
2
3
4 mol, and n(NH ) 4 mol. Here K (0.4) /0.2(0.4) 12.5. Now suppose we add
3
x
10 mol of N at constant T and P to give a system with n(N ) 12 mol, x(N ) 0.6,
2
2
2
x(H ) 0.2 x(NH ), and Q 8.33. . . . When the shift to the new equilibrium oc-
2
3
x
curs, one finds (Prob. 6.51) j 0.12608, n(N ) 11.8739, n(H ) 3.62175, n(NH )
3
2
2
4.2522, n tot 19.7478, and x(N ) 0.6013 0.6. Thus, addition of N to this sys-
2
2
tem at constant T and P produces a shift that further increases the intensive variable
x(N ). Hence, Le Châtelier’s principle can fail even when restricted to intensive vari-
2
ables. [These failures were pointed out in K. Posthumus, Rec. Trav. Chim., 52, 25
(1933); 53, 308 (1933).]
If the Le Châtelier “counteracting change” statement is carefully formulated and
restricted to changes in intensive variables brought about by infinitesimal changes
in the system and subsequent shifts in equilibrium, then it is valid [see J. de Heer,
J. Chem. Educ., 34, 375 (1957); M. Hillert, J. Phase Equilib., 16, 403 (1995); Z.-K. Liu
et al., Fluid Phase Equilib., 121, 167 (1996)], but changes in the real world are always
finite rather than infinitesimal.
Shifts in Systems with More Than One Reaction
Predicting the effect of a change such as an isothermal pressure increase in a system
with more than one reaction is tricky. Each of the reactions in (6.47) has more moles
of products than reactants, and we might therefore expect that an isothermal pressure
increase on a system with these two reactions in equilibrium would always shift both
reactions (1) and (2) in (6.47) to the left, the side with fewer moles. Thus, one might
expect an isothermal pressure increase to always increase the number of moles of
water vapor present at equilibrium. However, Fig. 6.11b shows that above 10 bar, an