In the Classroom
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Equilibria That Shift Left upon Addition of More Reactant Jeffrey E. Lacy Department of Chemistry, Shippensburg University, Shippensburg, PA 17257;
[email protected] Students of chemistry who have studied Le Châtelier’s principle in any general chemistry textbook (1, 2) and most physical chemistry textbooks (3, 4)1 are familiar with the result of adding more reactant to a reaction at equilibrium: the position of equilibrium shifts so that more of the product is formed. This result occurs for an isothermal system at constant volume. However, by reading the same textbooks no student will learn that a different result is possible if the isothermal system is at constant pressure instead of constant volume. For example, consider N2(g) + 3H2(g)
2NH3(g)
(1)
At a constant temperature and pressure, if the number of moles of nitrogen is increased to the extent that it is greater than one-half of the total number of moles of all species, rather than forming more product the equilibrium shifts toward the reactants and ammonia will decompose (5). We propose presentations that will enable students to perform a side-by-side comparison of these two conditions and, thereby, gain a better understanding of reactions at equilibrium. In addition, we have derived novel stoichiometric criteria for identifying equilibria that shift left upon addition of more reactant. General Chemistry At this level, descriptions of the constant volume condition are usually based on an algebraic analysis of the magnitude of the reaction quotient QC relative to that of the equilibrium constant KC for which both expressions are written in terms of molar concentrations. Using eq 1 as an example, the necessary expressions have the form Q C or KC =
[NH3 ]2 [ N 2 ][H2 ]3
(2)
However, to analyze the system at constant pressure, we must write these expressions in terms of mole fractions Xi and total pressure Pt such that KP = KX Pt∆ν, where ν is the stoichiometric coefficient. This is analogous to the expression KP = KC(RT )∆ν, with which most students will already be familiar. Given the expression for the equilibrium constant at constant pressure for eq 1
KP =
PNH32 PN PH 2
(3)
3 2
For constant total pressure, only KX will vary with added reactant. By substituting the explicit expressions for the mole fraction of each species Xi, as well as the definition for the total number of moles nt, we obtain
KX =
n N2 nt
KP =
1192
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3
=
2
2
nN2 nH23
3
)2 (5)
nt
Physical Chemistry The reaction quotient QX could serve as the basis for presentation of the constant pressure condition at this more advanced level as well. By taking the partial derivative of eq 5 with respect to the number of moles of added reactant (6, 7), students can derive a formula for determining the relative number of moles of a reactant that would be needed to shift the equilibrium toward the reactants. However, we failed in our attempt to derive a general expression based on this method. To derive a general expression and to ground this concept on a firm theoretical basis, the thermodynamic principle of Gibbs energy must be employed. Such an approach can be found as part of two extensive analyses of Le Châtelier’s principle (5, 8). In the following, we use the approach outlined in ref 5 as the more direct and facile basis for simplifying and extending the results. Given that the spontaneous change is to reduce the chemical potential of the added reactant µi, we express it in terms of partial pressure, pi, µ i = µ i° + RT ln
pi pi°
(6)
Since we must know how the chemical potential will change as the extent of reaction ξ changes, we derive (see the Supplemental MaterialW) ∂ pi ∂ ξ
(4)
X N2 X H23 Pt2
n H2
(
n NH32 n N + n H + n NH
The number of moles of nitrogen appears in both the numerator and denominator. Since the numerator is squared, addition of N2 will cause it to increase faster than the denominator so that at some point QX will become larger than KX (see the discussion of eq 9 below to quantify this point.) Thus, NH3 will need to decompose to return QX to equality with KX. On the other hand, adding H2 does not cause decomposition of the product: even though the numerator is squared, the number of moles of H2 in the denominator is cubed and QX will always decrease upon addition of H2.
we substitute Pi = Xi Pt and simplify to obtain
X NH32
2
n NH3 nt
= T, P, µ j
Pt n t2
(nt νi
− ni ∆ν )
(7)
where ∆ν = Σνi. For a spontaneous change, the numerator
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In the Classroom
of the slope on the left hand side of eq 7 will be negative. If the equilibrium shifts toward reactants, then the denominator also will be negative; thus, overall the slope will be positive. The right hand side of eq 7 will be positive only when
Recall that νi is negative when it is the stoichiometric coefficient of a reactant, so for eq 8 to be true ∆ν must also be negative. We proceed one step farther by rearranging to get nt νi ∆ν
(9)
We emphasize that this result was specifically derived for the reaction to proceed right to left from products to reactants in order to decrease the chemical potential of the added reactant, and so it applies only in that instance. If the desired result is for the equilibrium to shift from left to right, then the inequality in eq 8 must be reversed. Applying eq 9 to eq 1 we obtain n N2 >
(10)
3 nt
(11)
2
which is clearly impossible so that adding hydrogen at a constant pressure will always produce more ammonia. In sim-
www.JCE.DivCHED.org
(12)
Acknowledgments We thank all reviewers, but most especially reviewer #3, for the recommended revisions that improved the quality of this article. W
Supplemental Material
The derivation of eq 7 is available in this issue of JCE Online. Note 1. We list only a few of the textbooks that we reviewed. Literature Cited
nt 2
that is, the position of equilibrium will shift to the left when the added number of moles of nitrogen exceeds half of the total number of moles. On the other hand, in order for added hydrogen to shift the equilibrium to the left eq 9 yields n H2 >
νi < ∆ν and ∆ν < 0
(8)
n t νi > ni ∆ν
ni >
plest terms, the following criteria must be met for the reaction to shift left upon addition of reactant
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1. Chang, R. Chemistry, 6th ed.; McGraw-Hill: New York, 1998. 2. Petrucci, R. H.; Harwood, W. S.; Herring, F. G. General Chemistry; Prentice Hall: Upper Saddle River, NJ, 2002. 3. Barrow, G. M. Physical Chemistry, 6th ed.; McGraw-Hill: New York, 1996. 4. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 2nd ed.; John Wiley & Sons, Inc.: New York, 1997. 5. de Heer, J. J. Chem. Educ. 1957, 34, 375–380. 6. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2002. 7. Katz, L. J. Chem. Educ. 1961, 38, 375–377. 8. Canagaratna, S. G. J. Chem. Educ. 2003, 80, 1211–1219.
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