The Foundations of Chemistry

x

In this case a48.0, b19.6, and c1.47. Substituting these values gives

x



19.6

9

6.0

10.1

0.309 or 0.099

Solving a quadratic equation always yields two roots. One root (the answer) has physical

meaning. The other root, while mathematically correct, is extraneous; that is, it has no phys-

ical meaning. The value of xis defined as the number of moles of A per liter that react and

the number of moles of B per liter that react. No more B can be consumed than was initially

present (0.100 M), so x0.309 is the extraneous root. Thus, x0.099 is the root that has

physical meaning, and the extraneous root is 0.309. The equilibrium concentrations are

[A] (0.300x) M 0.201 M; [B] (0.100x) M 0.001 M;

[C]  [D] x M  0.099 M

You should now work Exercises 34, 44, and 46.

(19.6) 
( 1  9 .6)^2  4 (4 8 .0)( 1 .4 7 )



2(48.0)

b 
b^2  4 ac



2 a

Check Example 17-6:

Q

Q 49 Kc

Check Example 17-7

Q

Q 49 Kc

(0.099)(0.099)



(0.201)(0.001)

(0.175)(0.175)



(0.025)(0.025)

[C][D]



[A][B]

720 CHAPTER 17: Chemical Equilibrium

The following table summarizes Examples 17-6 and 17-7.

Initial Concentrations (M) Equilibrium Concentrations (M)

[A] [B] [C] [D] [A] [B] [C] [D]

Example 17-6 0.200 0.200 0 0 0.025 0.025 0.175 0.175

Example 17-7 0.300 0.100 0 0 0.201 0.001 0.099 0.099

Problem-Solving Tip:Solving Quadratic Equations — Which Root

Shall We Use?

Quadratic equations can be rearranged into standard form.

ax^2 bxc 0

All can be solved by the quadratic formula, which is

x (Appendix A)

This formula gives tworoots, both of which are mathematicallycorrect. A foolproof way

to determine which root of the equation has physical meaning is to substitute the value

of the variable into the expressions for the equilibrium concentrations. For the extra-

neous root, one or more of these substitutions will lead to a negative concentration,

which is physically impossible (there cannot be less than noneof a substance present!).

The correct root will give all positive concentrations. In Example 17-7, substitution of

the extraneous root x0.309 would give [A](0.3000.309) M0.009 Mand

[B](0.1000.309) M0.209 M.Either of these concentration values is impos-

sible, so we would know that 0.309 is an extraneous root. You should apply this check

to subsequent calculations that involve solving a quadratic equation.

b 
b^2  4 ac



2 a