A Use of Calculus in Freshman Chemistry At present many freshmen are well along in calculus even to the extent of being acquainted with maxima and minima. Such students naturally enjoy finding practical applications to other fields of this, to them, pure mathematics. Except in the field of thermodynamics, few such applications appear in freshman chemistry. However, we recently stumbled uoon one such amlication in a discussion of solubilitv constants. The followine .. - oroduct . .. oroblem was omed to the class. If 10 ml of 0.04 F Pb(N03)2 solution were mixed with 10 ml of 0.04 F hydrochloric acid, would PbClz precipitate? The K,, of PbClz is 1.7 X 10-J.Since [Pb2+][C1-l2= (0.02)3= 8.0 X 10-" is less than 1.7 X 10-\ the answer is obviously "no." At this point a student suggested that perhaps a precipitate would form if 20 ml of hydrochloric acid were used. Would it? Since (0.04/33((2/3)x 0.04)2= (4127)x 64 x 10-6 = 9.5 x 1 W 6 is less than 1.7 X I W 5 , again the answer is "no." The question $hen arose whether any combinations of volumes of these solutions could lead to a precipitate of PhC12. A general solution to this problem using the simple methods of calculus follows Let's add Vml of the hvdrochloric acid to 10 ml of the lead nitrate solution
.
~
~~
~~
~~
[Pb"]
=
0.04 X
+v
------
10
v
+
and [Cl-I = 0.04 10 v
What value of V leads to a maximum value of 6.4 X lo-' zero.
lnJ(10 +
q3. T O find this we equate the
derivative to
+
2i10 V) = 3v 20 = v It has already been shown ahove that this volume will not yield a precipitate and since this volume maximizes the function, no volume will yield a precipitate. As an exercise, the reader might apply the above method to the fallowing problem: Although the mixing of equal volumes of 0.02 F barium nitrate and 0.02 F sodium fluoride solutions will not result in the precipitation of BaFz (K,, = 1.7 X 1 W 6 )show that the mixture of certain relative volumes of these solutions will yield a precipitate, and find the ratio of volumes that result in maximum precipitation. Georgia Institute of Technology Atlanta, Georgia 30332
686
/
Journal of Chemical Education
William M. Spicer
