The SOLUTION of ELEMENTARY
CHEMICAL PROBLEMS W. M. SPICER
Georgia
School of Technology, Atlanta, Georgia
In working problems, students welcume any 'method which requires only mechanical operations a their part. Even instructors are too often satisfied with merely the correct answer. To maintain this attitude is to defeat the very purpose of problems, which is to teach the student to reason logically and rigorously from a premise to a conclusion. The object is not so m z h to Learn to work poblems, but to learn something by working them. Toward this end, we offer two suggestions. The first applies, no matter what method is used in solwing the problems; the second is concerned wi$z the advantages of the method of functions or variations as opposed to that of proportions. Seweral illustratiwe problems are worked by the function method.
This is not so. The essential difference is that the prohlem involves rigorous and exact reasoning from a premise to a conclusion. One must be explicit and definite. The main purpose iaassigning problems is to attempt to teach the student to reason rigorously and to make exact statements. The answer is only secondary; i t merely serves as a stopping point, or goal, of the reasoning. Toward this end, then, we suggest that the student be required, in working problems, to justify each step in writing by reference0 a law or definition, or by mathematical reasoning. To do this, the student need only ask himself, "Why?" a t each stop in the problem and then answer his query on the paper. If a student can work a given problem in this way, the instructor is justified in believing him able to solve any other problem of this type, but if the student works the problem by jotting down some figures and obtaining the correct answer, even if he works a number of these, the instructor can never be quite sure the student NE of the main difficulties faced by the students really knows what he is doing. We expect almost of elementary chemistry is that of working unanimous agreement with this first snggestion but problems. And if few learn to work problems almost as great a disagreement with the s e c o n d a t first. a t all, fewer still learn anything by working them. The second snggestion is essentially this. Instead of To this, I believe, the majority of instructors will teaching proportions, teach the concept of functions or agree, but can anything be done about i t ? In this re- variations. If the student discovers the proportion for gard, we wish to offer two suggestions. The first ap- himself, if he can jnstify it and wishes to use it, allow plies no matter what method is used; the second is him, but do not teach him proportions. We feel that proportions should be discarded for the concerned with the relative merits of different methods. Most students seem to think that the difference be- following reasons. tween a problem and a discussion question is that the (1) Students learn to use proportions mechanically problem involves a large amount of calculation toward without feeling or assurance and therefore often make the end of obtaining the correct numerical answer. mistakes.
0
(2) In working a problem mechanically, the most the student can get is the correct answer; he does not enrich his mind or enhance his ability to think. (3) In using proportions, stndents are using a formula whicb most of them cannot derive. (4) The idea of proportion is not a pregnant one; it is static and lifeless. As a student uses proportions more and more he learns no more about them, hecause the difficulty is not in their complexity. (5) Proportions do not come naturally to students. As an example, a class of fifty-three excellent students of freshman chemistry, who were steeped in the proportion idea, was given the problem, "If five gallons of gas cost $1.05, how much would seven gallons cost?" Only thirteen used proportions! The function or variation concept has the following recommendations. (1) The reasoning required in using this method is included in the reasoning necessary to justify the proportion. Our mistake here has been in allowing students to use proportions without justifying them, i. e., substituting in the proportion formula without being able to derive this formula. If we are dealing with two quantities, a and b, the a ratio of a to b is -. If this ratio remains constant, b i. e., if
make mistakes of the type so common with proportions (one ratio upside down) and mistakes that might occur are just as likely with proportions. We will now apply this method to some of the most common problems in freshman chemistry. (1) PROBLEMS BASED ON EQUATIONS What weight of hydrogen would be obtained by the action of dilute sulfuric acid on ten grams of zinc? It has been found by experiment that the weight of hydrogen so obtained is directly proportional to the weight of the zinc, i. e., Weight of hydrogen = k X weight of zinc
(4) We conld now solve our problem provided we knew the constant k, and we conld obtain k if we knew the weight of hydrogen obtained from some given weight of zinc. This information is given by the equation for the reaction, Zn
+ HzSOn= ZnSOh + Hz,
which tells us that one gram-atomic weight of zinc (65.4 grams) yields one gram-molecular weight of hydrogen (2.0 grams). Substituting these values in (4), we get,
and (4) can now be written, the quantities a and b are said to be proportional, and by the elimination of k, we write,
This is the proportion formula. Returning now to equation (I), we have,
This may he written in the form
This is read as "a is proportional to b," or "a varies as b," or "a is a linear function of b," or, even, "the ratio of a to b is constant." These steps, which are one less in number than those for proportions, are all that are necessary in using the function concept. (2) This method replaces a blind substitution in a formula by a reasoning process. It requires the student to think or at least gives him such an opportunity.' Proportions do not. (3) In learning to work problems from this new standpoint, the student would be learning a method of wide and fruitful application in mathematics and physics, as well as in chemistry, a method whicb he must learn if he is to continue even a little way in science. (4) In using this method it is almost impossible to
z
Weight of HZ = -X weight of zinc 65.4 . . z X 10 = 0.31 gram of Hz -65.4
-
.
A wide-awake student will immediately be concerned with the units and physical significance of this constant, k. He will soon see thqt k is simply the weight of hydrogen obtained from one gram of zinc and, further, that this chemical problem is essentially the same as many common practical problems with which he is already familiar; for example, such a prohlem as, "Find the cost of eighteen eggs." I believe that most stndents, if confronted with this problem, wonld begin with, Cost = number of dozens X cost per dozen = k X number of dozens And they wonld say, further, that in order to give a numerical solution, it would he necessary that they know the cost of some number of eggs, in order that they might obtain the cost per dozen, i. e., k, just as in the chemical problem i t was necessary to know the weight of hydrogen corresponding to some weight of zinc. To see this analogy is more important than t o he able to work thousands of such simple problems mechanically! (2) GAS L A W PROBLEMS A gas that occupies six liters at 700 mm. pressure,
occupies what volume a t 500 mm. pressure, the temperature remaining constant? Boyle found that, a t constant temperature, the volume of a gas is inversely proportioned to the pressure; i. e.,
i. e., M is 1 when V is 1 liter and w is 1 gram-molecular weight. Then
k = l and
This expression is true for any solute so long as w is in gram-molecular weights and V is in liters. Applying this to the above problem, we have,
M = - 4x - - 1000
Now we can find the value of k, knowing the value of P corresponding to a given value of V 700 X 6 = k Then P V = 6 X 700, so long as P is in mm. of Hg and V in liters. Now when
P
=
500 mm.,
6 X 700 = 8 . 4 liters v= 500
In using this method, notice that one starts from the beginning and works the problem through step by step; the only formulas substituted in are those of one's own derivation. (3) MOLAR CONCENTRATION OF SOLUTIONS What is the molar concentration of a solution containing four grams of sodium hydroxide in 300 cc. of solution? We are dealing here with three variables, the molar concentration, M, the volume of solution, V, and the weight of solute, w. How are these related? At constant V, if the weight of solute be douhle, the molar concentration will be double, that is to say, M is directly proportional to w or E
M a w But for a constant, w, if V he double, M will he halved, or M is inversely proportional to V , 1 M a -
v
Combining these we have
40
M a V
Again, we can obtain k if we know M for a given V and w. We know these values from the definition of M,
-
-1 molar 3
To solve this simple problem by use of proportions involves two proportions, thereby doubling the chance of error. (4) FREEZING POINT LOWERING Find the freezing point of a solution containing three grams of methyl alcohol in one hundred grams of water. The variables here are the freezing point lowering, At, weight of non-electrolyte, w, and weight of water, W. For a given weight of water, if w he doubles, At is doubles, therefore, But for a given w, if W be doubles, At is halved, therefore, 1 At a -
w:. ..
Then
Now it has been found experimentally that, At = 1.86 degrees when w = lgrslm molecular weight and W = 1000 grams
Therefore
k
=
1.86 X 1000 = 1
-
when w is in gram molecular weights and W i n grams. Then At = 1860 X - =
W
300
W
100 X 32
=
1.74 degrees
The freezing point will, therefore, be - 1.74'C. This method will he a little more difficult for the student a t first, especially if he has used proportions, but its fruits will be great in the end. After all, is i t not most important to furnish a student with a foundation for growth?
