THE CONCEPT OF THE POUND-MOLECULAR-VOLUME C. E. RONNEBERG, CRANECOLLEGE, CHICAGO, ILLINOIS
Teachers of the exact sciences are so nsed to thinking in terms of the c. g. s. system of units that often we little appreciate the fact that there are many situations in which it is better or even necessary to work with other units, as the f. p. s. system of units practically universally nsed in the industries of this country, Canada, and Great Britain. A problem from the field of chemical industry illustrates this point. A certain chamber acid plant utilizes 525,000 cubic feet of by-product sulfur d i o d e daily, expressed as @re sulfur dioxide at S. T . P . How many pounds of 96% sulfuric acid of density 1.84 does this nrolume of gas correspond to? Many teachers in order to solve this problem would first convert the volume of gas in cubic feet to liters and then solve the problem in the usual way as a problem of the weight-volume type based on an equation. The weight of acid obtained, of course, would be in grams, and this answer would then have to be recalculated to the answer in pounds. I t is worthwhile emphasizing that the above method of solving the problem is unnecessarily long and involves a reliance upon the c. g. s. system of units that to the writer seems unwarranted. From the standpoint of practical mathematics we should always remember that the shorter we can make a given solution for a problem, the less is the possibility of mak'mg arithmetical errors. The above type of problem may be solved in a simpler manner by the use of either the concepts of the "ounce-molecular-weight" and "ouncemolecular-volume," or the concepts of the "pound-molecular-weight" and "pound-molecular-volume," all of which correspond to the familiar ideas of the "gram-molecular-weight" and "gram-molecular-volume." One ounce-molecular-weight of oxygen, for example, is thirty-two ounces. The volume occupied by this weight of oxygen (S. T. P.) can be referred to as the ounce-molecular-volume and should be expressed in cubic feet. Dr. T. W. Richards, I believe, was the first to point out the very singular coincidence that the ounce-molecular-volume of a gas happens to be 22.4 cubicfeet. This should be apparent from the following calculations: 32.00 02. oxygen
= 32.00 X 28.35 g. (1 02. = 28.35 g.) = 907.0 g.
Volume of 32 oz. oxygen = 907'0 -X 1 (1 I. oxygen S. T. P. weighs 1.429 g.) 1 429 = 634.7 liters = 634.7 cm.' = X 35.31 (1 cubic meter = 35.31 cu. ft.) = 22.4 cubic feet.
6G
Hence as an application of Avogadro's theory we have the very unusual relationship that one ounce-molecular-weight of any gas (S. T . P.) occupies 22.4 cubic feet. The use of this concept is entirely analogous to the use of the concept of the gram-molecular-volume. 1659
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Perhaps even more useful in solving problems based upon equations when the common English units are involved, is to make use of the ideas of the pound-molecular-weight and pound-molecular volume, the volume in cubic feet occupied by one pound-molecular-weight of a gas a t S. T. P. This volume is numerically 359 cubic feet as shown below. Consider, for example, 2.016 pounds of hydrogen, which is one pound-molecular-weight. Weight in grams
= 2.016 X 453.6 (I = 914.5 a.
lb. = 453.6 g.)
914.5' X 1 (1 I. hydrogen S. T.P.weighs 0.0900 g.) Volume of hydrogen = 0.0900 = 10,160 liters = 10,160 ~ r n . ~ = 10'lfiO - X 35.31
1000
= 358.8 cubic feet.
This volume, of course, is also the volume of one-molecular-weight of any gas under standard conditiom. Because of the known variations in the gas laws, the above volume should be expressed as accurate to only three significant figures or 359 cubic feet. The use of the above derived concepts is similar to the use of the concepts of the gram-molecular-weight and the gram-molecular-volume. The following solutions of typical problems should make this clear. Problem I . What volume of sulfur dioxide ( S . T . P . ) could be obtained by burning two thousand pounds of pure sulfur? S 1 mole
First solution:
+
..
02---tSo~ 1 mole Number oz.-moles sulfur used = 2caJ x 16 = 1000 "Number ~ ~ . - m o lSO% e s possible = the same = lqOO Hence volume SO* = 1000 X 22.4 = 22,400 eubx feet
moo
Second solution: Number 1b.-molessulfur taken = 32 Number 1b.-molesSO2possible = the same 2000 Hence volume SOs = -X 359 = 22.400 cubic feet 32
Problem 2. In a fermentation vat carbon dioxide was being formed at a rate of twenty-fcue cubic feet per hour. What weight of sugar as dextrose is being converted to alcohol per hour? C6Ha06+2C,HsOH 1 mole Firsl solution:
+22C02 moles
25 22.4 1 25 0z.-moles dextrose fermented = - X 2 22.4 1 25 Hence weight dextrose fermented = - X - X 180 2 22.4 = 100.7 or.
0z:moles COXformed per hour = -
'=
-
16
or 6.28 lb.
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CONCEPT OF POUND-MOLECULAR-VOLUME
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25 . Second solulion: Lb.-moles CO, per hour = 1, smce one pound-mole of a gas 359 occupies 359 cubic feet 1 25 = - X Lb.-moles dextrose fermented 2 359 1 25 Hence weight dextrose fermented = - X - X 180 2 359 = 6.28 1b.
An inspection of the above two methods of solution shows that the method making use of the idea of the pound-molecular-weight and poundmolecular-volume is a little simpler mathematically, as the use of the idea of the ounce-molecular-weight requires somewhere in the solution the conversion of pounds to ounces or the reverse operation. However, i t is true that the use of the number 359 is a little awkward as it is a number which cannot be factored. Calling the pound-molecular-volume 360 cubic feet means introducing an error of only one part in 359 or less than 0.3%, an error which in most cases is negligible, as the known deviations in Boyle and Charles' laws in many cases are greater than this amount. The number 360, of course, is factorable in many ways, and for this reason i t might be advantageously substituted for the quantity 359. The sample solutions given above have been solved by the so-called mole method. It should be apparent that the equally correct proportion method of solving problems based upon equations could be used. For example, consider the problem given in the introductory paragraph. The equations for the series of reactions involved in the conversion of sulfur dioxide to sulfuric acid can be summed up as follows: 2Hz0
+ 2SO2 4-0%+ZHBOI
Z(360) cu. ft. --+2(98) lb. 720 cu. ft. --+ 196 Lb. 525,000 cu. ft. -+I Ib. 720 196 The proportion becomes: ---- = 525,000 x Hence z = (196)(525.000) = 142,900 lb, 720 100 This represents 100% acid; the weight of 96% mid would be 142,900 X - = 96 148,900 lb.
As far as the writer knows, there are very few chemistry texts that make any reference whatever to the ideas of the ounce-molecular-volume and the pound-molecular-volume. In view of the sinlplicity of these ideas and their usefulness, should this condition prevail? As long as the English units of measure are still so common, problems will continually arise involving these units. To solve these problems using c. g. s. units involves the use of conversion factors, which are easily forgotten. Solutions using the English units will be more direct and simpler. For instance, compare the following two solutions to the same problem.
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Problem: A steel tank of 8 cu. ft. capacity i s fclled with oxygen under a pressure of 100 lb. per sp. i n . and a temeerature of 6X°F. What is the weight of the oxygen i n the tank neglecting deeriations from the gas laws? First solution:
6S°F. = 20°C. 100
Pressure in atmospheres = = 6.80 14 7 273 6 8 8 X 293 X -i= 50.6 50.6 X 28.32 (1 cu. ft. = 28.32 1.) 1435 '435 X 32 (22.4 1. 0% = 32 g.) 22.4 2050 - X 2.20 = 4.51 Ib. 2050 1000
Volume gas S. T. P. in cubic feet Volume gas in liters
= =
Weight 0. in grams
=
=
Weight 0% in pounds Second solution:
=
=
Volume gas S.T.P.
=
273 6.8 8 X - X - = 50.6 cu. ft. 293 1
Weight oxygen
-
-X 32 (1 P.M.W.
toi,"
Answer
Oz, 359 cu. ft. = 32 lb.)
u""
= 4.51
1b.
Answer
A comparison of the two solutions shows that the second solution making use of the idea of the pound-molecular-volume is superior from the standpoint of simplicity, freedom from the use of awkward conversion factors, and because it involves fewer arithmetical operations. In concluding, may I emphasize that the writer is not putting forth a brief for the retention of the f. p. s. system of units? He merely wishes t o call attention to the fact that i t is unscientific t o slavishly rely on the use of the metric system of units in situations where the English units are in common use. I n such cases, as a matter of fact, the use of the metric units is often a distinct handicap. This is especially true when working with problems which involve volumes of gases expressed in cubic feet. In such cases, the concept of the ounce-molecular-weight or of the poundmolecular-weight might advantageously be used.
