ACCURACY OF CERTAIN PHYSICO-CHEMICAL EXPERIMENTS R~CHARD J. DEGRAY,LEHIGHUNIVERSITY.BETHLEHEM, PENNSYLVANIA
The theory of errors has been applied to a series of student determinations of molecular weights by freezing-point, boiling-point, and Victor Meyer methods. The data were'found to follow the theoretical distribution curve at h o t approximately. The presence of impurities i n the reagents was detected and the concentration thereof calculated successfully from this theory. A system of grading, based on the distribution law, i s suggested. The relatine accuracies of these three methods are calculated.
. . . . . .
A group of students handling special apparatus for the first time cannot be expected to obtain results of an accuracy comparable to that truly inherent in the method used. A course in this physical chemistry laboratory has as a primary object the instruction of the student in using such apparatus and methods and not the determination to the second decimal place, for instance, of the molecular weight of naphthalene. Sloppy and careless work should not he condoned on the above principle, however, and exceptional work should not lack proper reward. The questions thus arise: ',How inaccurate is 'sloppy' work, and how close are 'exceptional' results?" and "What mark should be given to intermediate qualities of work?" To answer these questions a record was kept of all results obtained in this laboratory during the second semester of the scholastic year 1930-31. The results capable of analysis were obtained from experiments for which the true values were known, and the best of these T r e the molecular-weight determinations. In this laboratory course the molecular weight of naphthalene is determined by using the Beckmann freezing-point apparatus and the old style Beckmann boiling-point apparatus; and the molecular weight of chloroform is determined by the Victor Meyer method. Each pair of students makes several determinations in each experiment. The eightyodd results from each method were analyzed. Since the true value was known, necessarily, all data sheets were initialed by the instructor before the men left the laboratory, and the calculations were checked both for accuracy and to insure that the data as taken were used. The results thus obtained should be a fair sample of the work to be expected from a student manipulating a piece of apparatus for almost the first time in his life. The naphthalene used had been recrystallized from alcohol. The benzene was of C.P. variety, stated to be free from thiophene and CS2. The chloroform also was C.P. The results were analyzed by the standard methods of theory of errors, the values calculated being the average, E, and r. The average is the arithmetical mean, E is the mean error, or the square root of the sum of the s q u m s of the errors, and r is the probable error-that is, between r and the average there is a probability of '/e that an error will occur. The 903
904
JOURNAL OF CHEMICAL EDUCATION
MAY,1932
theoretical distribution of errors was computed also, and was found to approximate the actual values, but the calculated distribution had a larger "spread," so these results will not be given. The following table gives the results of these calculations: Expcrimcnl
No. of Resulls
Aucvogc
E
Victor Meyer Boiling-Point Freezing-Point
83 08 93
120.40 126.87 127.55
3.74 9.12 3.96
2.51 6.51 2C7
The accuracy of a method may be judged by the value of r. Thus for the boiling-point method, only one-half of the results will be closer than * 6.5 units, or * 5.17,, whereas one-half of the values obtained by the freezing-point method will be within 2.67 units, or * 2.1%. The corresponding figures for the Victor Meyer method are + 2.51, and .L: 2.1'7,. That the boiling-point method is not as accurate as is the freezing-point method is well known, but the comparative accuracies obtained with the freezing-point and Victor Meyer methods are surprising. Before proceeding to further discussion of accuracy, let us define certain terms--not necessarily as standard diction, but merely for the purposes of this paper. I t has been stated that the true values of the molecular weights were known. We can see, however, that the presence of impurities or other disturbing factors may warp the apparent molecular weight, so that even if the experiment were absolqtely accurate the result would not check with the true value. The accuracy of the experimental method should be judged by comparison with this seconsquantity, which we shall call the "correct result" and not with the first quantity, which we have called the "true value." The two quantities are identical when all impurities and other disturbing factors are removed. According to the theory of errors, the mean error of the average, E,, gives a measure of the accuracy of the average of all determinations, of a quantity. Since we know the true values, the actual errors may be computed also: Erparsncnl
Victor Meyer Boiling-Point Freezing-Point
T T UIJaluc ~
A8erage
Aclual Enor
Eo
119.39 128.11 128.11
120.40 126.87 127.55
1.01 1.24 0.56
0.41 1.11 0.41
E,, the calculated error of the average, and the actual error do not check. The actual error depends upon the true value, and E, on the correct result. The discrepancy may be due to the difference between these twoquantities, caused by impurities as mentioned above. Thus the boiling-point and freezing-point methods used exactly the same benzene and naphthalene, and the difference between the actual error and the calculated error is 0.13 in one case and 0.15 in the other. The discrepancy, therefore, is constant,
VOL.9, NO. 5 ACCURACY OF PHYSICO-CHEMICAL EXPERIMENTS
. 905
and is the difference between the true value and the correct result. This enables us to calculate the correct result for these experiments as 128.11 - 0.14 (Av.), or 127.97. This discrepancy might he caused by a departure from the theory of errors. To ignore this and blame the presence of impurities may be made to seem even more reasonable by calculating the concentration of the supposed contamination. Thus since the naphthalene was recrystallized from alcohol it was believed to be the higher grade reagent, and the benzene was considered as containing the impurities. Toluene being the chief impurity of commercial benzene, the concentration of toluene necessary to warp the molecular weight and latent heats of benzene sufficiently to give a result of 127.97 for the molecular weight of naphthalene was calculated and found to be only 0.0097 mole per 1000 g. of benzene or 1.4 weight-per cent. of toluene. This is not unreasonable. For the Victor Meyer method the correct result may be calculated to be 120.40 - 0.41, or 119.99. The difference between this value and the true value, or 119.39, is 0.60 unit or 0.5%. This may be caused either by impurities or by imperfections of the gas laws, or both, so no calculations of percentage purity were made. The imperfections of the gas laws can account for a t least some of the 0.5%. That the methods of the theory of errors may be applied correctly to such data as these seems to be demonstrated amply. Grades on experimental work, therefore, may be based on such calculations with justice. Thus, since one-half of the results fall between 7 and th; correct result, one-half of the class should obtain results of a better accuracy than r. In other words, 7 should be the limit of error of the fiftieth percentile of the class. This percentile usually corresponds to a grade of 75 when 60 is passing. Hence a man turning in a result obtained by the boiling-point method which is A 5y0 from the average should receive a grade of 75, and those nearer proportionately higher marks. To receive a mark of 75 on the experimental work of the freezing-point determination, however, the result would have to come within .t 2% of the average. Similarly, a passing grade of 60 corresponds to the 20th percentile. The maximum acceptable error would be that of a probability of 0.80: this may be found in error tables to be 1.9r. Thus a man would be required to repeat the boiling-point experiment if his results were in error by more than 5.1 X 1.9, or 9.7Yo. The term "error" really refers to the departure of a value from the average of all values. To follow the above scheme of grading would necessitate averaging all the results of the year before grading any individual. Such a procedure is out of the question. The difference between the average and the true value should not be much if pure chemicals are used, so the error may be considered the departure from the true value; not from the correct result. With this understood, we may tabulate the error in
YO6
JOURNAL OF CHEMICAL EDUCATION
MAY,1932
per cent. which will receive a grade of 75 and the error which will receive a failureand command to repeat for the three experiments considered: Expnimcd
Moximnm AUmoblc Enor
Pmbable EIIDI
4.0% 9.7 4.0
2.1% 5.1 2.1
Victor Meyer Boiling-Point Freezing-Point
Before these calculations were made the acceptable errors for these experiments were those given by Findlay (I) which are Victor Meyer Boiling-Paint Freaing-Point
5% 0% 3-5%
These are general and elastic limits. To calculate the limit for the aRparatus and materials used in this particular laboratory is certainly better from both the instructor's and the students' viewpoints. As the data are compiled grades for other experiments will be calculated in the same manner. This method of grading may be applied wherever quantitative experiments are made, and need not be limited to physical chemistry. Literature Cited (1) FINDLAY,"Practical Physical Chemistry," 4th edition, Longmans, Green and Company, New York City, 1926, pp. 47,120,126.
