the computer bulletin board the slope of the graph, one r a n determine the half-life uf the isotope. The cumulative wunt method, described herein, requires the experimenter to measure the total accumulated counts as a function of the time. A mathematical relationship exists between the accumulated counts and the rate of radioactive decay. By using this relationship, the half-life can be readilv determined. One advantage of this method is the elimination of the need to reset the counter during the experiment. Theory The integrated first-order radioactive decay differential equation is given by eq 1, where I re~resentsthe intensity of the radiation at time t, lois the initial intensity, and k is the rate constant.
I(t) is often reported as the number of disintegrations per unit of time. Through integration of eq 1, from t = 0 to t , one obtains the total accumulated number of disintegrations as a function of time. This total is symbolized by the letter Q and is given by eq 2.
I
Q(t) = Ioe k'dt = Idk - Zoe-k'lk (2) The derivative of eq 2 is given in eq 3. Q'it) =
I
~
~
C
"
(3)
By using this relationship, eq 2 can be written as eq 4. Q(t) = I,lk
- Q'(t)lk
(4)
Equation 4 suggests that by plotting the accumulated count, Q, as a function of the derivative of the accumulated count, Q', one should obtain a straight line with a slope equal to the negative reciprocal of the decay (rate) constant.
ponential function. Equation 5 represents the form of this polynomial function where a is the intercept and b, c, d, e and f are coefficients obtained from the Lotus 1 - 2 3 regression command. The equation is then differentiated to obtain Q'(t). The half-life is calculated by multiplying the normalized half-life by the total time of theexperiment.
Summary This method is easy to implement and provides reasonable agreement with the accepted half-life of 2.55 min for barium-137 and 68 min for aallium-68. One obvious advantage of this method is the elimination of the resetting of the scaler. T o perform this experiment, one measures the accumulated count as a function of time, fits the aeeumulated count to a power series with narmalized time as the independent variable. The derivative of the accumulated count funetion is then taken with respect to time, and the values of it are correlated with the aceumulated count. From this correlation, the rate constant and the half-life can be determined. -~ Spreadsheet methods are extremely useful for data regression analysis. These techniques can be used in kinetics and also in calculating third law entropies.
Justln W. Dlehl a n d Larry M. Wler St. Bomventure University St. Bonaventure, NY 14778 One of the major methods used to monitor the scale-up of a chemical process is the calculation of the return of the original investment. It is very important for a chemist in an entry-level position in industry to understand this type of calculation. T o help our students with this, we have developed a s p r e a d s h e e t t e m p l a t e t o guide t h e m through a sample calculation. The template has three parts and is based on a common
organic synthesis of sulfanilamide. The ealculations in Parts I and I1 are based on a published exercise in Fieser's laboratory text (14).Usually laboratory manuals do not contain asimilartype of exercise, although a recent text gives a detailed discussion on the technique of process economics (15). Part I calculates the net chemical cost for production of sulfanilamide. The student is asked to provide the following information: the desired production level, laboratory bench data on reactants and products, and unit eost figures for reactants. The latter costs can be given to the student by the instructor, or the student can look them up in the Chemical Week price listings. The gallons per month processed, cost per month, and cast per pound of product are then calculated giving the net chemical cost per month and the chemical cost per pound of sulfanilamide. Part I1 ~ a l ~ u l a tthe e s total cost of produ~. tim. The student provides: rhe workmg vol. ume of the reactor, installed value of equip. ment, human-hours per batch and per thousand pounds of product, hourly wage, utilities' cost per batch, and estimates of repair, supplies, and occupancy costs per month. The cost for plant and supervisory labor, utilities, equipment items, waste disposal, and allocated expenses are calculated for the given level of production. These fixed casts are then added to the net chemical eost to arrive a t the total cost for production per month and per pound of sulfanilamide. In addition, the percent of the total cost due to each factor is displayed; this allows the student to see that labor, and not net chemical cost, is the major part of the total cost in most cases. The eost of waste disposal is an important consideration; in our template it adds five percent to the total cost. Part 111 calculates the return on investment following a procedure used at the E. I. DuPont Co. (16).Namely, %Return on = Investment (net profit per pcmnd)(yearly production1 {original inwatment + wrking rapitall
Experimental The barium-137 and the gallium-68 isotopes were obtained from conventional radioactive cows. Approximately 9-12 drops of eluant were used in each experiment. A Nucleus Model 500scalar was used to determine the total number of counts as a function of time. The total numher of counts and time were recorded photographically. Data from representative experiments are displayed in the table. The half-life, Tm,from each experiment is also displayed on the last row of the table. Treatment of Data This section illustrates the complete methodology of the technique. The first step involves normalizing the time of the experiment to unity and fitting the total accumulated counts iQ) to a fifth-order polynomial in which the normalized time is the independent variable. The normalized time is obtained by dividing each measured time by the time of the experiment. This improves the rate of convergence of the exA42
Journal
of Chemical Education
Ba-137 Time, min 0 0.26 0.5 0.75 1 1.28 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4.
Ga-68
0
Time, mi"
0
Time, min
0
0 1891 3206 4859 6093 7519 8640 9828 11005 12094 13006 13936 14829 15880 18370 17084 17779
0 1 2 3 4 5 8 7 8 9 10 11 12 13 14 15 16
0 619 1282 1885 2558 3075 3771 4469 5142 5719 6394 7095 7576 6370 8840 9462 9900
17 18 19 20 21 22 23 24 25 26 27
10530 11000 11645 12278 12850 13319 13825 14419 14980 15561 16030
T.,, = 2.80 min
63.5 min
the computer bulletin b o d The difference hetween the selling price per pound and the total cost per pound is the net profit per pound. The original investmentis assumed to be 30%of receivables per year; receivablesper year are the monies per year received if the product were sold for cost. The working capital, the money needed to run the business for one month, is the sum of chemical cost per month, 5% of receivables, and the allocated expenses for one month. The spreadsheet will calculate the return for a specified selling price, or it will calculate the required selling price to achieve a specified return. Such financial calculations are a very appropriate use of spreadsheets. With such templates, a student can decide whether it is economically feasible to continue the development of a particular chemical process, in this case, sulfanilamide production. I t should be noted that the return calculated by the spreadsheet is optimistic. Generally, there is a drop in the percent yield as the process is scaled up from bench to pilot plant to full-scale production. This makes the calculated return based on bench greater than the actual return. The spreadsheet program used to develop our template was Quattro, version 1.0, by Borland International. T h e R E TURN.WKQ template has also been converted to RETURN.WK1 format for possihle Lotus 1-2-3 users; the latter version is untried. Please note that you must already awn a spreadsheet program capable of reading one of these formats to use this template. To obtain a copy of RETURN, send a check for $5.M1, payable to St. Bonaventure University, to the authors, and specify the desired disk size (3.5 or 5.25 in.).
press).
12. Wil1ard.H. B.;Merritt,L.L..Jr.;Dean.J.A.;Settls,F. A . Jr. Inrtrumenid Methods of Anolvsia, 7th ed.; Wadswonh: Belmont, CA 1988:~401-421. 13. Hayes. R. L.; Butler, W. R.. JI. J. Cham. Educ. 1960,
37.690.
14. Fieser. L. F. Oreonie Experiments: Heath: Sostan,
1964:p 190.
15. Mort. C. P.Ezperimenlol
O~gonicChemiztly; Wiley:
New York, 1988:p 274. 16. Chem. Eng. News 1955. (5 September), 3676-3679.
