GAUSSIAN DISTRIBUTION SUPERIMPOSED ON ACTUAL DISTRIBUTION
INCREMENT
Figure 10. Experimentaland calculated Gaussian distributionsfor the water drop experiment. Automated Data Acquisition A short QUICKBASIC oroeram was written to acauire the data an> output it to fiie. Once the cable is a t t i h e d and the droo rate is adiusted to 1 dro~l20s, the oromam prompts for'the number of drops to becolle&d. ~ a t cola lection requires about 20 min and produces a spreadsheetreadable file t h a t is written to disk. Copies of t h e QUICKBASIC program and the spreadsheet shown in Figure 8 are available from the authors.
a
Spreadsheet Statistical Analysis The drop-weight data may be used to illustrate the concept of a distribution function. Advantages in experiments that perform this task and the necessary background material have appeared in this Journal (14).Figure 8 shows a spreadsheet page from a n abbreviated (35-point) data set. We call this a work sheet. At the conclusion of the water drop experiment, only the first two columns have data entries.
Column 1 contains the drop number. Column 2 contains the total accumulated weight after a drop has fallen and halanee stabilizationhas occurred. Column 3 calculates the mass of individual drops by subtracting successive mass readings in column 2. The minimum, maximum, average, and standard deviation of the individual drop masses are reported at the hottom ~ ~ -of ~- -column 3. These are calculated usine the soreadsheet's statistical functions. The average drop mas; is all that is reauired to calculate the surface tension of the liauid. The kmaining columns in the work sheet are used foi the statistical analysis of the data. The entries in column 4 are labeled EXPER. RESIDUAL and represent the difference between a single drop mass and the averme &ODmass. If classical ~rohahilitytheory applies, theseresid;als should approximate a daussi& distribution. The umer right three columns (5 through 7) are used to calculaiLthe number ofexperimental residuals that fall in a given value range column 5, FREQUENCY DISTRIBUTION BINS,. The results are displayed in a frequency distribution table tcolumn 6, EXPER. FREQ. 1)ISTKB... Thcsc rcsults are normalized by lvidingeach value hy the residual count (351, and the results arc shown in the column 7 rNORM. EXPI.:I(. PROB. DENSITYJ. ~~
~
The lower three columns on the right-hand side of the work sheet are used to calculate the eaussian distribution of residuals. The cnlculation uses the individual residual (Axi) and the standard deviation of the drop residuals (0).
Aeain. the entries in this column mav be normalized bv diviYdingby the sum of column entries (6.00035) to produck the entries in column 7 (NORM. GAUSSIAN PROB. DENSITY). Figure 10 compares the Gaussian probability densities with the exoerimental ~robabilitvdensities. This experimeni is ideal fordoing a propagation of errors calculation. Based on our micrometer measurements of the diameter of the drill, a reasonable estimate of the error in the diameter was determined to be 0.001 in. Because this error changes the value of the correction factor F by a negligible amount, i t was assumed constant. Using the standard equation for the propagation of errors below gives a value of q 0.34 dyneslcm.
Our value is outside the expected range of experimental error. This result and all of the student data had errors that produced a surface tension that is too high. Harkins (10) suggests that it takes at least 1min to reach thermodynamic equilibrium. Drops that do not attain equilibrium will have drop masses that are too large, leading to an error in the direction we observe.
Simulation of X-Ray Powder Diffraction Qian Pu University of Suzhou Suzhou 215006 People's Republic of China
X-ray powder diffraction is a very important experiment in physical chemistry (1548).Students determine the lattice type and the lattice constants of a cubic crystal from the X-ray diffraction pattern using a powder sample. Then they calculate the size of the unit cell and the density of the material. For lack of funds, many laboratories do not have X-ray equipment available. Each student is given an X-ray powder photograph from which the required measurements mav be carried out. Althoueh there is Dedaeo&dcal value in det&nining crystal structure by th;s met
