ity of obtaining computed F-ratios lower than the tabulated F-value and, therefore, of not finding a statistically significant lack of fit between the straight line model and the experimental points. In many cases, the analyst is urged to compare the slopes of different calibration lines as would be the case when comparing the slope of the calibration line obtained using aqueous standards with that obtained by following the addition standard technique. Dependinp on the presence or the absence of a statis&cally significant difference between the estimated variance of the residuals for both lines, the test is performed by computing different expressions of the t-value, which are compared, in each case, to the tabulated t-value with the corresponding degrees of freedom a t a chosen significance level (10). Many experimentalists, when calculating detection limits according to the IUPAC definition (191,continue to disregard the presence of random errors in the calibration line that is used to derive those limits. Winefordner and Long showed the effect of having errors in the slope orland intercept when computing detection limits by several approaches. The program computes detection limits according to their expression (20):
where sblstands for the standard deviation of the set of replicates corresponding to the analytical blank sample. As can be obsewed, the expression takes into account the statistics involved in the difference between a given response and the blank signal and also the uncertainty introduced by the presence of errors in both the slope and intercept of the calibration line. To summarize, the present computer program has been developed as a n aid for teaching linear calibration, using the straight line model, and method validation. It is suited to illustrate the effects of different data sets, containing a variety of random errors in the y-axis, on different statistical parameters of the regression. Due to the versatility and ease of handling of the program it should be considered as a complementary tool to the theoretical sessions. The incorporated graphical facilities help to understand the influence of errors on the calculated results. The program includes REM sentences to provide advice about the adequacy, meaning, mathematical expressions or importance of the statistical parameters computed. In spite of this, students should use the program, and play with it, only after having understood the theoretical concepts. Due to the rigorous treatment of many statistical tests incorporated, the program can be employed by all users that need to present adequate results of the linear regression process. The program, written in Turbo Basic, requires an IBM PC or one compatible with CGA graphic card or higher. It runs under operating system DOS 3.0 or higher. Either the listing or the machine-readable version, together with a brief introduction manual can be obtained from the authors. Acknowledgement
Financial support from the Spanish Ministry of Education and Science (CICYT project no. BP90-0453) has been given to R.B. and F.X.R.
232
Journal of Chemical Education
Spreadsheet Calculation of Vapor Pressures and Coexistence Curves Ian J. McNaught School of Chemistry University of Sydney Sydney, New South Wales, 2006 Australia
Thoughtful use of spreadsheets by an instructor can greatly increase a student's understanding of physical chemistry. By varying the conditions applied to a system it is possible to study easily the sensitivity of a given model to its parameters. Also by comparing models between themselves and against experimental data the strengths, weaknesses and applicability of various models can be explored. Many two parameter equations of state of the form
P =AV, 1: a, b) have been proposed for fluids. As noted by Maxwell (21) these equations can be used to predict the vapor pressure of a liquid as a function of temperature. Recently a program in BASIC to calculate the vapor pressure of a liquid as a function of temperature, based on the van der Waals equation of state, was described in this column (22). The spreadsheet described here calculates the parameters a and b from the critical pressure and temperature and then generates the vapor pressure as well as the molar volumes of liquid and vapor for a given temperature. This can be done for any of four different models of a fluid, namely those of van der Waals, Berthelot, Redlid-Kwong and Peng-Robinson. It offers the following advantages over the BASIC program: 1. it is much more rapid (typically two orders of magnitude
faster); 2. the range of temperatures over which the vapor pressure can be calculated is extended; 3. it permits exploration of several different equations of
state; 4. it can be integrated into a teaching module based on spreadsheets (23).
Numerous two parameter equations of state have been suggested. The best known is that of van der Waals (24) (eq 1)
Two others that do a betterjob of predicting vapor pressure are those of Berthelot (25) (eq 2)
and Redlich and Kwong (26) (eq 3)
Each of these equations consists of a repulsion pressure (the first term, representing a hard sphere with excluded volume b) and a n attraction pressure (the second term, with a being a measure of the interparticle attraction). The two parameters in these equations of state can be related to the critical Dressure. critical volume. and critical temperature by using the equation of state along with the observation that the first and second derivatives of Dressure with respect to volume are zero a t the critical h i n t (27).As has been discussed by Eberhart (281, for a given model the numerical values of the two parameters depend on which combination of these three constraining equations is used to calculate them. In this bpreadsheet the two
0.44
220
230
240
250
280 270 280 Temperature 1 K
290
3W
:
450
4W
590
550
BW
Temperature I K
Figure 4. Calculated/Experimentalvapor pressure for C02,
Figure 6. Experimental and calculated coexistence curves for H20.
parameters are calculated from the critical pressure and critical temperature (29). The physical rationalization for the terms in the van der Waals equation is appealing (the first term allows for the nonzero volume of the particles, the second term follows from a n attractive potential which vanes as the inverse sixth power of the interparticle distance (30)). The Maxwell wnstruction for the position of the vapor pressure line (equal areas in the two loops~is thermodnamicallv rigorous. These two aspects can lead students to con&se
