Straightening Plots:An Example

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…from CheLabWiki, an online resource for chemical-engineering laboratories located at www.chelabwiki.org; Site Revision #759; 6 January 2009.

To illustrate^[1] the process for straightening x-y plots, we consider the vapor pressures of water from its triple point to its critical point.^[2]

Original Data on Linear Axes

We begin by plotting the original data on linear axes; that is, we plot vapor pressures P^s against absolute temperature T. The result is in Figure 1. This is a highly nonlinear plot; in fact, the value of the linear correlation coefficient is r = 0.85.

Figure 1. Vapor pressures of water from the triple point (273.16 K) to the critical point (647.10 K). r = 0.85. Data from NIST Chemistry Webbook.

To straighten the plot, we compare the shape in Figure 1 with each quadrant of the Tukey Circle shown in Figure 2. The curve in Figure 1 has both (dP^s / dT) and (d²P^s / dT²) > 0; that is, it has the shape of the curve in the fourth quadrant of the Tukey Circle. Recall that the Tukey Circle tells us which directions to move on the progressions of functions in x and y so as to straighten a curve.

Figure 2. The Tukey Circle

Replot on Semilog Axes

The 4th quadrant of the Tukey Circle suggests that we could straighten the curve either by moving down the vertical progression of functions in y or by moving to the right on the horizontal progression of functions in x. We choose to move down the vertical progression from y to ln y, so we replot the data in semilog form:ln(P^s) vs T.

This new plot is shown in Figure 3. Note in the figure that we obtain ln(P^s) by using a log scale on the ordinate. The curve in Figure 3 still has some curvature; for this plot the linear correlation coefficient is now r = 0.97. But its curvature is opposite to that in the original plot of Figure 1. We have over-corrected in our attempt to straighten the curve in Figure 1.

Figure 3. Vapor pressures of water from Figure 1, replotted on semilog axes in an attempt to make the curve linear. r = 0.97.

Replot on Log-Log Axes

The shape of our new curve in Figure 3 corresponds to that in the second quadrant of the Tukey Circle in Figure 2. To straighten it, we must (according to Figure 2) move up in y or move to the left in x. Since we don’t want to undo what we have accomplished by moving in y, let’s now change the function for x. So we move from x to ln(x) on the progression of x-functions: we plot ln(P^s) vs ln(T), as in Figure 4.

Figure 4. Vapor pressures of water from Figure 1, plotted on log-log axes. r = 0.990.

Replot on Semilog Axes, But Use 1/T on Abscissa

The curve in Figure 4 has r = 0.990, so it is straighter than that in Figure 3, but it still retains some curvature. Therefore, we move farther to the left on the progression of functions in x, from ln(x) to 1/x. For example, if we plot ln(P^s ) vs 1/T, as in Figure 5, we obtain a nearly straight line. Indeed, this plot has r = 0.9997.

Figure 5. Vapor pressures of water from Figure 1, plotted on semilog axes of ln(P^s) vs 1/T. r = 0.9997.

The plot in Figure 5 is consistent with a fundamental thermodynamic relation, the Clausius-Clapeyron equation,^[3] which suggests that pure-component vapor pressures should form straight lines when the log of the vapor pressure is plotted against reciprocal absolute temperature. Our purely mathematical manipulations of the original vapor-pressure data produce a result that is consistent with a theoretical context provided by thermodynamics. A least-squares fit to the linear form shown in Figure 3 is presented and discussed on another page of this site.

See Also

References

1. ↑ J. M. Haile, Analysis of Data, Macatea Productions, Central, SC, 2003. ISBN 0-9728602-0-7.
2. ↑ W. G. Mallard and P. J. Linstrom, eds., NIST Chemistry Webbook, NIST Standard Reference Database Number 69, February 2000, National Institute of Standards and Technology, Gaithersburg, MD.
3. ↑ J. M. Haile, Lectures in Themodynamics: Heat and Work, Macatea Productions, Central, SC, 2002. ISBN 0-9715418-1-7