Page 238 - Physical Chemistry

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Equation (7.24) is generally accurate up to a few hundred atmospheres but fails Section 7.3
for larger pressure differences, due to the changes in S, V, and H with The Clapeyron Equation
fus
fus
fus
changes in T and P along the equilibrium line. For example, on the H O solid–liquid
2
equilibrium line, the following data are observed:
t 0°C 5°C 20°C
P/atm 1 590 1910
3
( H/ V)/(kJ/cm ) 3.71 3.04 1.84
fus
fus
3
( S/ V)/(J/cm -K) 13.6 11.3 7.26
fus
fus
For most substances, V V V solid is positive. Liquids are more compress-
fus
liq
ible than solids, so as P fus increases, V decreases faster than V solid and V decreases.
liq
fus
For a few substances, V is positive at low P fus values and becomes negative at high
fus
P . Here, the slope of the solid–liquid line changes sign at high pressures, producing
fus
a maximum in melting point at the pressure where V 0. Figure 7.7 shows the Solid
fus
melting-point line on the P-versus-T phase diagram of europium.
In applying the Clapeyron equation dP/dT H/(T V) to phase transitions Liquid
involving only condensed phases (solid–liquid or solid–solid), we approximated V
as constant and calculated V from the experimental densities of the two phases. In
applying the Clapeyron equation to transitions involving a gas phase (solid–gas or
liquid–gas), we neglected V of the condensed phase and approximated V as V , Figure 7.7
gas
where we used the ideal-gas approximation for the gas volume; these approximations
are valid well below the critical point. Melting point of europium versus
pressure. (The melting-point line
vap H from Linear and Nonlinear Least-Squares Fits of graphite also shows a
m
Example 7.6 shows how a spreadsheet is used to find H from vapor-pressure data. temperature maximum.)
m
vap
EXAMPLE 7.6 H from linear and nonlinear least-squares fits
vap m
Accurate vapor-pressure data for water are [H. F. Stimson, J. Res. Natl. Bur.
Stand., 73A, 493 (1969)]
t /°C 40 50 60 70 80
68
P/torr 55.364 92.592 149.510 233.847 355.343
where the temperatures and pressures are estimated to be accurate to within
3
4
about 10 % and 10 %, respectively, and t denotes the now obsolete
68
International Temperature Scale of 1968. Find H of water at 60°C.
vap m
If the three approximations that give Eq. (7.19) are made, then Eqs. (7.20) and
(7.22) show that a plot of ln (P/torr) versus 1/T will be a straight line with
slope H /R.We use appendix II of Quinn to convert the temperatures to ITS-
m
90 (Sec. 1.5). The data are entered into a spreadsheet (Fig. 7.8) and ln (P/torr) and
1/T are calculated in columns D and E. For columns B, D, and E, only the formulas
in row 3 need be typed; the others are produced using Copy and Paste (or in Excel
by dragging the tiny rectangle at the lower right of a selected cell). To use Excel
2003 to get the coefficients m and b that give the best least-squares fit to the straight
line y mx b, choose Data Analysis from the Tools menu and then choose
Regression in the scroll-down list and click OK. If Data Analysis is not on the
Tools menu, choose Add-Ins on the Tools menu, check Analysis ToolPak and click
OK. (In Excel 2007, click on the Data tab and then click on Data Analysis and
choose Regression. If Data Analysis is not on the Data tab, click the Office Button
at the upper left and then click Excel Options; click Add-Ins and select Excel Add-
Ins in the Manage box. Click Go. In the Add-Ins available box, click the Analysis
ToolPak check box and click OK.) In the dialog box that opens after Regression is