Page 240 - Physical Chemistry

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Section 7.4
As noted in Prob. 6.60, a least-squares fit assumes that the y data points have Solid–Solid Phase Transitions
about the same relative precision. Since it is the values of P and not the values of
ln (P/torr) that were measured and that have about the same relative precision,
the procedure of linearizing the data by taking logarithms is not the best way to
get an accurate H . The best way to treat the data is to minimize the sums of
m
the squares of the deviations of the P values. This is easily done in Excel using
the Solver (Sec. 6.4) with the parameters from the straight-line fit as the initial
guesses. We found from the linear fit that ln (P/torr) m(1/T) b,so P/torr
e b m/T . Enter the straight-line-fit values 20.43627 for b and 5141.24 for m
in cells F10 and G10. Enter the formula =EXP($F$10+$G$10/B3) in F3
and copy it to F4 through F7. The fitted and experimental values of P are in
columns F and C, respectively, so to get the squares of the residuals, we enter
=(F3-C3)^2 into G3 and copy it to G4 through G7. We put the sum of the
squares of the residuals into G8. Then we use the Solver to minimize G8 by
changing F10 and G10. The result is something like m 5111.26 and b
20.34821. The sum of the squares of the residuals has been reduced from 2.73 for
the linear-fit parameters to 0.98, so a much better fit has been obtained. With the
revised slope, we get H 42.50 kJ/mol, which is in better agreement with
m
the best literature value 42.47 kJ/mol.
[An alternative to using the Solver is to use a weighted linear least-squares
fit; see R. de Levie, J. Chem. Educ., 63, 10 (1986).]
Exercise
Set up the spreadsheet of Fig. 7.8 and use the Solver to verify the nonlinear-fit
m and b values given in this example.

7.4 SOLID–SOLID PHASE TRANSITIONS
Many substances have more than one solid form. Each such form has a different crys-
tal structure and is thermodynamically stable over certain ranges of T and P. This phe-
nomenon is called polymorphism. Polymorphism in elements is called allotropy.
Recall from Sec. 5.7 that in finding the conventional entropy of a substance, we must
take any solid–solid phase transitions into account. (Polymorphism is very common
with crystals of drugs. Which drug polymorph is obtained depends on the details of the
manufacturing process, and the most stable form at a given T and P might not be the
form that crystallizes. Different polymorphs may have different solubilities and sub-
stantially different biological activities. See A. M. Thayer, Chem. Eng. News, June 18,
2007, p. 17.)
Part of the phase diagram for sulfur is shown in Fig. 7.9a. At 1 atm, slow heating
of (solid) orthorhombic sulfur transforms it at 95°C to (solid) monoclinic sulfur. The
normal melting point of monoclinic sulfur is 119°C. The stability of monoclinic sul-
fur is confined to a closed region of the P-T diagram. Note the existence of three triple
points (three-phase points) in Fig. 7.9a: orthorhombic–monoclinic–vapor equilib-
rium at 95°C, monoclinic–liquid–vapor equilibrium at 119°C, and orthorhombic–
monoclinic–liquid equilibrium at 151°C. At pressures above those in Fig. 7.9a, 10
more solid phases of sulfur have been observed (Young, sec. 10.3).
If orthorhombic sulfur is heated rapidly at 1 atm, it melts at 114°C to liquid sulfur,
without first being transformed to monoclinic sulfur. Although orthorhombic sulfur is
thermodynamically unstable between 95°C and 114°C at 1 atm, it can exist for short
periods under these conditions, where its G is greater than that of monoclinic sulfur.
m