Physical Chemistry Third Edition

3.5 The Third Law of Thermodynamics and Absolute Entropies 145

If a metastable crystal exists in which each molecule can occur with equal probability

in either the equilibrium state or the reversed state, then

Ωorient 2 N (metastable CO crystal) (3.5-12)

whereNis the number of molecules in the crystal. The molar statistical entropy of the

metastable crystal near zero temperature is

Sst, mkBln(2NAv)NAvkBln(2)Rln(2) 5 .76JK−^1 mol−^1 (3.5-13)

This value agrees with the amount by which carbon monoxide appeared to deviate from

the third law.

Exercise 3.21

Pretend that you have synthesized 1.00 mol of CaCO 3 in which each carbonate ion has one^16 O

atom, one^17 O atom, and one^18 O atom. Assume that every calcium atom is the same isotope

and that every carbon atom is the same isotope. Calculate the entropy of the metastable crystal

near zero temperature, if nothing is known about the orientations of the carbonate ions except

that each equilibrium oxygen position is occupied by an oxygen atom of some isotope.

Trouton’s Rule

Trouton’s rule is an empirical rule for estimating entropy changes of vaporization.

It states that for “normal” liquids the molar entropy change of vaporization,∆vapSm,at

the normal boiling temperature (at 1.000 atm) is roughly equal to 10. 5 R≈

88JK−^1 mol−^1. If you need a value for∆vapSmor∆vapHmand do not have data,

you can use Trouton’s rule to obtain an estimate. Trouton’s rule underestimates the

entropy change of vaporization for liquids such as ethanol and water, in which there is

considerable molecular association. Trouton’s rule also badly overestimates the entropy

change of vaporization for hydrogen and helium. Modifications of Trouton’s rule have

been proposed, including a version that uses entropy changes of vaporization to form

gases with the same value of the molar volume instead of whatever molar volume cor-

responds to 1 atm pressure at the normal boiling temperature. This modified rule seems

to be more closely related to the formula for∆Sstin Eq. (3.4-19) than the original

version. The values for hydrogen and helium fall closer to those of other substances if

this modified rule is used. There is also a method in which contributions for different

groups of atoms in the molecule are considered.^9

EXAMPLE3.18

Use Trouton’s rule to estimate∆vapHfor methane from its normal boiling temperature,

− 164 ◦C. Compare with the correct value in Table A.7 of the Appendix.

Solution

∆vapHm≈(109 K)(88 J K−^1 mol−^1 )9600 J mol−^1

The value in Table A.7 is

∆vapHm(555.19Jg−^1 )(16.043 g mol−^1 )8907 J mol−^1

(^9) D. Hoshino, K. Nagahama, and M. Hirata,Ind. Eng. Chem. Fundam., 22 , 430 (1983).