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A Note on “A Thermodynamic Analysis to Explain the Boiling-Point Isotope Effect for Molecular Hydrogen” Lilia Meza-Montes Instituto de Fisica, UAP, Apdo Postal J-48, Puebla, Pue. 72570, Mexico Peter Hoffmann-Pinther* Department of Natural Sciences, University of Houston-Downtown, Houston, TX 77002-1001;
[email protected] Recently, two authors attempted to provide undergraduate college chemistry students with a pedagogical explanation for the normal boiling points of H2, D2, and T2 (1). In an extensive review of chemistry texts, they could not find an explanation for the increase of boiling point of these isotopes with mass. They considered the variation in the London forces for the molecules and found that this did not explain the difference in the boiling points. Therefore they looked to a difference in masses for the explanation. Discussion To explain the differences in the boiling points of molecular hydrogen, the authors considered a molecule in a one-dimensional simple harmonic oscillator potential. The corresponding energy levels depend inversely on the square root of the mass. Boiling occurs when a molecule surmounts the potential. Two constants were introduced and determined by two of the boiling temperatures. The third temperature was then predicted. This approach requires an infinite energy for a molecule to dissociate from the liquid. To overcome this difficulty, the authors took the energy required for the particle to dissociate from the liquid as the energy difference between the molecule outside the well (one of the constants) and its ground state in the well.
A simpler and clearer pedagogical approach is to consider the molecule in a potential that reflects the intermolecular forces that have been introduced, at least qualitatively, in many chemistry texts. The potential should provide an easily calculable dissociation temperature and should be such that it can be discussed qualitatively by the use of a diagram showing the potential and the energy levels. Two such potentials in use are the Lennard–Jones and the Morse potentials. Both of these potentials can be used to represent intermolecular forces in liquids (2) and provide finite dissociation energies. The Lennard–Jones potential in its basic form contains two terms proportional to inverse powers of r, the intermolecular distance, while the Morse potential uses two exponential terms. The latter potential introduces a small error in the well shape but this replacement has little effect on the molecular energy levels. When the Morse potential is inserted in the Schrödinger equation, the equation yields an analytic solution (3) that provides energy levels and their mass dependence. The energy levels can be adjusted with one free parameter (in the present case). The Morse potential is V(r) = V0(e᎑2a(r–r′) – 2e᎑a(r–r′))
where the product ar′ determines the sharpness of the minimum at r′, the equilibrium intermolecular distance, and V0 is positive. The potential is shown in Figure 1. The value of r′ is determined from the density of H2 and is taken to be the same for all three molecules. The energy levels associated with this potential are
En = n + 1 2
V
r'
r E0
-V0
Figure 1. The Morse potential as a function of the intermolecular distance r and the lowest energy level E0. The potential is finite at r = 0.
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(1)
2V 0 µ ah –
n + 1 ah 2 2µ
2
– V0
(2)
where µ is the reduced nuclear mass. The energy has an additional µ᎑1 dependence to that of the traditional harmonic oscillator. Boiling occurs when a molecule obtains sufficient kinetic energy through heating to surmount the potential. That energy is |E0 | (see Fig. 1). The boiling temperatures are determined from kT = |E0 |
(3)
where k is the Boltzmann constant and T is the temperature in kelvins. The value of V0 (used for all three molecules) is determined by using the lowest energy in eq 2 and the boiling point of H2, and by varying a to obtain the value of V0 that best reproduces the boiling points. The value of a is 1.035/r′ with r′ = 2.24 × 10᎑8 cm, which corresponds to the 5% of the minimum of the potential at r = 4.55r′. The value of V0 is 2.75 × 10᎑3 eV. The boiling points for the three molecular forms are given in Table 1.
Journal of Chemical Education • Vol. 78 No. 3 March 2001 • JChemEd.chem.wisc.edu
Information • Textbooks • Media • Resources Table 1. Normal and Calculated Boiling Temperatures for Three Forms of Molecular Hydrogen Normal Boiling Temperature/K
Calculated Value/K
H2
20.4
20.40
D2
23.5
23.51
T2
25.0
24.96
Molecular Form
Note: Data from ref 4.
Summary
Therefore the model leads to an easier understanding of the concepts by college students in beginning and advanced chemistry courses. Acknowledgments We thank S. E. Ulloa for discussions on the potentials and the Department of Physics and Astronomy at Ohio University for its hospitality and support. One of us (L.M.-M.) also acknowledges support from CONACyT–Mexico. Literature Cited
The boiling points for the three molecular forms of hydrogen are reproduced by a simple, more realistic model, with a single parameter, a. The model introduces quantum concepts that can be couched in qualitative terms by using a graph to illustrate the potential, the energy levels, and the finite energy required for boiling. The graph can easily be drawn using the minimum and the crossing at zero energy.
1. Baker, D. B.; Christmas; B. K. J. Chem. Educ. 2000, 77, 732. 2. Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic: New York, 1992; Chapter 7 3. Messiah, A. Quantum Mechanics, Vol. II; Wiley: New York, 1962; p 800. 4. Umland, J. B.; Bellama, J. M. General Chemistry, 2nd ed.; West: Minneapolis/St Paul, 1994; p 809.
JChemEd.chem.wisc.edu • Vol. 78 No. 3 March 2001 • Journal of Chemical Education
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