In the Classroom
A Thermodynamic Analysis to Explain the Boiling-Point Isotope Effect for Molecular Hydrogen D. Blane Baker*† Division of Natural Sciences, Ouachita Baptist University, Arkadelphia, AR 71998-0001;
[email protected] Byron K. Christmas Department of Natural Sciences, University of Houston–Downtown, One Main Street, Houston, TX 77002-1001
An isotope effect results from the dependence of a chemical or physical property of a substance upon which particular nucleus (isotope) of a given element is present within the substance’s constituent atoms or molecules. In response to questions in our general chemistry course, this article provides an explanation for the boiling-point isotope effect observed for molecular hydrogen isotopes (H2, D2, and T2); that is, an explanation for the increase in normal boiling point with increasing molecular mass. The discussion begins by showing how traditional analyses of relative boiling-point behavior are clearly inadequate for liquids such as molecular hydrogen. Subsequently, the analysis explaining the isotope effect is presented, along with pedagogical suggestions and considerations for instructors. To elucidate the relative boiling points of chemical systems, it is customary to introduce this topic with an explanation of intermolecular attractive forces. Upon mastering the basic factors that determine the relative strengths of these intermolecular attractive forces, the student can make reasonable predictions concerning relative boiling point behavior. For most substances, this method is straightforward, provided the student is familiar with the molecular structures, the relative dipole moments, and the relative polarizabilities of the species involved. A problem arises, however, with this qualitative approach when attempting to understand and explain the relative boiling-point behavior of molecular hydrogen. Boiling-point data for molecular hydrogen (1) are given in Table 1, illustrating the increase in normal boiling point with increasing molecular mass. The boiling-point differences exhibited by molecular hydrogen isotopes are actually rather surprising, since molecular hydrogen species are nonpolar and their polarizabilities are nearly identical. Thus, a reasonable—but incorrect—conclusion is that the boiling points of these species are identical, in contradiction to the known data. Discussion In an attempt to develop an explanation for the measured boiling points of molecular hydrogen isotopes, we surveyed the chemical pedagogical literature and numerous chemistry texts to determine how others had addressed this problem. Several texts refer to the boiling-point isotope effect and present boiling-point data for the isotopes of molecular hydrogen. However, of these, not a single one provides a direct explanation for the observed differences in boiling point. † Current address: Department of Physics, William Jewell College, Liberty, MO 64068-1896.
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Table 1. Normal Boiling Points for Molecular Hydrogen Molecular Substance Boiling Point/K Diprotium, H2
20.4
Dideuterium, D2
23.5
Ditritium, T2
25.0
Shriver, Atkins, and Langford (2) do address bond dissociation energy differences for these substances, but make no mention of their boiling point differences. Since no explanation for the boiling-point isotope effect was found, the problem was investigated initially by considering differences in molecular London forces for the various isotopes of molecular hydrogen. The dependence of London forces on the polarizability of the molecular species and the energy of their first electronic transition is such that the interaction energy between a pair of molecules is proportional to the energy of the first electronic transition and the square of the polarizability (3). A comparison of the polarizabilities of H2 and D2 reveals that the measured polarizability of D2 is slightly lower than that of H2 (3), while the energies of their first electronic transitions are nearly the same. Therefore, the boiling point of liquid D2 is expected to be slightly lower than the boiling point of liquid H2. However, since the observed data contradict this hypothesis, the physical mechanism responsible for the boilingpoint isotope effect clearly does not originate from differences in London forces. As an alternative approach, a thermodynamic analysis is performed under conditions of chemical equilibrium and at standard pressure. The result reveals a relation between the normal boiling point and the standard enthalpy of vaporization, which depends on molecular mass. The enthalpy of vaporization (per molecule) is determined by evaluating the energy necessary to remove a single hydrogen molecule from the liquid phase. In the liquid phase the molecule is assumed to reside within a potential energy well created by interactions with neighboring molecules, and in the gas phase the molecule is assumed to behave as an independent particle. For the energy analysis, the following assumptions are made concerning the potential energy well: 1. the potential energy well is one dimensional, harmonic, and finite in depth; 2. the potential energy well is identical for all isotopes of molecular hydrogen; 3. the energy required to escape from the potential energy well is simply the difference between the energy of the molecule when it is completely removed from the well and its initial energy within the well.
Journal of Chemical Education • Vol. 77 No. 6 June 2000 • JChemEd.chem.wisc.edu
In the Classroom
Assumption 1 is reasonable because the potential energy of the molecule depends upon the displacement in each of three dimensions in the same fashion. Furthermore, the harmonic potential energy well is appropriate for describing dispersion effects of neighboring molecules upon the molecule in question. Assumption 2 is valid because the London potentials for molecular hydrogen isotopes are nearly identical. And finally, assumption 3 is justified because, classically speaking, the molecule within the well must acquire sufficient energy to overcome the intermolecular attractive forces due to its neighbors. Evaluating the transition of a molecule from the liquid to the gaseous state requires an expression for the molecule’s energy within a harmonic potential. Solution of the Schrödinger equation for a particle in an infinite harmonic potential shows that for a given energy state, the energies depend on mass (as m1/2) and several constants. Since our goal is to determine the variation of boiling point with molecular mass m, the approximate ground state energy E0 of the molecule (near the potential minimum) is written simply as the product of a constant C and m 1/2: E0 = Cm 1/2 (1) According to assumption 3, the energy required for “dissociation” of the particle from the liquid is the difference between the energy of the particle when it is completely removed from the potential energy well and its initial (ground state) energy. Thus, the dissociation energy Dm of the particle is given by Dm = Doutside – E0 = Doutside – Cm 1/2 (2) where Doutside is the energy of the molecule when it is completely removed from the well. To proceed with a determination of the standard enthalpy of vaporization, a pure liquid hydrogen system in chemical equilibrium with its vapor is considered. Under equilibrium conditions and at a vapor pressure of one atmosphere, the change in Gibbs free energy is zero, and the temperature of the system is fixed at the boiling point of the liquid. For a system at equilibrium, the slope of the pressure-versus-temperature phase equilibrium line is equivalent to the ratio of the change in entropy to the change in volume by the Clausius–Clapeyron equation. An approximate solution to the Clausius–Clapeyron equation yields
∆H vap ∆S ln P = + P0 RT R o
o
(3)
where P is the vapor pressure of the liquid, P0 is the standard pressure (one atmosphere), ∆H vap ° is the enthalpy of vaporization at standard pressure, T is the absolute temperature, ∆S ° is the entropy of vaporization (at standard pressure), and R is the molar gas constant. Simplification of eq 3 reveals the functional dependence of the normal boiling point upon molecular mass. At its normal boiling point, the vapor pressure of the pure molecular hydrogen liquid is one atmosphere and the left-hand side of eq 3 vanishes. Thus, the standard enthalpy of vaporization is proportional to the absolute temperature, since ∆S ° is constant for the isotopes of molecular hydrogen (see discussion below). Further, the enthalpy term involves the change in internal energy of the system along with the
change in PV, the product of pressure P and volume V. Under constant pressure conditions, the work and ∆(PV ) terms cancel and the enthalpy change is simply the heat added during the transformation process. Consequently, the standard enthalpy of vaporization is equivalent to Dm and depends on mass according to the expression in eq 2. To determine how ∆S ° in eq 3 varies among the isotopes of molecular hydrogen, the functional dependence of the entropy change is considered. Assuming that the rotational and vibrational entropy terms are the same for the liquid and gas phases, ∆S ° depends only on the translational entropies of the two phases. The translational entropy (molar) of the gas phase depends on molecular weight M, absolute temperature T, and molar volume V via logarithmic terms (4 ). Likewise, the translational entropy of the liquid phase contains similar terms for the approximation that the liquid resembles a dense gas whose free volume Vf, throughout which the molecules can maneuver, is significantly reduced compared to that of the gas (5). An evaluation of ∆S ° gives an expression of the form ln(V/Vf ), since the other terms cancel for a fixed temperature and molecular weight (mass). As a result, ∆S ° is independent of mass and is constant for the isotopes of molecular hydrogen. Returning to the simplified expression for the enthalpy of vaporization, its equivalent Dm is equated to a constant times the absolute temperature. Now solving for temperature and renaming the constants in the above equations as A and B gives the normal boiling point Tbp as a function of molecular mass m so that Tbp(m) = A – Bm 1/2
(4)
Assuming A and B in eq 4 are positive values, the normal boiling point increases with increasing molecular mass, as expected. Finally, to show quantitative correlation among the boiling points, the constants A and B are determined from the known boiling points and molecular weights (in atomic mass units, amu) of two of the molecular hydrogen species. Evaluation of the constants A and B from the boiling points of diprotium (20.4 K) and dideuterium (23.5 K) and their molecular weights (masses) gives values of 31.0 K and 15.0 K(amu)1/2, respectively. Substituting the constants obtained for A and B into the expression for Tbp(m) predicts a value for ditritium of 24.9 K, which closely agrees with the known value of 25.0 K. Thus, the relation in eq 4 reveals the functional dependence of the normal boiling point on molecular mass and allows a quantitative correlation among the boiling points, as shown. Pedagogical Considerations The analysis developed here is motivated by students’ comments and questions and is designed for presentation to undergraduates. Ideally, a presentation of the material should be reserved for physical chemistry students, since a background in quantum mechanics as well as chemical thermodynamics is recommended. Within the physical chemistry curriculum, the material is particularly relevant following a treatment of the harmonic oscillator and equilibrium thermodynamics. As an alternative, the instructor may desire to present the results to general chemistry students. In this scenario, one
JChemEd.chem.wisc.edu • Vol. 77 No. 6 June 2000 • Journal of Chemical Education
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In the Classroom
could argue that the increase in boiling point with increasing molecular mass results from a decrease in the bound state energies of the molecules with increasing molecular mass (for a given potential energy well). As a result, greater temperatures are required to initiate boiling within liquids containing heavier molecular hydrogen isotopes. For completeness, the instructor should review the initial assumptions and clearly emphasize why the analysis is appropriate for treating molecules undergoing transformation from a liquid to a gas. Summary This analysis offers a suitable explanation for a simple, yet intriguing, question: how does one explain molecular hydrogen’s increase in normal boiling point with increasing molecular mass? At first glance, the boiling-point isotope effect is rather surprising, since traditional analyses comparing molecular London potentials predict little, if any, difference in boiling points for these species. However, an analysis of the standard enthalpy of vaporization ∆H vap ° reveals that the enthalpy change depends on molecular mass. In addition, the normal boiling point is related to ∆H vap ° via a standard equation for a system at equilibrium; thus, the boiling point depends on molecular mass, as observed.
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An important feature of the analysis is that it can be incorporated into the undergraduate chemistry curriculum. Qualitatively, the boiling-point behavior can be understood from the mass dependence of the energy required to transform a molecule from the liquid to the gas. For a quantitative treatment, an expression is derived for the normal boiling point as a function of molecular mass. In either case, the instructor should be able to provide the student with an explanation of a very interesting and exceptional phenomenon of nature that is not explained in the pedagogical chemical literature. Literature Cited 1. Umland, J. B.; Bellama, J. M. General Chemistry, 2nd ed.; West: Minneapolis/St. Paul, 1996; p 809. 2. Shriver, D. F.; Atkins, P.; Langford, C. H. Inorganic Chemistry, 2nd ed.; Freeman: New York, 1994; pp 374–376. 3. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; Wiley: New York, 1980; p 411. 4. Pauling, L. General Chemistry; Dover: New York, 1970; pp 384– 393. 5. Moore, W. J. Physical Chemistry, 2nd ed.; Prentice Hall: New York, 1955; pp 428–430.
Journal of Chemical Education • Vol. 77 No. 6 June 2000 • JChemEd.chem.wisc.edu
