Warren Platts

2007-Oct-30, 06:42 PM

Say you have some metallic hydrogen under the pressures and temperatures characteristic of the interior of gas giant planets. If you raise the metallic hydrogen to a zone where the PT is low enough (or conversely lower the zone as the planet cools), the metallic hydrogen atoms will recombine to form H2, releasing energy according the following formula:

H + H → H2 + energy

So the question is, how many Joules will be released when two moles of H combine to form one mole of H2?

At standard temperatures and pressures, my old organic chemistry textbook says that the energy of dissociation of the H-H bond is 104 kcal per mole. At 4.1868 kJ per kcal, this works out to 435.4 kJ/mol. This agrees with my CRC Handbook (85th edition) where it says (e.g. p. 9-55) that the bond strength (Do298) at 298oK of H-H is 435.990 kJ/mole.

Of course the temperatures typical of the interior of gas giants are in the thousands of degrees K. So, I would like to know Do6000. The CRC Handbook (p. 9-52) gives the following approximate relation:

Do298 = Do0 + (3/2)RT

where R is the gas constant. Since I the Handbook gives me Do298, I can solve for Do0 (the bond strength at absolute zero), which is 432 kJ/mole. Thus, by resubstituting the 432 kJ/mole for Do0 into the above equation, and solving for 6,000o K, I get a value of:

Do6000 ~ 500 kJ/mole

So far so good. The problem is I can't reconcile this figure with the following cryptic reference in my Jupiter book (Bagenal et al., Jupiter: The Planet, Satellites, and Magnetosphere [2004], p. 41):

[A]s the planet cools, a fraction of the mass of the envelope is converted from one phase to the other with an associated latent heat release (or absorption). The effect on the evolution is not very pronounced for a laten heat of ~0.5kB (Saumon et al. 1992 (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1992ApJ...391..827S&data_type=PDF_H IGH&whole_paper=YES&type=PRINTER&filetype=.pdf)).

where kB is Boltzmann's constant. The Saumon et al. (1992, 828) paper says that the latent heat figure is actually an "entropy jump . . . with the entropy of the metallic phase being higher than that of the molecular phase." If you work that figure out for one mole of H2, it's just Boltzman's constant times Avogadro's number which is 8.314 J mole-1 K-1, and that's just the ordinary gas constant R. The CRC Handbook (6-19) standard S for dihydrogen is 130.7 J mole-1, which is quite a bit higher than 8.314.

So which figure should a person interested in modeling the metallic-molecular phase transition use?

H + H → H2 + energy

So the question is, how many Joules will be released when two moles of H combine to form one mole of H2?

At standard temperatures and pressures, my old organic chemistry textbook says that the energy of dissociation of the H-H bond is 104 kcal per mole. At 4.1868 kJ per kcal, this works out to 435.4 kJ/mol. This agrees with my CRC Handbook (85th edition) where it says (e.g. p. 9-55) that the bond strength (Do298) at 298oK of H-H is 435.990 kJ/mole.

Of course the temperatures typical of the interior of gas giants are in the thousands of degrees K. So, I would like to know Do6000. The CRC Handbook (p. 9-52) gives the following approximate relation:

Do298 = Do0 + (3/2)RT

where R is the gas constant. Since I the Handbook gives me Do298, I can solve for Do0 (the bond strength at absolute zero), which is 432 kJ/mole. Thus, by resubstituting the 432 kJ/mole for Do0 into the above equation, and solving for 6,000o K, I get a value of:

Do6000 ~ 500 kJ/mole

So far so good. The problem is I can't reconcile this figure with the following cryptic reference in my Jupiter book (Bagenal et al., Jupiter: The Planet, Satellites, and Magnetosphere [2004], p. 41):

[A]s the planet cools, a fraction of the mass of the envelope is converted from one phase to the other with an associated latent heat release (or absorption). The effect on the evolution is not very pronounced for a laten heat of ~0.5kB (Saumon et al. 1992 (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1992ApJ...391..827S&data_type=PDF_H IGH&whole_paper=YES&type=PRINTER&filetype=.pdf)).

where kB is Boltzmann's constant. The Saumon et al. (1992, 828) paper says that the latent heat figure is actually an "entropy jump . . . with the entropy of the metallic phase being higher than that of the molecular phase." If you work that figure out for one mole of H2, it's just Boltzman's constant times Avogadro's number which is 8.314 J mole-1 K-1, and that's just the ordinary gas constant R. The CRC Handbook (6-19) standard S for dihydrogen is 130.7 J mole-1, which is quite a bit higher than 8.314.

So which figure should a person interested in modeling the metallic-molecular phase transition use?