In the Classroom edited by
Applications and Analogies
Ronald DeLorenzo Middle Georgia College Cochran, GA 31014
The Gravity of the Situation Damon Diemente 505 East Seventy-Ninth Street, New York, NY 10021
Every attraction and repulsion in nature is due to one of the four fundamental forces: gravitation, electromagnetism, the strong nuclear force, or the weak nuclear force. In highschool chemistry courses the two nuclear forces are scarcely mentioned, even in chapters devoted to atomic structure and radiochemistry, and most students may be unaware of the existence of these forces. But gravitation and electromagnetism are familiar, and students readily recognize both their existence and their importance. Without the steady pull of Earth’s gravity, a balance or a mercury barometer would not function, and there would be no law of conservation of mass, no table of atomic weights, no field of stoichiometry based on mass measurements. Chemical bonding, involving electrons and nuclei, ions and dipoles, electronegativity differences and charge-cloud densities, is clearly dependent on the electromagnetic force. Indeed, when discussing chemical bonding, teachers make the usually implicit assumption that electromagnetic forces alone need be considered; gravitational forces are simply ignored. Yet the reasons for this are not necessarily intuitive. Students realize that gravitational forces are expected to be small between particles as light as atoms, but they also know that these tiny masses are separated by minuscule distances, and that this latter condition favors strong gravitational attraction. How can we be sure that gravity makes no contribution to chemical bonding? This article presents a few straightforward and persuasive calculations demonstrating that gravitational attraction between atoms in tangential contact is many orders of magnitude weaker than the gravitational attraction between Earth and an atom on its surface, and that the gravitational attraction between two oppositely charged ions is likewise many orders of magnitude weaker than the electromagnetic attraction between them. Thus, the customary treatment is entirely justified: gravitational forces are trivially small at the atomic level. We begin with a calculation of the gravitation force of attraction between Earth and a hydrogen atom on its surface, readily performed with Newton’s law of universal gravitation, f G = ᎑ G m1 m2 / d 2
(1)
in which the masses m1 and m2 are of Earth and of the hydrogen atom, and the separation d between them is Earth’s radius. The necessary terrestrial values are conveniently found in most elementary astronomy books (1, 2); the atomic mass of hydrogen can be taken from any periodic table and converted from atomic mass units to grams by division by Avogadro’s number. Because we are more concerned here with order of magnitude than with close accuracy, all measurements in this article are quoted to one significant digit: gravitational constant: G = 7 × 10 ᎑ 11 N m 2/ kg 2 mass of Earth: me = 6 × 1024 kg mass of hydrogen atom: mH = 1 amu = 1/(6 × 1023) g = 2 × 10 ᎑ 27 kg radius of Earth: re = 6 × 106 m
We use these values in eq 1 to find the gravitational attraction between Earth and a hydrogen atom: f G(Earth,atom) = ᎑ G me m H / re 2 = ᎑2 × 10 ᎑26 N
(2)
where the negative sign indicates that the force is attractive. To calculate the gravitational attraction between two hydrogen atoms, we set each mass in eq 1 equal to mH, and we estimate the distance between the two atoms at one ångström unit, which is about as close as hydrogen atoms (i.e., the nuclei of adjacent atoms) ever get to each other (two Bohr radii, 2ao = 1.06 Å): dH = 1 Å = 10 ᎑10 m and we determine fG(atom,atom) = ᎑ G mH mH / dH 2 = ᎑3 × 10 ᎑44 N
(3)
By comparing eqs 2 and 3, we see that the gravitational attraction between Earth and a hydrogen atom is some 10 18 times greater than the gravitational attraction between two hydrogen atoms at the distance of closest approach. If instead of hydrogen atoms we consider atoms of atomic mass 100 amu, we find that Earth attracts the atom 10 16 times more strongly than the atoms attract each other at one ångström separation. Now, we know that Earth’s gravity is too weak to keep small atoms and molecules like He and H2 from escaping the atmosphere. The far weaker gravitational attraction between two atoms is therefore not expected to be great enough to keep them in each other’s vicinity. This suggests that gravitational attraction plays no significant part in chemical bonding. An even more persuasive case for the weakness of interatomic gravitational attraction can be built through direct comparison with a typical interatomic electromagnetic attraction. For this, we make use of Coulomb’s law (eq 4) to calculate the attraction between an anion of charge 1᎑ and a cation of charge 1+ at a distance of one ångström, these all being reasonable estimates for real chemical situations. Again, the necessary values can be found in most elementary physics (3) or chemistry textbooks: electromagnetic constant: ionic charges: charge separation:
k = 9 × 10 9 N m2 /C2 q + = ᎑q ᎑ = 2 × 10 ᎑19 C d = 1 Å = 10᎑10 m
electromagnetic attractive force: f E(ion, ion) = k q +q᎑ /d 2 = ᎑ 4 × 10᎑8 N
(4)
Comparing eq 3 with eq 4, we see that the electromagnetic attraction between the two ions is about 1036 times stronger than their gravitational attraction. We are indeed justified in ignoring gravity as a force to contend with in this example. Of course, we can narrow the gap of 36 orders of magnitude by making a different comparison, say between two atoms of mass 100 amu (which increases the gravitational attraction) that bond together by London dispersion forces (which are as much as 10 4 times weaker than 1+/1᎑ ionic
JChemEd.chem.wisc.edu • Vol. 76 No. 1 January 1999 • Journal of Chemical Education
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In the Classroom
attraction) (4). These alterations increase the estimated gravitational attraction by (102) 2 and decrease the estimated electromagnetic attraction by 104. Their combined effect on the gap of 36 orders of magnitude is to reduce it by eight orders, which puts the attraction due to London dispersion forces at 1028 times stronger than the gravitational attraction. Again, we are amply justified in ignoring the effects of gravity at the atomic and molecular level.
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Literature Cited 1. Struve, O. Elementary Astronomy; Oxford University Press: New York, 1959. 2. Kaler, J. Astronomy!; Addison Wesley: New York, 1997. 3. Zitzewitz, P.; Neff, R.; Davids, M. Physics, Principles and Problems; Glencoe/McGraw-Hill: Westerville, OH, 1995. 4. Zumdahl, S. Chemistry; Houghton Mifflin: Boston, 1997.
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
