edited by: JOHN J. ALEXANDER
exam que~tionexchange
University of Cincinnati Cincinnati. Ohio 45221
The Squeezed-Earth Problem Gale Rhodes University of Southern Maine
e = (8r:)'"
Portland, ME 04103
An essential aspect of teaching chemistry is helping the student to make logical connect& between the seen and the unseen-between the bulk, visible properties of matter and its atomic and molecular constituents. In no area of chemistry are the logical connections tighter than in the chemistry of the solid state. For many beginning students, the continuity, consistency, range, and power of chemical explanations first become clearly apparent in discussions of the crystal.and how its properties allow the chemist to measure atomic radii and interatomic distances. At the end of my coverage of solid-state chemistry, I sometimes pose the following problem, usually as a take-home exercise because it is too long for use as an exam prohlem and also because it can provoke classroom conversation on the application of chemical principles in a wide range of areas. I then discuss it in the next class, usingit tosummarize several aspects of the chemistrv of solids. Another nurnose of this assignment is to demon.;trutr that the neutrun and proton, the "hard svherrs" of general cl~emi.itrv.ultimatelv confront us with thesame conceptual difficultiesas does the electron.
. .~~~ ~
= rD(8j1/'
(3)
The volume (V) of the unit eell is
v = e3 = r,3((8)'")3
(4)
V = (0.5 X 10-13 ~ r nX) ((x)'")~ ~
(5)
Density of Closest-packed Neutrons
The face-centeredcubic cell contains four neutrons; therefore, its density ( p j is 4 neutrons =
unit cell
X
1/67 X l.0Z4g unit eell neutron 2.8 x 10-39 cm3
(7)
Radius (0 of the Spherical Neutron Crystal
~
The volume (V) of a sphere is
v = (413)sG
(9)
Therefore, The question is straightforward: What would be the radius of the earth if its mass were squeezed into a sphere having the same density as closest-packed neutrons? When I assign this prohlem, I briefly describe the transformation of a star by supernova compression and gravity into a neutron star', with electrons collapsing into the nucleus and combining with protons to form neutrons. I return to this topic when we discuss the solution. I also ask the students what physical constants they will need inprder to solve the prohlem, and I make sure to send them off knowing the mass of the earth2 (5.98 X loz7g), and the mass3 and diameter4 of the neutron (1.67 X g; 1 X 10-l3 cm). Finally, I point out that there is some uncertainty in the neutron's diameter (note that I give only one significant f i ~ u r e )because it, though 2000 times heavier than the electron, has easily meisurabre wavelike properties. I add that, in fact, the neutron, like the electron, has a spin of '/z and has all the quantum-mechanical foibles of the electron. Their last tidhit of food for thought is that there is uncertainty in hoth the magnitude and the meaning of the diameter of the neutron. Acceptable Solution Dimensions of the Unit Cell
The face diagonal (d) of the face-centered euhic unit cell is equal length to four neutron radii (r,, = 0.5 X 10-'3 cm). The unit cell edges ( e )form a right triangle with d as the hypotenuse: therefore,
in
' Clark, G. S. Sci. Amer. 1977, 237, (4),42.
Hartmann. W. K. "Astronomy: The Cosmic Journey": Wadsworth: Belmont, CA, 1978; p 112. Brady, J. E.: Humiston, G. E. "General Chemistry": Wiley: New York. 1982; p 68. Gamow. G. "Mr. Tompkins in Paperback": Cambridge University: London, 1969; p 137.
970
Journal of Chemical Education
Because V = massip,
Substitutingfor p from eq 8, and form, the mass of the earth,
The radius was given to one significant digit, so We would obtain the same radius whether we assumed hexagonal or cubic closest packing, but the latter is equivalent to face-centered cubic, and the face-centered cube makes the prohlem easier. In the next class, I return to a n intermediate result, the density of this neutron crystal, and to the neutron star itself, because the density obtained is 2.4 X 1015g/cm3, but astrophysicists estimate' that neutron stars are somewhat less dense: between 1 and 4 X l O I 4 g/cm3. I point out that the uncertainty in the diameter of the neutron is large enough to account for this discrepancy. Nevertheless, both nuclear matter and neutron stars are best modelled as liquids rather than as solids, suggesting that a neutron star stops short of collapsing into a packed array of neutrons. Then I raise this question: why cannot neutrons adopt a crystalline form such as cubic closest packing? Why are hoth the strong nuclear force and the incredible gravitational forces that produce the neutron star unable to produce this crystal of neutrons we are imagining? Can you answer this question before you read on? The key to this puzzle is that neutrons, having a spin of K, are fermions and obey the Pauli exclusion principle. For this
reason, a t densities near l O I 4 glcm3, they enter a degenerate state that prevents further collapse of the neutron star.' In the language of introductory chemistry, this means that if neutrons were,to adopt a closest-packed configuration, they would then be in identical quantum states, a condition forbidden for spin-lI2 particles. So matter is prevented from collapsing further, not by the charge repulsion that, in normal solids, separates packed atoms by well-defined distances, but by another form of "repulsion", a forbidden quantum-mechanical state.
The student, having met the Pauli principle in only one of its domains-the electronic energy levels of atoms and molecules-is usually quite surprised to encounter the same idea in such a different context. I hope that further reflection leads the student t o a deeper realization that the unseen world lies both below and above us, these two realms represented by the scales of the atom and the star. Between them, a t our rather narrow eye level, lie the few objects and properties we can experience directly and from which we must infer the unseen world.
Volume 63
Number 11 November 1966
971
