[math-fun] max negative charge on an ion -- chemists?
Plenty of atoms can be ionized to -1 charge. (I'm not sure if this is true of all of them. E.g. perhaps Helium and other noble gases cannot be; also perhaps Zinc, Magnesium, Beryllium, Nitrogen, Manganese are immune.) But no atom can be ionized to -2 charge -- i.e. no such ion is stable in isolation. There is, in fact, a naive approximation, called Thomas-Fermi-von Weizsacker, to quantum mechanics in which it is a theorem (by Benguria & Lieb) that the max possible negative charge you can put on an N-atom molecule, N>0, is at most N extra electrons. I would conjecture that asymptotically, the most negative charge you can put on an N-atom molecule is going to be constant*sqrt(N). This would be achieved for a hollow metal sphere, regarded as a giant "molecule," with thickness of order 1 atoms, radius r where N is of order r^2; charging to order sqrt(N) would give it a constant electric field at surface. But this conjecture is completely open. QUESTION: what is the most negative charge f(N) that can be put (stably) on an N-atom molecule for small values of N? E.g. sulfite (SO3) or carbonate (CO3) each with -2 charge both show f(4)>=2, assuming this ion actually is stable in isolation in vacuum. Arsenite AsO3 with charge -3 shows f(4)>=3 assuming same. The AsS (charge -3) and allylenide [C=C=C] with charge -4 ions each would be enough to refute that Thomas-Fermi-von Weisacker theorem for actual quantum mechanics, if these ions can exist stably in isolation (which I doubt, but they exist inside some ionic crystals). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
This paper [ http://www.chem.purdue.edu/kais/paper/ref101.pdf ] confirms the nonexistence of doubly charged atomic ions in the gas phase. Doubly charged molecules do exist. Experimentally found are Cn (n = 7-9), S2O6, PtX4, and PdX4 (X = Cl, Br). Other doubly charged molecules are theoretically predicted: alkali hadides MX3, BeC4, BeC6, Mg2X6, and small carbon clusters. CO3(2-) is unstable in the gas phase. The aim of this paper is to show theoretically that doubly charged He and Li can be stabilized in an intense laser field. With laser wavelength of 250 nm, a maximum detachment energy of about 1/8 eV is achieved at intensities of a few EW/cm^2. (E = exa = 10^18). -- Gene
________________________________ From: Warren Smith <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Saturday, March 2, 2013 2:39 PM Subject: [math-fun] max negative charge on an ion -- chemists?
Plenty of atoms can be ionized to -1 charge. (I'm not sure if this is true of all of them. E.g. perhaps Helium and other noble gases cannot be; also perhaps Zinc, Magnesium, Beryllium, Nitrogen, Manganese are immune.) But no atom can be ionized to -2 charge -- i.e. no such ion is stable in isolation.
There is, in fact, a naive approximation, called Thomas-Fermi-von Weizsacker, to quantum mechanics in which it is a theorem (by Benguria & Lieb) that the max possible negative charge you can put on an N-atom molecule, N>0, is at most N extra electrons. I would conjecture that asymptotically, the most negative charge you can put on an N-atom molecule is going to be constant*sqrt(N). This would be achieved for a hollow metal sphere, regarded as a giant "molecule," with thickness of order 1 atoms, radius r where N is of order r^2; charging to order sqrt(N) would give it a constant electric field at surface. But this conjecture is completely open.
QUESTION: what is the most negative charge f(N) that can be put (stably) on an N-atom molecule for small values of N?
E.g. sulfite (SO3) or carbonate (CO3) each with -2 charge both show f(4)>=2, assuming this ion actually is stable in isolation in vacuum. Arsenite AsO3 with charge -3 shows f(4)>=3 assuming same.
The AsS (charge -3) and allylenide [C=C=C] with charge -4 ions each would be enough to refute that Thomas-Fermi-von Weisacker theorem for actual quantum mechanics, if these ions can exist stably in isolation (which I doubt, but they exist inside some ionic crystals).
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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On Sat, Mar 2, 2013 at 5:39 PM, Warren Smith <warren.wds@gmail.com> wrote:
QUESTION: what is the most negative charge f(N) that can be put (stably) on an N-atom molecule for small values of N?
E.g. sulfite (SO3) or carbonate (CO3) each with -2 charge both show f(4)>=2, assuming this ion actually is stable in isolation in vacuum. Arsenite AsO3 with charge -3 shows f(4)>=3 assuming same.
Possibly relevant: http://oeis.org/A180638 "Phan Thanh Nam's upper bound on the number of non-relativistic electrons bound to a nucleus of charge n." http://arxiv.org/abs/1009.2367 Phan Thanh Nam, New bounds on the maximum ionization of atoms , arXiv:1009.2367, Sep 14, 2010. limit: 1.22z + 3z^(1/3) where z is the proton count http://pra.aps.org/abstract/PRA/v29/i6/p3018_1 E. H. Lieb, Bound on the maximum negative ionization of atoms and molecules, Phys. Rev. A 29, 3018-3028 (1984). limit: 2z+1, but superseded by Nam -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
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Warren Smith