- #1
Sonderval
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- 11
Consider a monatomic gas of hydrogen (just to make the example as simple as possible) at a temperature T. If I use Boltzmann statistics, I would say that the probability of finding any arbitrary atom at energy E should be proportional to
##g_i e^{-E_i/(k_BT)} / Z(T)##
where ##g_i## is the state degeneracy (number of states at each level).
Thinking about this, I utterly confused myself in two ways, so I hope someone can help me out.
1. Since the states become more and more dense at higher quantum numbers n (only looking at bound states), it seems as if even at small temperatures, most atoms should be in an excited state with very large n: If I use 1Ry as the energy difference between ground state and a highly excited state (which should be a reasonable approximation), I can always find enough states close to the ionisation energy that the summed probability of each of these states is larger than that of the ground state. Obviously I'm making a mistake here, but I have no clue what it is.
2. Consider just the first to shells. Is the ratio of the probabilty of finding an atom in the ground or in the first excited state given by ##e^{-(E_2-E_1)/(k_BT)}## or do I need an additional factor of 4 because the second shell has one s and three p orbitals?
##g_i e^{-E_i/(k_BT)} / Z(T)##
where ##g_i## is the state degeneracy (number of states at each level).
Thinking about this, I utterly confused myself in two ways, so I hope someone can help me out.
1. Since the states become more and more dense at higher quantum numbers n (only looking at bound states), it seems as if even at small temperatures, most atoms should be in an excited state with very large n: If I use 1Ry as the energy difference between ground state and a highly excited state (which should be a reasonable approximation), I can always find enough states close to the ionisation energy that the summed probability of each of these states is larger than that of the ground state. Obviously I'm making a mistake here, but I have no clue what it is.
2. Consider just the first to shells. Is the ratio of the probabilty of finding an atom in the ground or in the first excited state given by ##e^{-(E_2-E_1)/(k_BT)}## or do I need an additional factor of 4 because the second shell has one s and three p orbitals?