At what Kelvin temperature will the thermal energy of electrons be greater than the energy binding an n = 1 electron to the proton in a hydrogen atom?
SolutionThe energy of the n = 1 electron is 1 / 1^2 * -13.6 eV = - 13.6 eV = - 13.6 eV * ( 1.6 * 10^-19 J / eV) = - 21.76 * 10^-19 J.
It therefore requires 21.76 * 10^-19 J of added energy to separate this electron from the atom. Thus the electron is 'bound' to the atom, requiring this much energy to separate it.
At temperature T the average thermal energy of particles is 3/2 k T, where k = 1.38 * 10^-23 J / ( particle Kelvin ) is the Boltzmann gas constant, equal to R / Navagodro.
Therefore if the average thermal energy of particles is equal to the energy we have
so
[ 2/3 * 21.76 * 10^-19 J / particle ] / [1.38 * 10^-23 J / (particle Kelvin ) ]
= 105.1208 * 10^3 Kelvin.
The result is that above this temperature the n = 1 electrons of a hydrogen atom will become separated from the atom.
Above the n=1 temperature of 105.1208 * 10^4 K all the electrons in hydrogen atoms will tend to be separated from the protons and we will have an electron-proton plasma, with a number of unique physical properties. Plasmas constitute a fourth state of matter, in addition to the liquid, solid and gaseous states.
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