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Question
What will be the energy corresponding to the first excited state of a hydrogen atom if the potential energy of the atom is taken to be 10 eV when the electron is widely separated from the proton? Can we still write En = E1/n2, or rn = a0 n2?
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Solution
Energy of nth state of hydrogen is given by
`E_n = -13.6/n^2 eV`
Energy of first excited state (n = 2) of hydrogen, `E_1 = -13.6/4 eV = 3.4 eV`
This relation holds true when the refrence point energy is zero.Usually the refrence point energy is the energy of the atom when the electron is widely separated from the proton.In the given question, the potential energy of the atom is taken to be 10 eV when the electron is widely separated from the proton so here our refrence point energy is 10 eV. Earlier The energy of first excited state was -3.4 eV when the refrence point had zero energy but now as the refrence point has shifted so The energy of the first excited state will also shift by the corresponding amount.Thus,
E1, = -3.4 eV-10 eV = -13.4 eV
We still write En = E1/n2, or rn = a0 n2 because these formulas are independent of the refrence point enegy.
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