Question

Supposing the energy of the fourth shell for hydrogen atom is −50 a.u (arbitrary unit). What would be its ionization potential?
(A) 50
(B) 800
(C) 15.4
(D) 20.8

Answer 1

Hint: An ion with only one electron is equivalent to the hydrogen atom but having a positive charge of Z instead of one. The Bohr theory applies to such a system.

Step-by-Step Solution:
Let us first analyse the Bohr model of a Hydrogen atom and how we can use it to help answer this question.
The Bohr Model of the hydrogen atom first proposed the planetary model, but later an assumption concerning the electrons was made. The assumption was the quantization of the structure of atoms. Bohr proposed that electrons orbited the nucleus in specific orbits or shells with a fixed radius. Only those shells with a radius provided by the equation below were allowed, and it was impossible for electrons to exist between these shells.
By one of Bohr’s postulates, we know that energy level of a shell of the Hydrogen atom is given by:
\[E=-13.6\dfrac{{{Z}^{2}}}{{{n}^{2}}}\]
Now, we also know that the Ionisation potential of a particular shell is the negative of its total energy.
By the above energy equation, we can say that energy of the first shell is 16 times that of the fourth shell. 
So, the energy of the first shell should be – (50×16) au.
Having now also obtained the relationship between IP and energy,
We conclude that the IP of H atom will be +800 au.
Therefore, the answer to this question is b).

Note: Remember that the ionization energy or ionization potential is the energy necessary to remove an electron from the neutral atom and is always the additive inverse of the energy level in question. Also remember that the value of Z in this question is 1 due to the atomic number of Hydrogen being equal to 1.