Question

An excited state of doubly joined Lithium (\[L{i^{2 + }}\] ) has an orbital radius that is about 1.33 times that of the ground state of hydrogen H in (Bohr’s theory). The ratio of energy of the two states, E(\[L{i^{2 + }}\])/E(H) is
A. 2.25
B. 4.5
C. 1
D. 9

Answer 1

Hint: To find the ratio we need to use the equation for energies and radius of orbit given by Bohr's atomic theory. According to Bohr's model an electron will absorb energy in the form of photons to get excited to a higher energy level. The energy thus obtained is always a negative number and the ground state (n = 1) has the negative value.

Formula used:
According to the Bohr's atomic theory,
\[r = \dfrac{{{n^2}{h^2}}}{{4{\pi ^2}m{e^2}}} \times \dfrac{1}{Z}\]
Where n is an integer
\[r\] is the radius of orbit
n is the principal quantum number of the orbit
Z is the atomic number
Energies for an electron in the shell given by Bohr's model is given as,
\[E = - 13.6\dfrac{{{Z^2}}}{{{n^2}}}\]

Complete step by step solution:
Given that Lithium (\[L{i^{2 + }}\] ) has an orbital radius 1.33 times that of the ground state of hydrogen (H).
i.e., \[{r_{L{i^{2 + }}}} = 1.33{\rm{ }}{r_H}\]
As we know that radius of orbital in Bohr’s model is,
\[r = \dfrac{{{n^2}{h^2}}}{{4{\pi ^2}m{e^2}}} \times \dfrac{1}{Z}\]

The ratio of Lithium (\[L{i^{2 + }}\] ) and hydrogen (H) is written as
\[\dfrac{{{r_{L{i^{2 + }}}}}}{{{r_H}}} = \dfrac{{{n_{L{i^{2 + }}}}}}{{{n_H}}} \times \dfrac{{{Z_H}}}{{{Z_{L{i^{2 + }}}}}} \\ \]
Using \[{r_{L{i^{2 + }}}} = 1.33{\rm{ }}{r_H} \\ \]
\[\dfrac{{1.33{\rm{ }}{r_H}}}{{{r_H}}} = \dfrac{{{n_{L{i^{2 + }}}}}}{{{n_H}}} \times \dfrac{{{Z_H}}}{{{Z_{L{i^{2 + }}}}}} \\ \]
\[\Rightarrow {n_{L{i^{2 + }}}} = 3.99 \approx 4\]

Also, we know that the energies for an electron in the shell given by Bohr's model is
\[E = - 13.6\dfrac{{{Z^2}}}{{{n^2}}}\]
For the energy of lithium and hydrogen,
\[{E_{L{i^{2 + }}}} = - 13.6\dfrac{{{3^2}}}{{{1^2}}}\]
And \[{E_H} = - 13.6\]
Now the ratio of energy of the two states is
\[\therefore \dfrac{{{E_{L{i^{2 + }}}}}}{{{E_H}}} = \dfrac{9}{1} \\ \]
Therefore, the ratio of energy of the two states, E(\[L{i^{2 + }}\])/E(H) is 9.

Hence option D is the correct answer.

Note: The Bohr model of an atom came into existence with some modification of Rutherford’s model of an atom. Bohr’s theory modified the atomic structure of the model by explaining that electrons will move in fixed orbitals or shells and each of the orbitals or shells has its fixed energy.