The energy level diagram for a hydrogen-like | vedclass

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Question

(A) The energy of the $n$-th orbit for a hydrogen-like atom is given by $E_n = -13.6 \frac{Z^2}{n^2} \, 	ext{eV}$.From the given energy level diagram

Vedclass · Accepted Answer

$0.265\, \mathring{A}$

Vedclass · Answer

(A) The energy of the $n$-th orbit for a hydrogen-like atom is given by $E_n = -13.6 \frac{Z^2}{n^2} \, 	ext{eV}$.From the given energy level diagram,for the ground state $(n = 1)$,the energy is $E_1 = -54.4 \, 	ext{eV}$.Substituting these values into the formula: $-54.4 = -13.6 \frac{Z^2}{1^2}$.Solving for $Z^2$: $Z^2 = \frac{54.4}{13.6} = 4$,which gives $Z = 2$.The radius of the $n$-th Bohr orbit is given by $r_n = 0.53 \frac{n^2}{Z} \, \mathring{A}$.For the first Bohr orbit $(n = 1)$ and $Z = 2$,the radius is $r_1 = 0.53 \frac{1^2}{2} = 0.265 \, \mathring{A}$.Thus,the correct option is $A$.

The energy level diagram for a hydrogen-like atom is shown in the figure. The radius of its first Bohr orbit is

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