Question 103: Atoms & Nuclei

Question

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n, where n >> 1. Which of the following statement(s) is(are) true?

Accepted Answer

The radius of $n^{	ext {th }}$ orbital for a hydrogen-like atom of atomic number $Z$ is given by $r_n={{{n^2}{a_0}} \over Z}$, where $a_0=0.53 $$$\mathop A\limits^o $$ is the Bohr's radius.
$$ 	herefore $$ $$
	ext { Radius of orbit, } r \propto \frac{n^2}{Z}
$$
The relative change in the radii of two consecutive orbitals is
$$
\frac{r_{n+1}-r_n}{r_n}=\frac{\frac{(n+1)^2}{Z}-\frac{n^2}{Z}}{\frac{n^2}{Z}}=\frac{2 n+1}{n^2} \approx \frac{2}{n} \quad(\because n>>1)
$$
The energy of the n
th
orbital is given by
$${E_n} = {{ - 13.6{Z^2}} \over {{n^2}}}eV$$
$$ 	herefore $$ $$
E_n \propto \frac{-Z^2}{n^2}
$$
The relative change in the energy of
two consecutive orbitals is
$$
\begin{aligned}
& \frac{E_{n+1}-E_n}{E_n}=\frac{\frac{-Z^2}{(n+1)^2}+\frac{Z^2}{n^2}}{\frac{-Z^2}{n^2}}=-\frac{(2 n+1)}{(n+1)^2} \\
& \approx-\frac{2 n}{n^2}=-\frac{2}{n} \quad(n>>1)
\end{aligned}
$$
The angular momentum of the $n^{	ext {th }}$ orbital is given by $L_n={{nh} \over {2\pi }}$. The relative change in the angular momentum of two consecutive orbitals is
$$
\frac{L_{n+1}-L_n}{L_n}=\frac{(n+1) {{h} \over {2\pi }}-{{nh} \over {2\pi }}}{{nh} \over {2\pi }}=\frac{1}{n}
$$

Answer

Relative change in the radii of two consecutive orbitals does not depend on Z.

Answer

Relative change in the radii of two consecutive orbitals varies as 1/n

Answer

Relative change in the energy of two consecutive orbitals varies as 1/n3

Answer

Relative change in the angular momenta of two consecutive orbitals varies as 1/n

Atoms & Nuclei

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n, where n >> 1. Which of the following statement(s) is(are) true?

Choose the correct answer: