Question

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

• A. Relative change in the radii of two consecutive orbitals does not depend on $Z$
• B. Relative change in the radii of two consecutive orbitals varies as $1/n$
• C. Relative change in the energy of two consecutive orbitals varies as $1/n^3$
• D. Relative change in the angular momenta of two consecutive orbitals varies as $1/n$

Answer 1

Answer: (A), (B), (D)

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

From Bohr's model, we know that

$r_n\propto \frac{n^2}{Z};E_n\propto \frac{Z^2}{n^2}\&L_n=\frac{nh}{2\pi }$

So, relative change in the radii of two consecutive orbitals,

$\frac{r_n-r_{n-1}}{r_n}=1-\frac{r_{n-1}}{r_n}=1-\frac{(n-1{)}^2}{n^2}$

$\Rightarrow \frac{r_n-r_{n-1}}{r_n}=\frac{2n-1}{n^2}\approx \frac{2}{n}(\because n>>1)$

So, relative change in radii does not depend on $Z$.

Now, relative change in the energy of two consecutive orbitals,

$\frac{E_n-E_{n-1}}{E_n}=1-\frac{E_{n-1}}{E_n}=1-\frac{n^2}{(n-1{)}^2}\approx \frac{-2}{n}(\because n>>1)$

Now, relative change in the angular momenta of two consecutive orbitals,

$\frac{L_n-L_{(n-1)}}{L_n}=1-\frac{L_{n-1}}{L_n}=1-\frac{n-1}{n}=\frac{1}{n}$

Hence, options $\text{(A), (B)}$ and $\left(\text{D}\right)$ are the correct answer.