Question

Consider Lyman, Balmer, Paschen, Brackett & P-fund series of H atom and He⁺ ion. There are given few pairs of these series. Identify the pairs in which there is no overlapping of wavelengths of series.

• A. Lyman series of H & Balmer series of He⁺
• B. Lyman series of H & Paschen series of He⁺
• C. Balmer series of H & Brackett series of He⁺
• D. Lyman series of H & P-fund series of He⁺

Answer 1

Answer: (B), (D)

(b) and (d)

Answer 2

To determine which pairs of spectral series have no overlapping wavelengths, we need to calculate the wavelength ranges for each series for both the Hydrogen atom ($Z=1$) and the Helium ion ($Z=2$). The Rydberg formula for the wavelength ($\lambda$) of spectral lines is given by: $\frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$ where $R$ is the Rydberg constant, $Z$ is the atomic number, $n_f$ is the final principal quantum number, and $n_i$ is the initial principal quantum number ($n_i > n_f$).

The spectral series are defined by their final energy level ($n_f$):

• Lyman series: $n_f = 1$
• Balmer series: $n_f = 2$
• Paschen series: $n_f = 3$
• Brackett series: $n_f = 4$
• P-fund series: $n_f = 5$

The shortest wavelength ($\lambda_{min}$) for a series occurs for the transition from $n_i = \infty$ to $n_f$: $\lambda_{min} = \frac{n_f^2}{RZ^2}$ The longest wavelength ($\lambda_{max}$) for a series occurs for the transition from $n_i = n_f + 1$ to $n_f$: $\lambda_{max} = \frac{n_f^2 (n_f+1)^2}{RZ^2 (2n_f+1)}$ Let $K = 1/R$ for convenience.

For Hydrogen atom (H, $Z=1$):

• Lyman ($n_f=1$): Range is $[K, 4K/3]$ (approx. $[1.00K, 1.33K]$)
• Balmer ($n_f=2$): Range is $[4K, 36K/5]$ (approx. $[4.00K, 7.20K]$)
• Paschen ($n_f=3$): Range is $[9K, 144K/7]$ (approx. $[9.00K, 20.57K]$)
• Brackett ($n_f=4$): Range is $[16K, 400K/9]$ (approx. $[16.00K, 44.44K]$)
• P-fund ($n_f=5$): Range is $[25K, 900K/11]$ (approx. $[25.00K, 81.82K]$)

For Helium ion (He⁺, $Z=2$): $RZ^2 = 4R$. So, the effective constant is $4R$, and $K_{He^+} = K/4$.

• Lyman ($n_f=1$): Range is $[K/4, K/3]$ (approx. $[0.25K, 0.33K]$)
• Balmer ($n_f=2$): Range is $[K, 9K/5]$ (approx. $[1.00K, 1.80K]$)
• Paschen ($n_f=3$): Range is $[9K/4, 36K/7]$ (approx. $[2.25K, 5.14K]$)
• Brackett ($n_f=4$): Range is $[4K, 100K/9]$ (approx. $[4.00K, 11.11K]$)
• P-fund ($n_f=5$): Range is $[25K/4, 225K/11]$ (approx. $[6.25K, 20.45K]$)

Now, let's check for overlaps in the given pairs:

(a) Lyman series of H & Balmer series of He⁺: H Lyman: $[1.00K, 1.33K]$ He⁺ Balmer: $[1.00K, 1.80K]$ Overlap exists because $1.00K \le 1.80K$ and $1.00K \le 1.33K$.

(b) Lyman series of H & Paschen series of He⁺: H Lyman: $[1.00K, 1.33K]$ He⁺ Paschen: $[2.25K, 5.14K]$ No overlap exists because $1.33K < 2.25K$.

(c) Balmer series of H & Brackett series of He⁺: H Balmer: $[4.00K, 7.20K]$ He⁺ Brackett: $[4.00K, 11.11K]$ Overlap exists because $4.00K \le 11.11K$ and $4.00K \le 7.20K$.

(d) Lyman series of H & P-fund series of He⁺: H Lyman: $[1.00K, 1.33K]$ He⁺ P-fund: $[6.25K, 20.45K]$ No overlap exists because $1.33K < 6.25K$.

Therefore, the pairs with no overlapping wavelengths are (b) and (d).