Question

Find the ratio of wave number (\( \nu \)) of the 1st line of Balmer series and Brackett series for Hydrogen-like species.

• A. \( \frac{1}{0.09} \)
• B. \( \frac{0.81}{5} \)
• C. \( \frac{5}{0.81} \)
• D. \( 0.09 \)

Answer 1

The Correct Option is C

textbf{Step 1: Formula for wave number.}
For a hydrogen-like species, the wave number (\( \nu \)) for the first line of the Balmer series is given by: \[ \nu_{\text{Balmer}} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \] where \( R_H \) is the Rydberg constant. For the first line of the Brackett series: \[ \nu_{\text{Brackett}} = R_H \left( \frac{1}{4^2} - \frac{1}{5^2} \right) \] Step 2: Compute the wave numbers.
The ratio of the wave numbers is: \[ \frac{\nu_{\text{Balmer}}}{\nu_{\text{Brackett}}} = \frac{\left( \frac{1}{2^2} - \frac{1}{3^2} \right)}{\left( \frac{1}{4^2} - \frac{1}{5^2} \right)} = \frac{ \frac{1}{4} - \frac{1}{9} }{ \frac{1}{16} - \frac{1}{25} } \] Step 3: Simplify the expression.
Simplifying the numerator and denominator: \[ \frac{ \frac{5}{36} }{ \frac{9}{400} } = \frac{5}{0.81} \] \[ \boxed{\frac{5}{0.81}} \]