Chapter 8, Problem 8A.1ST

Interpretation Introduction

Interpretation:

From the given wavelength, the quantum number of higher energy level of Brackett series has to be calculated.

Concept-Introduction:

Rydberg equation:

  $\frac{\text{1}}{\text{λ}}={\text{ R}}_{\infty }\left(\frac{\text{1}}{{\text{n}}_{\text{1}}{}^{\text{2}}}-\frac{\text{1}}{{\text{n}}_{\text{2}}{}^{\text{2}}}\right)$

This equation is known as the Rydberg equation. Here, $\text{λ}$ is the wavelength of a line in the spectrum; ${\text{R}}_{\infty }$ is the Rydberg constant ($\text{1}{\text{.09737316 × 10}}^{\text{7}}{\text{ m}}^{-1}$ for $\text{H}$ atom); and ${\text{n}}_{\text{1}}{\text{ and n}}_{\text{2}}$ are positive integers, where ${\text{n}}_{\text{2}}{\text{ > n}}_{\text{1}}$.

Explanation of Solution

The quantum number of higher energy level of Brackett series is calculated as,

    $\begin{array}{l}{\text{n}}_{\text{1}}\text{ = 4}{\text{n}}_{\text{2}}\text{ = ?}\\ \\ \frac{\text{1}}{\text{λ}}{\text{= R}}_{\infty }\left(\frac{\text{1}}{{\text{n}}_{\text{1}}{}^{\text{2}}}\text{-}\frac{\text{1}}{{\text{n}}_{\text{2}}{}^{\text{2}}}\right)\\ \\ \frac{\text{1}}{\text{4}\text{.50}\text{×}{\text{10}}^{\text{-7}}\text{m}}\text{=}\text{(1}\text{.097}\text{×}{\text{10}}^{\text{7}}{\text{m}}^{\text{-1}}\text{)}\left(\frac{\text{1}}{4^{\text{2}}}\text{-}\frac{\text{1}}{{\text{n}}_{\text{2}}}\right)\\ \\ \frac{\text{1}}{\text{4}\text{.50}\text{×}{\text{10}}^{\text{-7}}\text{m}}\text{=}\text{(1}\text{.097}\text{×}{\text{10}}^{\text{7}}\text{)}\left(\text{0}\text{.0625}\text{-}\frac{\text{1}}{{\text{n}}_{\text{2}}}\right)\\ \\ {\text{n}}_{\text{2}}\text{=}5\end{array}$

The quantum number of upper energy level in Brackett series is five.