Question

If an electron jumps from the 3rd orbit to the 2nd orbit, its wavelength is \(\lambda\). Then the wavelength of the electron when it jumps from the 4th orbit to the 3rd orbit in terms of \(\lambda\) is:

Hint:

To find the relationship between wavelengths in different transitions, compare the terms \(\frac{1}{n_1^2} - \frac{1}{n_2^2}\) using the Rydberg formula.

Answer 1

Step 1: Apply the Rydberg formula for wavelength. The wavelength \(\lambda\) of an emitted photon during an electronic transition is calculated using the Rydberg formula: \[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), \] where \(R\) is the Rydberg constant, \(n_1\) is the final orbit number, and \(n_2\) is the initial orbit number (\(n_2>n_1\)).

Step 2: Calculate the wavelength for a transition from the 3rd orbit (\(n_2 = 3\)) to the 2nd orbit (\(n_1 = 2\)). This wavelength is denoted as \(\lambda\). Substituting these values into the Rydberg formula yields: \[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right). \] Simplifying the expression gives: \[ \frac{1}{\lambda} = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{9 - 4}{36} \right) = R \cdot \frac{5}{36}. \]

Step 3: Calculate the wavelength for a transition from the 4th orbit (\(n_2 = 4\)) to the 3rd orbit (\(n_1 = 3\)). This wavelength is denoted as \(\lambda'\). The formula becomes: \[ \frac{1}{\lambda'} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right). \] Simplifying this expression results in: \[ \frac{1}{\lambda'} = R \left( \frac{1}{9} - \frac{1}{16} \right) = R \left( \frac{16 - 9}{144} \right) = R \cdot \frac{7}{144}. \]

Step 4: Establish the relationship between \(\lambda'\) and \(\lambda\). Divide the equation for \(\lambda'\) by the equation for \(\lambda\): \[ \frac{1}{\lambda'} \div \frac{1}{\lambda} = \frac{R \cdot \frac{7}{144}}{R \cdot \frac{5}{36}}. \] Performing the simplification: \[ \frac{\lambda}{\lambda'} = \frac{\frac{7}{144}}{\frac{5}{36}} = \frac{7}{144} \cdot \frac{36}{5} = \frac{7 \cdot 36}{144 \cdot 5} = \frac{7}{20}. \] Rearranging to solve for \(\lambda'\): \[ \lambda' = \frac{20}{7} \lambda. \]

Step 5: Conclude the wavelength of the electron transition. The wavelength for an electron transitioning from the 4th orbit to the 3rd orbit is: \[ \boxed{\frac{20}{7} \lambda}. \]