Chapter 7: Problem 67

Consider only the transitions involving the first four energy levels for a hydrogen atom: a. How many emissions are possible for an electron in the \(n=4\) level as it goes to the ground state? b. Which electronic transition is the lowest energy? c. Which electronic transition corresponds to the shortest wavelength emission?

Short Answer

Expert verified

a. The number of emissions possible for an electron in the \(n=4\) level as it goes to the ground state is 3. b. The electronic transition with the lowest energy is \(n=4\) to \(n=3\), with an energy of \(-0.47 eV\). c. The electronic transition corresponding to the shortest wavelength emission is the \(n=4\) to \(n=1\) transition, with an energy of \(-12.1 eV\).

Step by step solution

Identify the possible electronic transitions for an electron at n=4 level.

The electron can undergo multiple transitions from the n=4 level going to the ground state (n=1 level). They are listed below: 1. n=4 to n=3 2. n=4 to n=2 3. n=4 to n=1

Calculate the number of possible emissions from n=4 to the ground state.

There are 3 possible emissions an electron in the n=4 level can have while going to the ground state. They are: 1. n=4 to n=3 2. n=4 to n=2 3. n=4 to n=1 #a.# So, the number of emissions possible for an electron in n=4 level as it goes to the ground state is 3.

Determine the energy of each electronic transition.

Use the Rydberg formula to calculate the energy of each transition: \(E = -13.6 eV * (\frac{1}{n_f^2} - \frac{1}{n_i^2}) \) Where: - E is the energy of the transition - n_i is the initial energy level - n_f is the final energy level Calculate the energy for each transition: 1. n=4 to n=3 \(E = -13.6 eV * (\frac{1}{3^2} - \frac{1}{4^2}) = -0.47 eV\) 2. n=4 to n=2 \(E = -13.6 eV * (\frac{1}{2^2} - \frac{1}{4^2}) = -3.4 eV\) 3. n=4 to n=1 \(E = -13.6 eV * (\frac{1}{1^2} - \frac{1}{4^2}) = -12.1 eV\)

Identify the lowest energy electronic transition.

Based on the calculated energies, the lowest energy electronic transition is the n=4 to n=3 transition with an energy of -0.47 eV. #b.# The electronic transition with the lowest energy is n=4 to n=3.

Identify the transition with the shortest wavelength emission.

For electronic transitions, the shortest wavelength emission corresponds to the highest energy transition. Among the transitions from n=4 to the ground state, the transition with the highest energy is n=4 to n=1 which has an energy of -12.1 eV. #c.# The electronic transition corresponding to the shortest wavelength emission is the n=4 to n=1 transition.

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Short Answer

Step by step solution

Identify the possible electronic transitions for an electron at n=4 level.

Calculate the number of possible emissions from n=4 to the ground state.

Determine the energy of each electronic transition.

Identify the lowest energy electronic transition.

Identify the transition with the shortest wavelength emission.

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