Question

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

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.

Key concepts

Energy Levels

In an atom, the energy levels refer to the distinct regions where electrons reside. These are quantized, meaning an electron can only exist at specific energy states, like steps on a staircase. For a hydrogen atom, these levels are given by the principal quantum number, denoted as \(n\). As \(n\) increases, the energy levels are further from the nucleus, resulting in higher energy states. The main energy levels are often visualized starting from \(n=1\), which is the ground state, all the way up to higher levels like \(n=2\), \(n=3\), etc. Transitioning between these levels involves an electron gaining or losing energy. If an electron jumps from a higher level to a lower one, it emits energy, often in the form of light.

Rydberg Formula

The Rydberg Formula is essential for calculating the energy change when an electron transitions between different energy levels in a hydrogen atom. It is expressed as:\[ E = -13.6 \, \text{eV} \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right) \] Here:

• \(E\) represents the energy change during the transition.
• \(n_i\) is the principal quantum number of the initial energy level, where the electron starts.
• \(n_f\) is the principal quantum number of the final energy level, to which the electron moves.

This formula predicts the spectral lines of hydrogen, helping determine the energies of emitted photons during transitions. The negative sign indicates the energy release when moving to a lower energy level. The larger the difference between the initial and final levels, the greater the energy released, which can be observed as light.

Wavelength Emission

Wavelength emission refers to the release of light as electrons transition between energy levels. When electrons drop to a lower energy state, they emit energy as light, which we see as spectral lines at specific wavelengths. The wavelength \(\lambda\) of this light is inversely related to the energy \(E\) of the transition, determined by the equation:\[ E = \frac{hc}{\lambda} \]Where:

• \(h\) is Planck's constant \((6.626 \times 10^{-34} \, \text{Js})\).
• \(c\) is the speed of light \((3.00 \times 10^8 \, \text{m/s})\).
• \(\lambda\) is the wavelength in meters.

A shorter wavelength emission occurs when more energy is released, which corresponds to the most significant drop in energy levels. Thus, for an electron transitioning from \(n=4\) to \(n=1\), the highest energy transition results in the shortest wavelength.