Problem 155 - Chemistry

Question

Atoms of an element have only two accessible excited states. In an emission experiment, however, three spectral lines were observed. Explain. Write an equation relating the shortest wavelength to the other two wavelengths.

Step-by-Step Solution

Answer

The three spectral lines are observed due to three different electron transitions to the ground state. The shortest wavelength will be produced by the transition from the highest excited state to the ground state. The relationship between the shortest wavelength and the other two wavelengths is given by \(1 / λ_A = 1 / λ_B + 1 / λ_C\).

Step 1: Understand the Electronic Transitions

There are two energy states that an electron can occupy above the ground state. Essentially, this will give rise to three possible transitions: from the highest energy level to the ground state (Transition A), the next highest energy level to the ground state (Transition B), and between the two excited states to the ground state (Transition C). The transition that involves the greatest energy change will correspond to the shortest wavelength, due to the inverse relationship between energy and wavelength in the equation \(E = h \times c / λ\), where \(E\) is energy, \(h\) is Planck's constant, \(c\) is the speed of light and \(λ\) is the wavelength.

Step 2: Identify the Shortest Wavelength

The greatest energy change (and therefore shortest wavelength) will occur for Transition A, from the highest excited state directly to the ground state.

Step 3: Relate the Wavelengths

The energy of each transition can also be related to the energies of the other transitions. This is due to energy conservation where the energy released in one transition should equal to the sum of the energies released in the other transitions. Therefore, the energy (and thus the frequency and inversely the wavelength) of Transition A can be written as the sum of the energies of Transitions B and C. Using the relationship \(E = h \times c / λ\), we have \(h \times c / λ_A = h \times c / λ_B + h \times c / λ_C\). The Planck's constant and the speed of light can be cancelled from both sides, leaving an equation that relates the wavelengths: \(1 / λ_A = 1 / λ_B + 1 / λ_C\).

Key Concepts

Electronic Transitions

When electrons in an atom jump between different energy states, these movements are known as electronic transitions. These jumps occur when an electron moves from one energy level to another, releasing or absorbing energy. Think of electrons like kids on monkey bars; they can only be on certain bars (energy levels) and they jump from one bar to another, but not in between.
For electronic transitions in an element with two accessible excited states, three possible transitions can be observed:

• From the highest excited state to the ground state
• From the lower excited state to the ground state
• Between the two excited states

Each of these transitions results in a different spectral line, observed as unique wavelengths of light, because each transition involves a distinct energy change.

Excited States

In atoms, electrons reside at specific energy levels, or shells. Normally, they occupy the lowest energy level available, known as the ground state. However, when these electrons absorb energy, they can move to higher energy levels, known as excited states. These states are less stable and electrons will eventually fall back to lower levels, releasing energy as they do.
Imagine exciting an electron to a higher perch like lifting a ball on a hill; it's not stable and wants to roll back down to its original place. When electrons transition back to their ground state from an excited state, they emit energy in the form of light, producing spectral lines.

Energy Levels

The concept of energy levels is crucial in understanding why we see specific spectral lines for an atom. Each energy level represents a potential energy state for electrons in an atom. These levels are quantized, meaning electrons can only exist in specific energy levels rather than a continuous range.

• Ground State: The lowest energy level where electrons can reside.
• Excited States: Higher energy levels where electrons can move temporarily.

These energy levels determine the energy involved in electronic transitions, and hence the wavelengths of light emitted. When an electron jumps back to a lower energy level from a higher level, the energy difference between these levels dictates the wavelength and frequency of the emitted light.

Wavelength Equation

The relationship between energy changes in electronic transitions and observed spectral lines is captured through the wavelength equation. For this purpose, the equation used is derived from the energy of a photon, given by:\[E = \frac{h \times c}{λ} \]where:

• \(E\) is the energy of the photon
• \(h\) is Planck's constant
• \(c\) is the speed of light
• \(λ\) is the wavelength of the emitted light

In the context of our problem, the energy conservation during transitions results in the equation relating wavelengths:\[\frac{1}{λ_A} = \frac{1}{λ_B} + \frac{1}{λ_C}\]Here, \(λ_A\) is the wavelength from the highest to the lowest state, \(λ_B\) and \(λ_C\) are from the other transitions. This equation shows how these wavelengths are interconnected, reflecting the conservation of energy.