Problem 38 - Chemistry: The Central Science in SI Units, Global Edition

Question

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=3,(\mathbf{b})\) from an orbit of radius 0.529 to one of radius \(0.476 \mathrm{nm},(\mathbf{c})\) from the \(n=9\) to the \(n=6\) state.

Step-by-Step Solution

Answer

For the electronic transitions in hydrogen: (a) From n = 2 to n = 3, energy is absorbed. (b) From an orbit of radius 0.529 nm to one of radius 0.476 nm, energy is emitted. (c) From the n = 9 to the n = 6 state, energy is emitted.

Step 1: Transition (a): n = 2 to n = 3

The electron is transitioning from n = 2 to higher energy level n = 3. In this case, the electron is jumping to a higher energy state, and hence the atom needs to absorb energy for this transition to occur. Therefore, energy is absorbed in this case.

Step 2: Transition (b): Orbit of radius 0.529 nm to orbit of radius 0.476 nm

In this case, we need to use the relation between the orbital radius and principal quantum number in a hydrogen atom: \(r_n = a₀n^2\), where \(a₀ \approx 0.529\) nm is the Bohr radius. First, let's find the initial principal quantum number, \(n_i\): \[ n_i^2 = \frac{r_i}{a₀} = \frac{0.529\, \text{nm}}{0.529\, \text{nm}} = 1. \] So, \(n_i = 1\); Next, let's find the final principal quantum number, \(n_f\): \[ n_f^2 = \frac{r_f}{a₀} = \frac{0.476\, \text{nm}}{0.529\, \text{nm}} \approx 0.81. \] So, \(n_f \approx 0.9\); Since \(n_f < n_i\), the electron is transitioning to lower energy level. Thus, the atom releases energy for this transition to occur, and energy is emitted.

Step 3: Transition (c): n = 9 to n = 6

The electron is transitioning from higher energy level n = 9 to lower energy level n = 6. In this case, the electron is falling to a lower energy state, and hence the atom needs to release energy for this transition to occur. Therefore, energy is emitted in this case. To summarize: (a) Energy is absorbed. (b) Energy is emitted. (c) Energy is emitted.

Key Concepts

Energy Absorption

When we talk about energy absorption, we are referring to an electron in an atom taking in energy. This allows it to move from a lower to a higher energy level. Imagine an electron is initially at an energy level, say level 2 - often denoted by the principal quantum number, \( n = 2 \). When this electron absorbs energy, it has enough "boost" to climb to a higher energy level, like \( n = 3 \).

• Such transitions are crucial in processes like the formation of spectral lines.
• This absorption results in a visible "jump" or transition in the energy state of the electron.

In the Bohr model of the hydrogen atom, these energy levels are distinct, just like rungs on a ladder. Each "jump" requires a specific amount of energy, which is unique for different levels. If an electron transitions from \( n = 2 \) to \( n = 3 \), as in the exercise above, energy is absorbed because the atom needs that energy to accomplish this transition.

Energy Emission

Energy emission occurs when an electron drops from a higher energy level to a lower one. In this process, the atom releases energy. The energy difference is usually released in the form of light. Imagine an electron descending from a high state such as \( n = 9 \) to a lower energy state like \( n = 6 \).

• This release of energy is often what we see as light or a photon being emitted.
• Electronic transitions that involve emission are responsible for phenomena like luminescence.

When an electron emits energy, it is essentially giving back its "climb down the ladder". In the original exercise, when the electron transits from \( n = 9 \) to \( n = 6 \), the energy difference is released, indicating the emission of energy.

Bohr Model

The Bohr model of the atom offers a simple yet elegant framework to understand electronic transitions such as absorption and emission of energy. It suggests that electrons move in circular orbits around the nucleus, and these orbits correspond to specific energy levels.

• Each orbit in the model is associated with a quantized energy level, labeled by the principal quantum number, \( n \).
• The model accounts for the stability of atoms and the observed spectra of emitted light.

In this model, transitions between these energy levels involve discrete changes in energy, either absorbing or releasing energy, depending on whether the electron is moving up or down in energy levels.
For instance, when transitioning from a radius of 0.529 nm to 0.476 nm, as shown in the step-by-step solution, the electron is moving to a lower energy state, hence energy is emitted. The Bohr model provides a straightforward method to calculate the energy differences using the formula \( r_n = a_0n^2 \), where \( a_0 \) is the Bohr radius. This is the underlying principle employed in calculating transitions and interpreting whether the energy is emitted or absorbed.