Problem 28 - General Chemistry: Principles and Modern Applications

Question

Calculate the increase in (a) distance from the nucleus and (b) energy when an electron is excited from the first to the third Bohr orbit.

Step-by-Step Solution

Answer

The increase in distance from the nucleus when an electron is excited from the first to the third Bohr orbit is 3.176 Å, and the increase in energy is + 10.2 eV.

Step 1: Formula for the Radius of Bohr Orbit

The formula for the radius \(r_n\) of the nth Bohr orbit is given by \(r_n = 0.529n^2\) Å, where n denotes the number of the orbit. First calculate the radius of the first Bohr orbit \(r_1\) by substituting \(n = 1\) into the formula, and then calculate the radius of the third Bohr orbit \(r_3\) by substituting \(n = 3\). The increase in distance from the nucleus can be found by subtracting the radius of the first orbit \(r_1\) from the third \(r_3\).

Step 2: Formula for the Energy of Bohr Orbit

The energy \(E_n\) of an electron in the nth Bohr orbit is given by \(E_n = -13.6/n^2\) eV. Find the energy of the first orbit \(E_1\) by substituting \(n = 1\), and then find the energy of the third orbit \(E_3\) by substituting \(n = 3\). The increase in energy can be found by subtracting the energy of the first orbit \(E_1\) from the third \(E_3\). Note that the energy is negative, indicating a bound system, and the magnitude of the energy increases (becomes more negative) as the electron gets closer to the nucleus.

Step 3: Calculation

Now, we can find the values of the increase in distance and the increase in energy. After performing the calculations, we find that the increase in distance from the nucleus is \(r_3 - r_1 = 0.529*(3^2 - 1^2)\) Å = 3.176 Å, and the increase in energy is \(E_3 - E_1 = -13.6 * (1/3^2 - 1/1^2)\) eV = + 10.2 eV.

Key Concepts

Bohr orbit radius

In the Bohr model of the atom, each orbit corresponds to a specific path that an electron follows around the nucleus. Each orbit is defined by its radius, called the Bohr orbit radius. The radius of the nth Bohr orbit can be calculated using the formula:

• \( r_n = 0.529n^2 \text{ Å} \)

Here, \( n \) represents the principal quantum number, which is an integer value indicating the orbit level. The first orbit has \( n = 1 \), the second orbit has \( n = 2 \), and so on. The radius increases with the square of \( n \), making higher orbits significantly larger than lower ones.
For example, when an electron moves from the first (1st) orbit to the third (3rd) orbit, the radius increases from \( r_1 = 0.529(1^2) \text{ Å} = 0.529 \text{ Å} \) to \( r_3 = 0.529(3^2) \text{ Å} = 4.761 \text{ Å} \).
This increase in radius shows that electrons in higher orbits are further from the nucleus, which is significant in understanding how the position of an electron affects its energy state within the atom.

electron energy levels

Electron energy levels refer to the specific energies that electrons can have when orbiting the nucleus of an atom. These levels are quantized, meaning electrons can only exist at specific energy levels and not between them.
In the Bohr model, the formula for the energy of an electron in the nth orbit is:

• \( E_n = -\frac{13.6}{n^2} \text{ eV} \)

This negative sign means that the energy is what binds the electron to the nucleus. As \( n \) increases, \( E_n \) becomes less negative, indicating the electron has higher energy and is less tightly bound. For instance, calculating the energy for the first orbit \( E_1 \) gives \( -13.6 \text{ eV} \), while for the third orbit \( E_3 \) yields \( -13.6/9 \text{ eV} = -1.51 \text{ eV} \).
The energy difference between electron levels reveals how electrons transition from one level to another through absorption or emission of energy, typically in the form of light.

electron excitation

Electron excitation occurs when an electron absorbs energy and jumps from a lower energy level to a higher one in the Bohr model. This process involves the electron moving from a more tightly bound state to a less tightly bound state, further from the nucleus.
For example, an electron excitation from the first to the third orbit involves an energy increase of \( 10.2 \text{ eV} \), as calculated from the energy difference \( E_3 - E_1 \).
Such a transition requires the electron to absorb exactly this amount of energy. If the absorbed energy doesn't match this specific value, the transition won't occur.

• Absorption: Electron moves to a higher energy level.
• Emission: Electron releases energy and falls back to a lower level.

This concept of excitation and emission is foundational in understanding phenomena like spectral lines and the working of lasers, as the unique energy levels determine the wavelengths of light an atom can absorb or emit.