Problem 19 - Chemistry and Chemical Reactivity

Question

The energy emitted when an electron moves from a higher energy state to a lower energy state in any atom can be observed as electromagnetic radiation. (a) Which involves the emission of less energy in the H atom, an electron moving from \(n=4\) to \(n=2\) or an electron moving from \(n=3\) to \(n=2 ?\) (b) Which involves the emission of more energy in the H atom, an electron moving from \(n=4\) to \(n=1\) or an electron moving from \(n=5\) to \(n=2 ?\) Explain fully.

Step-by-Step Solution

Answer

(a) Transition from \( n=3 \) to \( n=2 \) emits less energy. (b) Transition from \( n=4 \) to \( n=1 \) emits more energy.

Step 1: Understanding the Formula for Energy Transition

The energy change when an electron transitions between energy levels can be calculated using the formula \( \Delta E = -13.6 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \), where \( n_i \) is the initial energy level and \( n_f \) is the final energy level. The constant 13.6 eV is the ionization energy of hydrogen.

Step 2: Calculate Energy for Transition from n=4 to n=2

Substitute \( n_i = 4 \) and \( n_f = 2 \) in the energy change formula: \( \Delta E = -13.6 \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = -13.6 \left( \frac{1}{4} - \frac{1}{16} \right) = -13.6 \left( \frac{4-1}{16} \right) = -13.6 \left( \frac{3}{16} \right) = -2.55 \text{ eV} \).

Step 3: Calculate Energy for Transition from n=3 to n=2

Substitute \( n_i = 3 \) and \( n_f = 2 \) in the formula: \( \Delta E = -13.6 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = -13.6 \left( \frac{1}{4} - \frac{1}{9} \right) = -13.6 \left( \frac{9-4}{36} \right) = -13.6 \left( \frac{5}{36} \right) = -1.89 \text{ eV} \).

Step 4: Compare Energies for Part (a)

The energy emitted for the transition from \( n=4 \) to \( n=2 \) is \(-2.55 \text{ eV}\), and for the transition from \( n=3 \) to \( n=2 \) is \(-1.89 \text{ eV}\). Since \(-1.89 \text{ eV}\) is less than \(-2.55 \text{ eV}\), the transition from \( n=3 \) to \( n=2 \) emits less energy.

Step 5: Calculate Energy for Transition from n=4 to n=1

Substitute \( n_i = 4 \) and \( n_f = 1 \) in the formula: \( \Delta E = -13.6 \left( \frac{1}{1^2} - \frac{1}{4^2} \right) = -13.6 \left( 1 - \frac{1}{16} \right) = -13.6 \left( \frac{16 - 1}{16} \right) = -13.6 \left( \frac{15}{16} \right) = -12.75 \text{ eV} \).

Step 6: Calculate Energy for Transition from n=5 to n=2

Substitute \( n_i = 5 \) and \( n_f = 2 \) in the formula: \( \Delta E = -13.6 \left( \frac{1}{2^2} - \frac{1}{5^2} \right) = -13.6 \left( \frac{1}{4} - \frac{1}{25} \right) = -13.6 \left( \frac{25-4}{100} \right) = -13.6 \left( \frac{21}{100} \right) = -2.856 \text{ eV} \).

Step 7: Compare Energies for Part (b)

The energy emitted for the transition from \( n=4 \) to \( n=1 \) is \(-12.75 \text{ eV}\), and for \( n=5 \) to \( n=2 \) is \(-2.856 \text{ eV}\). Since \(-12.75 \text{ eV}\) is more than \(-2.856 \text{ eV}\), the transition from \( n=4 \) to \( n=1 \) emits more energy.

Key Concepts

Energy Levels

In quantum mechanics, energy levels refer to the specific energies that an electron within an atom can have. Electrons exist in distinct energy levels or "shells" around the nucleus of an atom. Each level is associated with a principal quantum number, denoted as \( n \), which can be any positive integer.

The energy of each level in the hydrogen atom is given by the equation \[ E_n = -13.6 \frac{1}{n^2} \text{ eV} \]where \( 13.6 \text{ eV} \) is the energy needed to ionize the hydrogen atom from its ground state (\( n=1 \)). This formula shows that energy becomes less negative as \( n \) increases, meaning the electron is higher up and more energized.

Moving between these levels involves either absorbing or emitting energy. When an electron jumps to a higher energy level, it absorbs energy, and when it falls to a lower one, it emits energy as light or other radiation.

Hydrogen Atom

The hydrogen atom is the simplest and most well-studied atom in physics. It consists of one proton at its nucleus and one electron orbiting around it. This straightforward setup makes hydrogen a crucial model for understanding atomic structures and behaviors.

The allowed energy levels for the hydrogen atom are determined by quantum mechanics. For hydrogen, the energy levels are quantized, meaning only specific energy values are permissible, which is crucial for the electron transitions that lead to light emission. The hydrogen atom model helps explain different phenomena such as atomic spectra, where distinct lines are produced due to electron transitions between different energy levels.

Electron Transitions

Electron transitions are the movements of electrons between energy levels within an atom. These transitions can release or require energy and are governed by the changes in an electron's quantum state. The amount of energy involved in these transitions determines the type and behavior of electromagnetic radiation emitted or absorbed.

Key points about electron transitions include:

• When electrons drop from a higher energy level to a lower one, they emit energy, often in the form of visible light.
• The energy emitted is variable and depends on the difference between the two energy levels.
• A larger energy difference corresponds to higher frequency light (e.g., ultraviolet), whereas a smaller difference corresponds to lower frequency light (e.g., infrared).

This principle underlies the concept of spectra and laser technology.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of very small particles like electrons in atoms. It provides a fundamental understanding of how energy levels and transitions operate.

In quantum mechanics:

• Energy levels are quantized, meaning they occur in fixed steps rather than a continuous spectrum.
• Electrons behave both as particles and waves, a concept known as wave-particle duality.
• The position and momentum of an electron cannot be known precisely at the same time, as per the Heisenberg Uncertainty Principle.

These principles give rise to the complex behaviors seen in atomic-scale phenomena, as demonstrated in exercises involving electron transitions and energy emission in hydrogen atoms.