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 \(\mathrm{H}\) 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 \(\mathrm{H}\) 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) From \( n=3 \) to \( n=2 \) emits less energy; (b) From \( n=4 \) to \( n=1 \) emits more energy.

Step 1: Calculate Energy Levels

In the hydrogen atom, the energy of an electron in a given orbit is given by the formula \( E_n = -13.6 \frac{1}{n^2} \) eV, where \( n \) is the principal quantum number. We need to calculate the energy levels for \( n=1,2,3,4,5 \).

Step 2: Calculate Energy for Transition (a)

For the transition from \( n=4 \) to \( n=2 \), calculate the energy change \( \Delta E = E_2 - E_4 \). Use the formula, \( E_2 = -13.6 \left(\frac{1}{2^2}\right) = -3.40 \) eV and \( E_4 = -13.6 \left(\frac{1}{4^2}\right) = -0.85 \) eV. Therefore, \( \Delta E = -3.40 - (-0.85) = -2.55 \) eV.

Step 3: Calculate Energy for Transition (b)

For the transition from \( n=3 \) to \( n=2 \), calculate the energy change \( \Delta E = E_2 - E_3 \). Use the formula, \( E_3 = -13.6 \left(\frac{1}{3^2}\right) = -1.51 \) eV. Therefore, \( \Delta E = -3.40 - (-1.51) = -1.89 \) eV. Compare the transitions: \(-2.55\) eV vs \(-1.89\) eV. The transition from \( n=3 \) to \( n=2 \) emits less energy.

Step 4: Calculate Energy for Transition (c)

For the transition from \( n=4 \) to \( n=1 \), calculate the energy change \( \Delta E = E_1 - E_4 \). Use the formula, \( E_1 = -13.6 \) eV and \( E_4 = -0.85 \) eV. Therefore, \( \Delta E = -13.6 - (-0.85) = -12.75 \) eV.

Step 5: Calculate Energy for Transition (d)

For the transition from \( n=5 \) to \( n=2 \), calculate the energy change \( \Delta E = E_2 - E_5 \). Use the formula, \( E_5 = -13.6 \left(\frac{1}{5^2}\right) = -0.544 \) eV. Therefore, \( \Delta E = -3.40 - (-0.544) = -2.856 \) eV. Compare the transitions: \(-12.75\) eV vs \(-2.856\) eV. The transition from \( n=4 \) to \( n=1 \) emits more energy.

Key Concepts

Hydrogen Atom

The hydrogen atom is perhaps one of the simplest yet most fundamental models in atomic physics. It consists of a single electron orbiting around a single proton. The behavior of this electron forms the basis for understanding more complex atoms as well. The hydrogen atom provides a model for studying how electrons interact within other elements. This atom's simplicity allows scientists to study its energy levels, quantum states, and transition processes.

In essence, the way an electron moves in a hydrogen atom gives us insights into how electrons behave within more complex atoms. By observing the hydrogen atom, we uncover the fundamental principles of atomic energy and structure.

Energy Levels

Energy levels in atoms describe the specific energies that an electron can have while orbiting the nucleus. In the hydrogen atom, these energy levels are quantized, meaning they are like steps on a ladder, and electrons can only exist on these predefined steps. The energy of these levels is calculated using the formula: \[ E_n = -13.6 \frac{1}{n^2} \text{ eV} \]where \(E_n\) is the energy at the nth level, and \(n\) is the principal quantum number.

This formula shows us that the energy becomes increasingly negative as \(n\) lowers, pulling the electron closer to the nucleus. Electrons can jump between these levels, and during such transitions, energy is either absorbed or emitted. The amount of energy emitted or absorbed during these transitions determines the type of electromagnetic radiation that is observed.

Quantum Numbers

Quantum numbers are essential when describing the position and energy of an electron in an atom. In the case of a hydrogen atom, the principal quantum number \(n\) is the most crucial one, as it determines the energy level of the electron. Every possible electron state in an atom is uniquely specified by a set of quantum numbers:

• Principal quantum number (n): Determines the overall energy level.
• Azimuthal quantum number (l): Relates to the shape of the electron's orbital.
• Magnetic quantum number (m_l): Relates to the orientation of the orbital.
• Spin quantum number (m_s): Describes the electron's spin direction.

Understanding quantum numbers helps us describe and predict an electron's behavior and chemistry, providing the basis for the electron configuration of atoms.

Electromagnetic Radiation

Electromagnetic radiation is energy emitted as electrons transition between different energy levels. This radiation spans a range of wavelengths, including visible light, ultraviolet, and infrared, depending on the magnitude of the energy change. When an electron in a hydrogen atom drops from a higher energy level to a lower one, it emits energy in the form of electromagnetic radiation. This happens because moving to a lower energy state releases the energy previously "stored" in the form of excited electron states. The specific amount and type of energy released determine the radiation's place on the electromagnetic spectrum.

Understanding electromagnetic radiation from hydrogen atoms helps in exploring phenomena such as spectral lines in astronomy, which reveal the composition of stars and galaxies. In summary, such transitions give rise to light and radiation, enriching our exploration of the universe.