Problem 33 - Chemistry: The Central Science

Question

Is energy emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=4\) to \(n=2,(\mathrm{~b})\) from an orbit of radius \(2.12 \AA\) to one of radius \(8.46 \AA\), (c) an electron adds to the \(\mathrm{H}^{+}\) ion and ends up in the \(n=3\) shell?

Step-by-Step Solution

Answer

In summary, for the given electronic transitions in hydrogen: (a) energy is emitted as the electron moves from n=4 to n=2, (b) energy is absorbed when the orbit radius changes from 2.12 Å to 8.46 Å, and (c) energy is emitted when an electron adds to the H+ ion and ends up in the n=3 shell.

Step 1: Case (a): Transition from n=4 to n=2

Let's calculate the energy change during the transition: \(\Delta E = E_f - E_i = -\cfrac{13.6\,\mathrm{eV}}{n_f^2} - \left(-\cfrac{13.6\,\mathrm{eV}}{n_i^2} \right)\) \(\Delta E = -\cfrac{13.6\,\mathrm{eV}}{2^2} + \cfrac{13.6\,\mathrm{eV}}{4^2}\) \(\Delta E = -3.4\,\mathrm{eV} + 0.85\,\mathrm{eV} = -2.55\,\mathrm{eV}\) The negative result indicates that energy is emitted in this transition.

Step 2: Case (b): Orbit radius changes from 2.12 Å to 8.46 Å

To determine the principal quantum numbers corresponding to the initial and final orbits, we can use the formula for the radius: \(r_n = a_0n^2 = 0.529n^2 \AA\) Solve for n: \(n_i^2 = \cfrac{2.12\,\mathrm{\AA}}{0.529\,\mathrm{\AA}}\) \(n_i^2 = 4\) So, initial quantum number \(n_i = 2\). For final quantum number: \(n_f^2 = \cfrac{8.46\,\mathrm{\AA}}{0.529\,\mathrm{\AA}}\) \(n_f^2 = 16\) So, final quantum number \(n_f = 4\). Now, let's calculate the energy change during the transition: \(\Delta E = E_f - E_i = -\cfrac{13.6\,\mathrm{eV}}{n_f^2} - \left(-\cfrac{13.6\,\mathrm{eV}}{n_i^2} \right)\) \(\Delta E = -\cfrac{13.6\,\mathrm{eV}}{4^2} + \cfrac{13.6\,\mathrm{eV}}{2^2}\) \(\Delta E = -0.85\,\mathrm{eV} + 3.4\,\mathrm{eV} = 2.55\,\mathrm{eV}\) The positive result indicates that energy is absorbed in this transition.

Step 3: Case (c): Electron adds to H+ ion and ends up in n=3 shell

For an electron adding to H+ ion, the initial energy level would be infinity since the electron is very far from the nucleus. Using the energy level formula: \(E_i = -\cfrac{13.6\,\mathrm{eV}}{n_i^2} = -\cfrac{13.6\,\mathrm{eV}}{\infty} = 0\,\mathrm{eV}\) Now calculate the final energy level when the electron is in the n=3 shell. \(E_f = -\cfrac{13.6\,\mathrm{eV}}{n_f^2} = -\cfrac{13.6\,\mathrm{eV}}{3^2}\) \(E_f = -1.51\,\mathrm{eV}\) Now let's calculate the energy change during the transition: \(\Delta E = E_f - E_i = -1.51\,\mathrm{eV} - 0\,\mathrm{eV} = -1.51\,\mathrm{eV}\) The negative result indicates that energy is emitted in this transition.

Key Concepts

Energy Emission and Absorption

When electrons make transitions between different energy levels in a hydrogen atom, energy either gets absorbed or emitted. This process depends on whether the electron is moving to a higher or lower energy state.

Elecrons absorb energy when they move from a lower energy level to a higher one. This transition requires an input of energy as the electron must overcome the attractive force from the nucleus to reach a higher orbit. For instance, moving from the second to the fourth energy level absorbs energy, which can be visualized as climbing uphill.

• Absorption occurs with transitions to higher energy states.
• Energy addition can come from light or other forms of electromagnetic radiation.

Conversely, electrons emit energy when they fall from a higher energy level to a lower one. This emission releases energy because the electron moves closer to the nucleus, revealing a loss of potential energy, similar to sliding downhill. This energy is released often in the form of light.

• Emission happens during transitions to lower energy states.
• Light energy is often emitted as photons during these transitions.

Grasping these concepts of absorption and emission is crucial for understanding how spectral lines are created and why each element has its unique emission spectrum.

Principal Quantum Numbers

The concept of principal quantum numbers is fundamental to quantum mechanics and atomic physics. In the hydrogen atom, each electron occupies a unique energy level or orbit, characterized by the principal quantum number, denoted as \(n\).

This quantum number not only defines the size of the orbit but also its energy, as energy levels increase with \(n\). Lower \(n\) values refer to orbits closer to the nucleus, while higher values indicate orbits further out. For instance, \(n=1\) represents the smallest orbit, which is closest to the nucleus and has the lowest energy.

• The principal quantum number \(n\) dictates the electron's orbital size and energy.
• Lower \(n\) signifies orbits near the nucleus with less energy.
• Higher \(n\) values relate to orbits that are larger and carry more energy.

Each increase in \(n\) signifies a significant step up in energy level, and while electrons need energy to move to a higher \(n\), moving to a lower \(n\) results in energy emission. Familiarizing yourself with this number allows a deeper understanding of electron configurations and the size of atoms.

Hydrogen Atom Electron Transitions

Electron transitions within a hydrogen atom involve movement between specific energy levels, dictated by quantum rules. These transitions can be predicted using the principal quantum number and are key to understanding atomic spectra and energy exchanges.

In practical terms, when an electron in a hydrogen atom transitions from one orbit (defined by \(n_i\)) to another (defined by \(n_f\)), energy changes are calculated. The transition formula \(\Delta E = E_f - E_i\) helps determine if the energy is gained or lost.

• Transiting from a higher \(n\) to a lower \(n\) emits energy.
• Moving from a lower \(n\) to a higher \(n\) absorbs energy.

These transitions are central to the production of spectral lines and the emission of light when electrons drop to lower energy states. Observing hydrogen spectrum lines involves detecting photons released during such transitions, a foundational principle in spectroscopy. Understanding these transitions enables insight into atomic structure and physicochemical behaviors, essential for both chemistry and physics.