Question

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=6,(b)\) from an orbit of radius \(4.76 \AA\) to one of radius \(0.529 \AA,(\mathrm{c})\) from the \(n=6\) to the \(n=9\) state.

Short answer

In the given electronic transitions for hydrogen: (a) Energy is absorbed when transitioning from n=2 to n=6. (b) Energy is emitted when transitioning from an orbit of radius \(4.76 \AA\) to one of radius \(0.529 \AA\). (c) Energy is absorbed when transitioning from n=6 to n=9.

Step-by-step solution

(a) From n=2 to n=6 Transition.

For this transition, first calculate the energy of the electron in the initial state (n=2) and the final state (n=6) using the formula mentioned above. Energy in initial state (n=2): \[E_2 = - \frac{13.6\: eV}{2^2} = -3.4\: eV\] Energy in final state (n=6): \[E_6 = - \frac{13.6\: eV}{6^2} = -0.378\: eV\] Comparing the energies, since the electron is moving to a higher energy state, energy is absorbed in this transition.

(b) From \(r_1=4.76\AA\) to \(r_2=0.529\AA\) Transition:

We'll need to find the energy levels corresponding to the given radii. The radius of an electron orbit in the Bohr's model is given by: \[r_n = 0.529 \cdot n^2\: Å\] Let r_1 and n_1 represent the initial radius and energy level, and r_2 and n_2 represent the final radius and energy level. We have: \(r_1 = 4.76 \AA\) \(r_2 = 0.529 \AA\) Now, we will find n_1 and n_2 from the radii given. \[n_1^2 = \frac{r_1}{0.529} = 9\] \[n_1 = 3\] \[n_2^2 = \frac{r_2}{0.529} = 1\] \[n_2 = 1\] Now, find the energy of the initial state (n=3) and the final state (n=1) using Bohr's energy level formula. Energy in initial state (n=3): \[E_3 = - \frac{13.6\: eV}{3^2} = -1.511\: eV\] Energy in final state (n=1): \[E_1 = - \frac{13.6\: eV}{1^2} = -13.6\: eV\] Comparing the energies, we notice that since the electron is moving to a lower energy state, energy is emitted in this transition.

(c) From n=6 to n=9 Transition.

For this transition, calculate the energy of the electron in the initial state (n=6) and the final state (n=9) using the formula mentioned above. Energy in initial state (n=6): \[E_6 = - \frac{13.6\: eV}{6^2} = -0.378\: eV\] Energy in final state (n=9): \[E_9 = - \frac{13.6\: eV}{9^2} = -0.168\: eV\] Comparing the energies, we notice that since the electron is moving to a higher energy state, energy is absorbed in this transition. To sum up, (a) Energy is absorbed. (b) Energy is emitted. (c) Energy is absorbed.

Key concepts

Electron Transitions

Electron transitions are fascinating occurrences where electrons move between different energy states or levels within an atom. In the context of the Bohr model of the hydrogen atom, these transitions occur when electrons jump from one 'orbit' or level to another. Each orbit corresponds to a specific energy level, denoted by the principal quantum number, \(n\). The movements of electrons between these energy levels correlate with either the absorption or emission of energy.

• When an electron moves to a higher energy level, it absorbs energy.
• Conversely, when an electron transitions to a lower energy level, it emits energy.

This gain or loss of energy manifests as photons, which are particles of light. The energy change can be calculated through the energy difference between the initial and final states using the equation for the energy of an electron in a particular state.

Energy Levels

The concept of energy levels is integral to understanding how electrons are situated within an atom. In the Bohr model, each energy level corresponds to a specific orbit around the nucleus and is characterized by the quantum number \(n\). These levels are like the steps of a ladder, and electrons can occupy only these defined levels, no in-between states.

• The lower the energy level (closer to the nucleus), the more negative and more stable it is.
• As energy levels increase (farther from the nucleus), the levels are less negative, indicating higher energy potential and lower stability.

The energy associated with each level in a hydrogen atom can be calculated using the formula:
\[ E_n = - rac{13.6\, \text{eV}}{n^2} \]This formula shows that energy becomes less negative as \(n\) increases, which reflects the higher energy state of electrons located in outer orbits.

Hydrogen Atom

The hydrogen atom, with its single electron and proton, serves as an essential model for understanding atomic structure and electron transitions. In the Bohr model of the hydrogen atom, the electron revolves around the nucleus in distinct circular orbits or shells.

• This simple hydrogen system exemplifies fundamental principles due to the atom's simplicity.
• The concept of quantized energy levels, where electrons have set energy levels, was revolutionary and remains foundational in atomic physics.

The Bohr model explains many spectroscopic properties of hydrogen, helping us understand emission and absorption spectra, which results from electrons transitioning between these levels.

Energy Absorption and Emission

Energy absorption and emission are key processes in the functioning of atoms, particularly phenomena like spectroscopy. When an electron absorbs energy, it transitions to a higher energy level, leading to an absorption spectrum. This occurs because specific wavelengths of light are absorbed by electrons, creating distinct spectral lines for each element.

Conversely, as electrons return to a lower energy level, they release energy, producing an emission spectrum. These emissions manifest as photons in discrete wavelengths, each corresponding to specific energy changes as defined by electron transitions.

• Energy absorbed is proportional to the energy difference between the initial and final states of the electron.
• Similarly, energy emitted equals this difference when electrons relax back to lower energy levels.

Ultimately, the study of these processes provides insight into the electron configuration and energy dynamics within atoms, allowing us to decode the material composition and behaviors of substances through their interaction with light.