Question

Consider only transitions involving the \(n=1\) through \(n=5\) energy levels for the \(\mathrm{H}\) atom (see Figures 6.7 and 6.10) (a) How many emission lines are possible, considering only the five quantum levels? (b) Photons of the highest frequency are emitted in a transition from the level with \(n=\)__________ to a level with \(n=\)__________. (c) The emission line having the longest wavelength corresponds to a transition from the level with \(n=\)___________ to the level with \(n=\)_____________

Short answer

(a) 10 emission lines; (b) 5 to 1; (c) 2 to 1.

Step-by-step solution

Determine total possible transitions

To determine the possible emission lines among energy levels from \(n=1\) to \(n=5\), consider that an electron can transition from any higher energy level to any lower level. Calculate these transitions combinatorially. For 5 levels, each level \(n\) has \(n-1\) possible lower levels to transition to. Thus:- From \(n=5\): Transitions to \(n=4, 3, 2, 1\)- From \(n=4\): Transitions to \(n=3, 2, 1\)- From \(n=3\): Transitions to \(n=2, 1\)- From \(n=2\): Transition to \(n=1\)Sum the possible transitions: \(4 + 3 + 2 + 1 = 10\)

Identify transition with the highest frequency

For the highest frequency (and therefore highest energy) transition, an electron must fall between two levels with the largest energy difference. The energy difference \(\Delta E\) between levels \(n_i\) and \(n_f\) is given by:\[ \Delta E = 13.6 \times \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \]The largest energy change occurs for the transition from \(n=5\) to \(n=1\), as the initial and final levels are as far apart as possible.

Determine transition with the longest wavelength

The longest wavelength corresponds to the smallest energy transition. Given the relationship \( \Delta E = hu = \frac{hc}{\lambda} \), a smaller \(\Delta E\) results in a longer wavelength. A transition from \(n=2\) to \(n=1\) represents the smallest energy difference for transitions that generate photons, among the given quantum levels, thus corresponding to the longest wavelength.

Key concepts

Quantum Levels

Quantum levels, sometimes called energy levels, are specific regions around an atom's nucleus where electrons reside. These levels are quantized, meaning electrons can only exist in certain discrete values instead of a continuous spectrum. In the hydrogen atom, these energy levels are symbolized by the principal quantum number, denoted as \(n\). Each of these quantum levels represents a unique orbit that an electron can occupy, and each level has a sublayer representing various electron spins and angular momentum values.
In a hydrogen atom, these energy levels can range from \(n=1\) to \(n=\infty\), where \(n=1\) represents the ground state, or the lowest energy state of the atom. The higher the value of \(n\), the further the electron is from the nucleus, and therefore, the higher the energy level. Importantly, when an electron transitions between these energy levels, it either absorbs or emits energy in the form of a photon.

Emission Lines

Emission lines are unique spectral signatures that occur when electrons transition between different energy levels within an atom. In the case of a hydrogen atom, when an electron falls from a higher energy level to a lower one, it emits energy in the form of a photon. This photon has a specific frequency that corresponds to the energy difference between these two levels.
Because each element has its own distinct set of energy levels, the emission lines act like fingerprints for each element. For hydrogen, the emission lines are observed in different series, such as the Lyman and Balmer series, which are visible under certain conditions.
The total possible number of emission lines from a set of energy levels can be determined by calculating how many possible downward transitions there are. For instance, with levels from \(n=1\) to \(n=5\), there are 10 possible emission lines as calculated by considering all possible paths an electron can take to drop from a higher level to a lower one.

Energy Levels

Energy levels in the hydrogen atom are characterized by the principal quantum number \(n\). These levels define the amount of energy an electron possesses when in a particular orbit around the nucleus. The formula to determine the energy of a specific level is given by:\[E_n = -13.6 \text{ eV} \times \frac{1}{n^2}\]This equation indicates that as \(n\) increases, the energy becomes less negative, meaning the electron has higher potential energy and is less tightly bound to the nucleus.
When electrons transition between these energy levels, they either absorb or release energy. An electron moving from a lower to a higher energy level must absorb energy, often from a photon, while a descent from a higher to a lower level results in the emission of a photon. This relationship between energy levels and electron transitions is fundamental to understanding atomic spectra and the emission of light.

Photon Frequency

The frequency of a photon emitted or absorbed during an electron's transition between energy levels is directly related to the energy change involved in the process. The energy change \(\Delta E\) during a transition from an initial level \(n_i\) to a final level \(n_f\) is described by the formula:\[ \Delta E = 13.6 \times \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\]This energy difference dictates the photon's frequency through the relationship \(\Delta E = hu\), where \(h\) is Planck's constant and \(u\) is the frequency.
The frequency of the photon emitted is proportional to the energy difference:\(\Delta E\), since \(u = \frac{\Delta E}{h}\). Thus, larger energy differences result in higher frequency photons, which correspond to transitions occurring over larger quantum level differences. For example, a transition from \(n=5\) to \(n=1\) would result in the highest possible frequency emission, as it involves the greatest energy change within the specified levels.