Question

Calculate the longest and shortest wavelengths of light emitted by electrons in the hydrogen atom that begin in the \(n=6\) state and then fall to states with smaller values of \(n\).

Short answer

The longest wavelength of light emitted is approximately \(8.10 \times 10^{-7}\) m when the electron transitions from n=6 to n=5, while the shortest wavelength of light emitted is approximately \(9.69 \times 10^{-8}\) m when the electron transitions from n=6 to n=1.

Step-by-step solution

Determine the Transition for the Longest Wavelength

We need to find the smallest energy difference among all possible transitions. The smallest energy difference occurs when the electron moves from n=6 (the initial state) to n=5 (the next smallest quantum number).

Calculate the Longest Wavelength

Now we will use the Rydberg formula with n1 = 6 and n2 = 5: \[\frac{1}{\lambda_{long}} = R_H \left(\frac{1}{5^2} - \frac{1}{6^2}\right)\] Plug in the Rydberg constant value: \[\frac{1}{\lambda_{long}} = (1.097 \times 10^7 \,\text{m}^{-1}) \left(\frac{1}{25} - \frac{1}{36}\right)\] Calculate the reciprocal of the wavelength: \[\frac{1}{\lambda_{long}} = 1.235 \times 10^6 \,\text{m}^{-1}\] Now find the longest wavelength by taking the reciprocal: \[\lambda_{long} = \frac{1}{1.235 \times 10^6 \,\text{m}^{-1}} \approx 8.10 \times 10^{-7} \,\text{m}\]

Determine the Transition for the Shortest Wavelength

We need to find the largest energy difference among all possible transitions. The largest energy difference occurs when the electron moves from n=6 (the initial state) to n=1 (the smallest quantum number).

Calculate the Shortest Wavelength

Now we will use the Rydberg formula with n1 = 6 and n2 = 1: \[\frac{1}{\lambda_{short}} = R_H \left(\frac{1}{1^2} - \frac{1}{6^2}\right)\] Plug in the Rydberg constant value: \[\frac{1}{\lambda_{short}} = (1.097 \times 10^7 \,\text{m}^{-1}) \left(\frac{1}{1} - \frac{1}{36}\right)\] Calculate the reciprocal of the wavelength: \[\frac{1}{\lambda_{short}} = 1.032 \times 10^7 \, \text{m}^{-1}\] Now find the shortest wavelength by taking the reciprocal: \[\lambda_{short} = \frac{1}{1.032 \times 10^7 \,\text{m}^{-1}} \approx 9.69 \times 10^{-8} \,\text{m}\]

Final Answer

The longest wavelength of light emitted is approximately \(8.10 \times 10^{-7}\) m, while the shortest wavelength of light emitted is approximately \(9.69 \times 10^{-8}\) m.

Key concepts

Rydberg formula

To calculate the wavelength of light emitted when an electron in a hydrogen atom transitions between energy levels, we use the Rydberg formula. This formula helps us determine the spectral lines of hydrogen. The formula is expressed as:\[ \frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]Where:

• \(\lambda\) represents the wavelength of light emitted.
• \(R_H\) is the Rydberg constant, approximately equal to \(1.097 \times 10^7 \text{ m}^{-1}\).
• \(n_1\) is the energy level the electron is transitioning into.
• \(n_2\) is the energy level the electron is transitioning from.

The formula shows that the wavelength inversely depends on the difference in the inverses of the squares of the principal quantum numbers involved. This dependency is why electrons that transition to levels closer together (small energy difference) emit longer wavelengths, whereas those transitioning to more distant levels (large energy difference) emit shorter wavelengths.

hydrogen atom

The hydrogen atom, made up of a nucleus with a single proton surrounded by an electron, is the simplest atom. Understanding the behavior of electrons in a hydrogen atom provides a fundamental model for learning how electrons transition between different energy levels.In a hydrogen atom, the energy levels are quantized, meaning only specific energy levels are permitted. These levels correspond to different orbits that an electron can inhabit, described by the principal quantum number, \( n \). When an electron in a hydrogen atom jumps from a higher energy level to a lower energy level, it emits a photon. The energy associated with these levels can be calculated and predicts the wavelengths that the hydrogen electron transitions will emit. These emitted wavelengths form the hydrogen spectrum, a series of spectral lines that can be observed as light is refracted through a prism.

quantum transitions

Quantum transitions involve electrons moving from one energy level to another within an atom. In the context of a hydrogen atom, these transitions occur when electrons absorb or emit specific amounts of energy.

• When an electron absorbs energy, it moves to a higher energy level (an excited state).
• Conversely, when an electron loses energy, it transitions to a lower energy level (a less excited or ground state), releasing a photon.

The movement of electrons between these quantized levels is the basis of quantum transitions. Moreover, the amount of energy involved in these transitions is represented by a difference between the initial and final energy states. It dictates the frequency and wavelength of the light emitted or absorbed. These transitions within the hydrogen atom help create the line spectrum that can be observed as part of its unique signature.

energy difference

The energy difference between two quantum states in an atom is a crucial factor in determining the wavelength of the emitted or absorbed light. According to the Planck-Einstein relation, this energy difference \(\Delta E\) is directly related to the frequency \(u\) of the emitted photon:\[\Delta E = hu\]Here, \(h\) is Planck's constant, \(6.626 \times 10^{-34} \text{ Js}\). The energy difference can also be linked to the wavelength \(\lambda\) of the photon using the relation:\[\Delta E = \frac{hc}{\lambda}\]Where \(c\) is the speed of light, \(3.00 \times 10^8 \text{ m/s}\).In the example of the hydrogen atom, larger energy differences yield shorter wavelengths of emitted light due to the inverse relationship between energy and wavelength. This principle helps in identifying the spectral lines observed in hydrogen's emission spectrum, as it shows precise transitions due to the quantized nature of energy levels in atoms.