Question

Consider an electron for a hydrogen atom in an excited state. The maximum wavelength of electromagnetic radiation that can completely remove (ionize) the electron from the \(\mathrm{H}\) atom is \(1460 \mathrm{~nm} .\) What is the initial excited state for the electron \((n=?) ?\)

Short answer

The initial excited state of the electron in the hydrogen atom for the given maximum wavelength of \(1460 \mathrm{~nm}\) is \(n_i = 3\).

Step-by-step solution

Rydberg formula

We will use the Rydberg formula which relates the wavelength of electromagnetic radiation (λ) with the initial (n_i) and final (n_f) energy levels of the electron in a hydrogen atom: \[ \frac{1}{\lambda}=R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \] where \(R_H\) is the Rydberg constant for hydrogen, approximately equal to \(1.097 \times 10^7 \mathrm{~m^{-1}}\).

Ionization condition

For ionization to occur, the electron needs to completely escape the hydrogen atom, which implies that the final energy level (\(n_f\)) is infinite. Thus, we can rewrite the Rydberg formula in terms of the initial energy level (\(n_i\)) only: \[ \frac{1}{\lambda}=R_H \left(0 - \frac{1}{n_i^2}\right)=-R_H\frac{1}{n_i^2} \]

Solve for the initial energy level

Now, insert the value of the maximum wavelength (\(\lambda = 1460 \mathrm{~nm}\)) and the Rydberg constant for hydrogen into the formula: \[ \frac{1}{1460 \times 10^{-9} \mathrm{~m}}=-1.097 \times 10^7 \mathrm{~m^{-1}}\frac{1}{n_i^2} \] First we will isolate \(n_i^2\): \[ n_i^2 = -\frac{1.097 \times 10^7 \mathrm{~m^{-1}}}{1460 \times 10^{-9} \mathrm{~m}} \] Now calculate the value of \(n_i^2\): \[ n_i^2 = \frac{1.097 \times 10^7 \mathrm{~m^{-1}}}{1460 \times 10^{-9} \mathrm{~m}} \approx 7.511 \] Finally, take the square root to find the initial energy level, \(n_i\): \[ n_i = \sqrt{7.511}\approx 2.74 \]

Conclusion

Since the initial energy level (n) must be an integer value, and we calculated n ≈ 2.74, we can conclude that the initial energy level of the electron for the given wavelength of electromagnetic radiation is \(n_i = 3\).

Key concepts

Hydrogen Atom Ionization

Ionization of a hydrogen atom takes place when an electron is given enough energy to break free from the attractive force of the nucleus. This process converts the atom into a positively charged ion. The ionization energy is the minimum energy required to remove the electron from its ground state, where it is most strongly bound to the nucleus. In quantum mechanics, an electron orbits the nucleus at discrete energy levels or shells.

When an electron is at an excited state, it is already at a higher energy than the ground state. Providing energy equal to the difference between the current excited state and infinity (which represents complete separation of the electron) will ionize the atom. In the textbooks' case, the wavelength of 1460 nm is the threshold wavelength — the maximum wavelength which is still capable of causing ionization. By using the Rydberg formula, we can calculate the initial energy level from which an electron would need to be ionized at this wavelength.

Excited State

An excited state of an electron in a hydrogen atom refers to any energy level above the ground state (n=1). When an electron absorbs energy, it can 'jump' to a higher energy level, or orbital, which is quantized and denoted by an integer value 'n'. The larger the value of 'n', the farther away the electron is from the nucleus and the higher its potential energy. Excited states are not permanent; electrons tend to return to a lower energy level, releasing energy in the form of electromagnetic radiation.

In the exercise, the initial excited state is what we're trying to determine. Based on the provided wavelength of the electromagnetic radiation capable of ionizing the atom, the Rydberg formula allows us to work backwards to find this initial state, which is indicated to be an integer value. The calculated value of 2.74 suggests that the electron's initial state was the third energy level (n=3), as states are whole numbers and we round up from the calculated value.

Electromagnetic Radiation Wavelength

The wavelength of electromagnetic radiation is a key factor in determining the amount of energy carried by that radiation. It is inversely proportional to the energy; thus, radiation with a longer wavelength has less energy. The spectrum of electromagnetic radiation includes a wide range of wavelengths, from short gamma rays to very long radio waves.

In the context of the hydrogen atom, when an electron transitions between energy levels, it must emit or absorb radiation with a wavelength that corresponds to the energy difference between these levels. The Rydberg formula relates this wavelength to the initial and final energy levels of the atomic transition. In our problem, the maximum wavelength (1460 nm) is that which just allows the electron to be ionized from the hydrogen atom — any longer, and the radiation would not carry enough energy to liberate the electron from its energy state.