Question

Consider a transition in which the electron of a hydrogen atom is excited from \(n=1\) to \(n=\infty\) . (a) What is the end result of this transition? (b) What is the wavelength of light that must be absorbed to accomplish this process? (c) What will occur if light with a shorter wavelength that in part (b) is used to excite the hydrogen atom? (d) How are the results of parts (b) and (c) related to the plot shown in Exercise 6.88\(?\)

Short answer

(a) The end result of the transition is that the electron becomes unbound from the hydrogen atom, turning it into a free electron and an ion. (b) The wavelength of light needed for this transition is \(9.11 \times 10^{-8}\) m. (c) If a shorter wavelength is used, the electron will absorb more energy, making it more likely to be removed from the atom. (d) Parts (b) and (c) are related to the plot in Exercise 6.88, which shows energy levels and transitions becoming more closely spaced as n approaches infinity.

Step-by-step solution

Calculate the energy required for the transition

To find the energy difference, we will use the Rydberg formula, which calculates the energy levels of hydrogen: \[E_n = -\dfrac{13.6 \, \text{eV}}{n^2}\] Where \(E_n\) is the energy of the hydrogen atom at the level \(n\). We must find the energy difference between the ground state \((n=1)\) and the state with \(n=\infty\): \[E_\infty - E_1 = \left(-\dfrac{13.6\text{ eV}}{\infty^2}\right) - \left(-\dfrac{13.6\text{ eV}}{1^2}\right)\]

Evaluate the energy difference

Now, we will simplify the expression: \[E_\infty - E_1 = 0 - \left(-13.6\, \text{eV}\right) = 13.6 \, \text{eV}\] The energy difference is 13.6 eV.

Convert energy difference to wavelength

Next, we will convert this energy difference in eV to wavelength in meters using the formula: \[\lambda = \dfrac{hc}{E}\] Where \(\lambda\) is the wavelength, \(h\) is the Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\)), \(c\) is the speed of light in a vacuum (\(3.00 \times 10^8 \, \text{m/s}\)), and \(E\) is the energy in joules. First, convert the energy difference to joules: \[13.6 \ \text{eV}\times \frac{1.6\times10^{-19}\,\text{J}}{1\,\text{eV}}= 2.18\times 10^{-18}\,\text{J}\] Now, calculate the wavelength: \[\lambda = \dfrac{(6.63 \times 10^{-34} \, \text{Js}) (3.00 \times 10^8 \, \text{m/s})}{2.18\times 10^{-18}\,\text{J}} = 9.11 \times 10^{-8} \, \text{m}\]

Answer the questions

(a) The end result of this transition is that the electron will no longer be bound to the hydrogen atom and will become a free electron. The hydrogen atom will become an ion. (b) The wavelength of light that must be absorbed to accomplish this process is \(9.11 \times 10^{-8} \, \text{m}\). (c) If light with a shorter wavelength than part (b) is used, the electron will absorb more energy than required, making it more likely that it will be removed from the atom. (d) Parts (b) and (c) are related to the plot shown in Exercise 6.88 because the energy levels and corresponding transitions become more closely spaced as n approaches infinity, which is shown in Exercise 6.88. As the wavelength of light used to excite the atom becomes shorter, more energy is given to the electron, causing the electron to be ejected from the atom.

Key concepts

Rydberg formula

The Rydberg formula is essential for understanding the transitions of electrons within a hydrogen atom. It is primarily used to calculate the energy levels of hydrogen and can be expressed as follows:
\[E_n = -\dfrac{13.6 \, \text{eV}}{n^2}\]
This formula helps us determine the energy of an electron at a specific energy level, denoted by \(n\). In our context, these energy levels are crucial for calculating the energy difference during electron transitions.

• The negative sign indicates that energy is required to free the electron from the atom.
• \(n\) represents the principal quantum number, indicating the different energy levels.

Understanding the Rydberg formula is a fundamental step in exploring the behavior of electrons in hydrogen atoms and assessing how they absorb or emit light during transitions.

Energy levels

Energy levels are the allowed states an electron can occupy in an atom, each characterized by a specific amount of energy. In hydrogen atoms, these levels are quantized, and the formula for these levels is derived from the Rydberg formula.
In the case where an electron transitions from \(n=1\) to \(n=\infty\), it involves a shift from the lowest energy state to a completely free state of an electron (ionization).

• At \(n=1\), the electron is in the ground state, confined and most stable.
• At \(n=\infty\), the electron escapes, indicating no longer bound by the atom's nucleus.

The concept of energy levels is crucial to understanding electron transitions, particularly when assessing the energy change that corresponds with moving from one level to another.

Wavelength calculation

To determine the wavelength of light that an electron needs to absorb for a transition, we use the energy calculated from the Rydberg formula and apply a separate equation.
Wavelength is found using the energy-wavelength relationship:
\[ \lambda = \dfrac{hc}{E} \]
This involves:

• \(h\), Planck's constant: typically \(6.63 \times 10^{-34} \, \text{Js}\).
• \(c\), speed of light: \(3.00 \times 10^8 \, \text{m/s}\).
• \(E\), energy in joules from the Rydberg formula.

By converting the energy difference from eV to joules and substituting these values into the equation, we calculate the specific wavelength, such as the light wavelength enabling the transition to ionize the hydrogen atom to a free electron state.

Electron excitation

Electron excitation is the process where an electron absorbs energy and moves from one energy level to a higher one. In the context of hydrogen atoms, this phenomenon is significant when shifting between varied states.
For instance, if a hydrogen atom's electron is excited from \(n=1\) to \(n=\infty\):

• The electron absorbs sufficient energy to escape the atom's nucleus.
• It transitions from being bound to becoming a free electron.

This can occur through photon absorption, where the wavelength needs to match the energy difference for the specific level jump. Understanding this concept allows us to predict how hydrogen atoms interact with light, crucial for fields like spectroscopy and astrophysics.