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\(?\)

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 \(n=\infty\): \[E_\infty - E_1 = \left(-\dfrac{13.6\text{ eV}}{\infty^2}\right) - \left(-\dfrac{13.6\text{ eV}}{1^2}\right)\]

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.

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: \[\lambda = \dfrac{(6.63 \times 10^{-34} \, \text{Js}) (3.00 \times 10^8 \, \text{m/s})}{2.18\times 10^{-18}\,\text{J}} = 9.11 \times 10^{-8} \, \text{m}\]

(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.