Question

An electron is excited from the \(n=1\) ground state to the \(n=3\) state in a hydrogen atom. Which of the following statements are true? Correct the false statements to make them true. a. It takes more energy to ionize (completely remove) the electron from \(n=3\) than from the ground state. b. The electron is farther from the nucleus on average in the \(n=3\) state than in the \(n=1\) state. c. The wavelength of light emitted if the electron drops from \(n=3\) to \(n=2\) will be shorter than the wavelength of light emitted if the electron falls from \(n=3\) to \(n=1\). d. The wavelength of light emitted when the electron returns to the ground state from \(n=3\) will be the same as the wavelength of light absorbed to go from \(n=1\) to \(n=3\). e. For \(n=3\), the electron is in the first excited state.

Short answer

The following statements are true after correcting the false ones: a. It takes less energy to ionize (completely remove) the electron from \(n=3\) than from the ground state. b. The electron is farther from the nucleus on average in the \(n=3\) state than in the \(n=1\) state. c. The wavelength of light emitted if the electron drops from \(n=3\) to \(n=2\) will be longer than the wavelength of light emitted if the electron falls from \(n=3\) to \(n=1\). d. The wavelength of light emitted when the electron returns to the ground state from \(n=3\) will be the same as the wavelength of light absorbed to go from \(n=1\) to \(n=3\). e. For \(n=3\), the electron is in the second excited state.

Step-by-step solution

Statement a

In a hydrogen atom, the ionization energy is the energy required to remove the electron completely from the atom. As the principal quantum number (n) increases, the electron moves farther from the nucleus, and the attractive force between the nucleus and the electron decreases. Therefore, it takes less energy to ionize the electron from a higher energy state. Thus, statement a is false. The correct statement should be: "It takes less energy to ionize (completely remove) the electron from n=3 than from the ground state."

Statement b

The electron is indeed farther from the nucleus on average in the n=3 state than in the n=1 state. This is consistent with the fact that as the principal quantum number (n) increases, the electron orbits get larger and more distant from the nucleus. Thus, statement b is true.

Statement c

To verify statement c, we need to use the formula for the wavelength of emitted light during an electron's transition, which is given by: \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \) where \(R_H\) is the Rydberg Constant, \(n_f\) is the final quantum number, and \(n_i\) is the initial quantum number. For the transition of the electron from n=3 to n=2, the formula becomes: \( \frac{1}{\lambda_{32}} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \) For the transition of the electron from n=3 to n=1, the formula becomes: \( \frac{1}{\lambda_{31}} = R_H \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \) When comparing the two formulas, it's clear that \( \frac{1}{\lambda_{31}} > \frac{1}{\lambda_{32}} \), which means that \( \lambda_{32} > \lambda_{31} \). Therefore, statement c is false, and the corrected statement should be: "The wavelength of light emitted if the electron drops from n=3 to n=2 will be longer than the wavelength of light emitted if the electron falls from n=3 to n=1."

Statement d

When an electron returns to the ground state, the energy of the emitted photon is equal to the energy difference between the final and initial energy levels. The same energy is required to excite an electron from the ground state to the higher energy state. Since the energy of the photon is related to its wavelength, the emitted wavelength when the electron returns to the ground state from n=3 is the same as the absorbed wavelength to go from n=1 to n=3. Thus, statement d is true.

Statement e

The wording of statement e is slightly incorrect. When the electron is in the n=3 state, it is actually in the second excited state (since the ground state is n=1 and the first excited state is n=2). Thus, statement e is false and should be corrected to: "For n=3, the electron is in the second excited state."