Question

An electron in a hydrogen atom is excited from the ground state to the \(n=4\) state. Comment on the correctness of the following statements (true or false). (a) \(n=4\) is the first excited state. (b) It takes more energy to ionize (remove) the electron from \(n=4\) than from the ground state. (c) The electron is farther from the nucleus (on average) in \(n=4\) than in the ground state. (d) The wavelength of light emitted when the electron drops from \(n=4\) to \(n=1\) is longer than that from \(n=4\) to \(n=2\) (e) The wavelength the atom absorbs in going from \(n=1\) to \(n=4\) is the same as that emitted as it goes from \(n=4\) to \(n=1\)

Short answer

Statements (a), (b), and (d) are false; (c) and (e) are true.

Step-by-step solution

Understanding Energy Levels

The energy levels of an electron in a hydrogen atom are characterized by the principal quantum number \( n \). The ground state has \( n=1 \), the first excited state has \( n=2 \), and so on. The fourth energy level corresponds to \( n=4 \).

Analyzing Statement (a)

Statement (a) claims that \( n=4 \) is the first excited state. This is false because the first excited state occurs at \( n=2 \), following the ground state \( n=1 \).

Analyzing Statement (b)

Statement (b) claims it takes more energy to ionize the electron from \( n=4 \) than from the ground state. This is false; it takes less energy to remove the electron from \( n=4 \) as it is already at a higher energy level than the ground state.

Analyzing Statement (c)

Statement (c) states that the electron is farther from the nucleus on average in \( n=4 \) than in the ground state. This is true. Higher energy levels correspond to larger average distances from the nucleus.

Analyzing Statement (d)

Statement (d) involves comparing wavelengths of emitted light. The energy difference between \( n=4 \) to \( n=1 \) is greater than \( n=4 \) to \( n=2 \), resulting in a shorter wavelength for \( n=4 \to n=1 \). Therefore, this statement is false, as it claims the opposite.

Analyzing Statement (e)

Statement (e) posits that the wavelength absorbed when moving from \( n=1 \) to \( n=4 \) is the same as emitted from \( n=4 \) to \( n=1 \). This is true because energy level transitions are symmetric, and the energy (and thus wavelength) of absorbed and emitted light is the same for the same transition.

Key concepts

Energy Levels

The concept of energy levels in a hydrogen atom revolves around the principal quantum number, denoted as \( n \). Each value of \( n \) corresponds to a specific energy level or "shell" that an electron can occupy. The lowest energy level, called the "ground state," is when \( n = 1 \).

As \( n \) increases, the energy levels become progressively higher, meaning the electron is further from the nucleus. Excited states occur when \( n > 1 \). For example, \( n = 2 \) is the first excited state, not \( n = 4 \); this is a common error. Higher energy levels mean the electron has absorbed energy to move away from the nucleus.

Electron Transitions

Electron transitions refer to the movement of electrons between these energy levels, which occur when an electron absorbs or emits energy. When an electron absorbs energy, it moves to a higher energy level (further from the nucleus), a process called excitation. Conversely, when an electron releases energy, it falls to a lower energy level, a process known as emission.

The specific amount of energy associated with these transitions dictates the frequencies or wavelengths of electromagnetic radiation, most commonly seen in the form of light, that is absorbed or emitted. Understanding these transitions helps in explaining phenomena like atomic spectra.

Ionization Energy

Ionization energy is the energy required to completely remove an electron from its atom or ion. In the case of a hydrogen atom, this means stripping the electron away from the pull of the nucleus, effectively sending it into infinity where it no longer interacts with the atom.

This energy is higher for electrons closer to the nucleus due to the stronger electrostatic force. Therefore, it is easier to ionize an electron from a higher energy state, such as \( n = 4 \), than from the ground state (\( n = 1 \)). Higher energy states require less energy to ionize because the electron is already further away from the nucleus.

Quantum Number

The quantum number is a foundational concept in quantum mechanics used to qualify characteristics such as energy levels. The principal quantum number \( n \) determines the energy level and ultimately the average distance of the electron from the nucleus. The larger the \( n \), the higher the energy level, and the more space the electron occupies, further removing it from the attractive force of the nucleus.

These numbers begin at 1 and proceed upward, linking directly to each distinct energy state of an electron in an atom. They are crucial in predicting the behavior and arrangement of electrons within an atom.

Wavelength Emission

The concept of wavelength emission is directly linked to electron transitions. When an electron transitions from a higher to a lower energy level, it emits energy in the form of electromagnetic radiation. The energy change between levels determines the wavelength of this radiation.

The wavelength is shorter if the energy difference is large and longer if the energy difference is smaller. Thus, a transition from \( n=4 \) to \( n=1 \) releases more energy than from \( n=4 \) to \( n=2 \), resulting in a shorter wavelength for the larger drop. This means emitted wavelengths can tell us much about the initial and final energy levels of an electron transition.