An electron is excited from the ground state to the n=3...

An electron is excited from the ground state to the $n=3$ state in a hydrogen atom. Which of the following statements are true? Correct any false statements. a. It takes more energy to ionize (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 ground state. c. The wavelength of light emitted if the electron drops from $n=3$ to $n=2$ is 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 $n=3$ is the same as the wavelength of light absorbed to go from $n=1$ to $n=3.$ e. The first excited state corresponds to $n=3.$

An electron is excited from the ground state to the $n=3$ state in a hydrogen atom. Which of the following statements are true? Correct any false statements.
a. It takes more energy to ionize (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 ground state.
c. The wavelength of light emitted if the electron drops from $n=3$ to $n=2$ is 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 $n=3$ is the same as the wavelength of light absorbed to go from $n=1$ to $n=3.$
e. The first excited state corresponds to $n=3.$

Key Concepts

Ionization Energy

Ionization energy is the energy required to remove an electron completely from an atom. Because electrons in higher energy levels (i.e., higher n values) are less tightly bound to the nucleus (their energy states are closer to zero), less energy is needed to ionize an electron from an excited state compared to one in the ground state.

Excited State Classification

The term 'excited state' refers to any electron configuration that has an electron in a higher energy level than the ground state. The first excited state is the level immediately above the ground state (n=2 for hydrogen), so any state beyond that, such as n=3, is a higher excited state and not the first excited state.

Absorption and Emission Processes

Electron transitions involve both absorption (when an electron gains energy and jumps to a higher level) and emission (when it releases energy and drops to a lower level). The wavelengths of photons involved in these processes will generally differ because the energy differences between specific levels vary.

Energy Quantization

In quantum mechanics, electrons in atoms are restricted to specific energy levels determined by the principal quantum number (n). In the hydrogen atom, these energies are quantized according to a formula that relates them inversely to the square of n, allowing only fixed discrete energy values rather than a continuum.

Orbital Radius and Quantum Number

The average distance of an electron from the nucleus in an atom increases with the principal quantum number (n). In simple terms, electrons in higher energy levels are found further from the nucleus, a concept quantifiable by expressions involving n² that relate directly to the orbital radius.

Photon Emission and Energy Transitions

When electrons transition between energy levels, the difference in energy is emitted or absorbed as photons. The relationship between the energy of a photon and its wavelength is given by E = hc/?, meaning that larger energy differences correspond to shorter wavelengths of light.

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