Calculate the wavelength and energy in kilojoules necessary to completely remove an electron from the second shell (m = 2) of a hydrogen atom (R∞ = 1.097 * 10-2 nm-1).

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Identify the initial and final energy levels for the electron transition. In this case, the electron is transitioning from the second energy level (n_i = 2) to being completely removed from the atom (n_f = \(\infty\)).

Use the Rydberg formula to calculate the wavelength (\(\lambda\)) of the photon that will be emitted or absorbed during this transition: \( \frac{1}{\lambda} = R_{\infty} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \). Since n_f = \(\infty\), the term \( \frac{1}{n_f^2} \) becomes 0.

Calculate the energy of the photon using the relationship between energy (E), Planck's constant (h), and the speed of light (c): \( E = \frac{hc}{\lambda} \).

Convert the energy from joules to kilojoules by dividing by 1000, as 1 kilojoule equals 1000 joules.

Summarize the results: the wavelength and energy calculated will be the values necessary to completely remove an electron from the second shell of a hydrogen atom.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bohr Model of the Hydrogen Atom

The Bohr model describes the hydrogen atom as having electrons in specific energy levels or shells. Each shell corresponds to a quantized energy state, with the second shell (n=2) being further from the nucleus than the first. The energy of an electron in a shell can be calculated using the formula E = -R_H/n², where R_H is the Rydberg constant. This model is essential for understanding electron transitions and energy calculations in hydrogen.

Photon Energy and Wavelength Relationship

The energy of a photon is directly related to its wavelength through the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship indicates that shorter wavelengths correspond to higher energy photons. Understanding this concept is crucial for calculating the energy required to remove an electron from an atom, as it links the energy of the photon to the wavelength emitted or absorbed during electron transitions.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion in its gaseous state. For hydrogen, the ionization energy from the second shell can be calculated using the energy levels derived from the Bohr model. This concept is vital for determining the energy needed to completely remove an electron from a specific shell, which is a key part of the question regarding the hydrogen atom.

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