The ionization energy of a certain element is 412 kJ/mol....

The ionization energy of a certain element is 412 kJ/mol. When the atoms of this element are in the first excited state, however, the ionization energy is only $126 \mathrm{kJ} / \mathrm{mol} .$ Based on this information, calculate the wavelength of light emitted in a transition from the first excited state to the ground state.

Key Concepts

Ionization Energy

Ionization energy is the minimum energy required to remove an electron from a gaseous atom or ion. It is an intrinsic property that reflects the strength of the attraction between the nucleus and the electrons, and it is usually reported in units such as kilojoules per mole, emphasizing the energy needed for many atoms rather than just a single atom.

Electronic Energy Levels and Excited States

Atoms possess discrete energy levels, with the lowest energy configuration known as the ground state and higher energy configurations referred to as excited states. When an atom is excited, one of its electrons is promoted to a higher energy level, which in turn affects properties such as the ionization energy because less energy is required to remove an electron from an excited state compared to the ground state.

Energy Difference and Transition Emission

The transition of an electron from a higher energy level (excited state) down to a lower energy level (ground state) results in the emission of a photon. The energy of this photon corresponds to the difference between the energy levels, and in the context of ionization energies, this difference can be deduced by subtracting the ionization energy from the excited state from that of the ground state.

Planck's Equation and the Energy-Wavelength Relationship

Planck's equation, E = hc/?, connects the energy of a photon (E) with its wavelength (?) by using Planck's constant (h) and the speed of light (c). This relationship is fundamental in quantum mechanics and is used to determine the wavelength of light emitted or absorbed during electronic transitions in atoms.

Unit Conversion (kJ/mol to J/photon)

Often, energies are provided in units such as kilojoules per mole, which describe energy on a per mole basis. To apply physical equations like Planck’s equation, it is necessary to convert these values to the energy per individual photon. This involves converting kilojoules to joules and then dividing by Avogadro's number to obtain the energy in joules for a single photon.

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