Consider the electronic structure of the element bismuth. (a) The first ionization energy of bismuth is Ei1 = +703 kJ/ mol. What is the longest possible wavelength of light that could ionize an atom of bismuth?

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Identify the relationship between energy and wavelength using the equation: \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is Planck's constant (6.626 \(\times\) 10^{-34} \(\text{ J s}\)), \( c \) is the speed of light (3.00 \(\times\) 10^8 \(\text{ m/s}\)), and \( \lambda \) is the wavelength.

Convert the ionization energy from kJ/mol to J/atom. Since 1 mol contains Avogadro's number of atoms (6.022 \(\times\) 10^{23} \(\text{ atoms/mol}\)), use the conversion: \( 703 \text{ kJ/mol} \times \frac{1000 \text{ J}}{1 \text{ kJ}} \times \frac{1 \text{ mol}}{6.022 \times 10^{23} \text{ atoms}} \).

Substitute the converted energy value into the equation \( E = \frac{hc}{\lambda} \) to solve for \( \lambda \).

Rearrange the equation to solve for wavelength: \( \lambda = \frac{hc}{E} \).

Calculate the wavelength using the values for \( h \), \( c \), and the converted \( E \) to find the longest possible wavelength of light that could ionize an atom of bismuth.

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Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom in its gaseous state. For bismuth, the first ionization energy is 703 kJ/mol, indicating the energy needed to remove the outermost electron. This concept is crucial for understanding how much energy is necessary to ionize an atom and relates directly to the wavelength of light that can provide this energy.

Photon Energy and Wavelength Relationship

The energy of a photon is inversely related to its wavelength, described by 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 means that longer wavelengths correspond to lower energy photons. To find the longest wavelength capable of ionizing bismuth, one must calculate the wavelength that corresponds to the ionization energy.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics, valued at approximately 6.626 x 10^-34 J·s. It relates the energy of a photon to its frequency and wavelength. Understanding this constant is essential for calculating the energy of photons and determining the wavelength of light needed to achieve specific energy transitions, such as ionization.