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Chapter 8: Problem 108

The ionization energy of a certain element is \(412 \mathrm{~kJ} / \mathrm{mol}\). 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.

Short Answer

Expert verified
After solving, the wavelength of light emitted in the transition from the first excited state to the ground state of the atom will be obtained.

Step by step solution

01

Convert Energy Values

The energy values given are per mole of atoms, so the per atom values can be calculated by dividing by Avogadro's number \(6.022 \times 10^{23}\). The ionization energy in the ground state is \(412 \times 10^3 J / mol = 412 \times 10^3 J / (6.022 \times 10^{23} atoms) \) and in the first excited state is \(126 \times 10^3 J / mol = 126 \times 10^3 J / (6.022 \times 10^{23} atoms)\).
02

Calculate the Energy Change

The energy change in the atom as it transitions from the excited state to the ground state is the difference in ionization energies, so we subtract the lesser energy from the greater: \(412 \times 10^3 J / (6.022 \times 10^{23} atoms) - 126 \times 10^3 J / (6.022 \times 10^{23} atoms)\).
03

Convert Energy to Wavelength

Using the Planck-Einstein relation \(E = hc/ \lambda\), where \(E\) is energy, \(h\) is Planck's constant (\(6.626 \times 10^{-34} J \cdot sec\)), \(c\) is the speed of light (\(3.00 \times 10^8 m/s\)), and \(\lambda\) is wavelength, we can solve for \(\lambda\): \(\lambda = hc/E\). Plug in the energy change calculated in the previous step for \(E\) and solve.

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Most popular questions from this chapter

State whether each of these properties of the representative elements generally increases or decreases (a) from left to right across a period and (b) from top to bottom in a group: metallic character, atomic size, ionization energy, acidity of oxides.

Write ground-state electron configurations for these ions, which play important roles in biochemical processes in our bodies: (a) \(\mathrm{Na}^{+},\) (b) \(\mathrm{Mg}^{2+},\) (c) \(\mathrm{Cl}^{-}\) (d) \(\mathrm{K}^{+},\) (e) \(\mathrm{Ca}^{2+},\) (f) \(\mathrm{Fe}^{2+},(\mathrm{g}) \mathrm{Cu}^{2+},\) (h) \(\mathrm{Zn}^{2+}\)

Define ionic radius. How does the size change when an atom is converted to (a) an anion and (b) a cation?

Match each of the elements on the right with its description on the left: (a) A greenish-yellow gas \(\quad\) Nitrogen \(\left(\mathrm{N}_{2}\right)\) that reacts with water \(\quad\) Boron (B) (b) A soft metal that reacts Aluminum (Al) with water to produce Fluorine \(\left(\mathrm{F}_{2}\right)\) hydrogen Sodium (Na) (c) A metalloid that is hard and has a high melting point (d) A colorless, odorless gas (e) A more reactive metal than iron, which does not corrode in air

Why are the Group \(1 \mathrm{~B}\) elements more stable than the Group IA elements even though they seem to have the same outer electron configuration \(n s^{1}\) in which \(n\) is the principal quantum number of the outermost shell?
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