Ground-state ionization energies of some one-electron...

Ground-state ionization energies of some one-electron species are $$\begin{array}{rlrl}\mathrm{H} & =1.31 \times 10^{3} \mathrm{~kJ} / \mathrm{mol} & \mathrm{He}^{+}=5.24 \times 10^{3} \mathrm{~kJ} / \mathrm{mol} \\ \mathrm{Li}^{2+} & =1.18 \times 10^{4} \mathrm{~kJ} / \mathrm{mol} \end{array}$$(a) Write a general expression for the ionization energy of any one-electron species.(b) Use your expression to calculate the ionization energy of $\mathrm{B}^{4+}$ (c) What is the minimum wavelength required to remove the electron from the $n=3$ level of $\mathrm{He}^{+} ?$ (d) What is the minimum wavelength required to remove the electron from the $n=2$ level of $\mathrm{Be}^{3+\gamma}$

Ground-state ionization energies of some one-electron species are
$$\begin{array}{rlrl}\mathrm{H} & =1.31 \times 10^{3} \mathrm{~kJ} / \mathrm{mol} & \mathrm{He}^{+}=5.24 \times 10^{3} \mathrm{~kJ} / \mathrm{mol} \\
\mathrm{Li}^{2+} & =1.18 \times 10^{4} \mathrm{~kJ} / \mathrm{mol}
\end{array}$$(a) Write a general expression for the ionization energy of any one-electron species.(b) Use your expression to calculate the ionization energy of $\mathrm{B}^{4+}$
(c) What is the minimum wavelength required to remove the electron from the $n=3$ level of $\mathrm{He}^{+} ?$
(d) What is the minimum wavelength required to remove the electron from the $n=2$ level of $\mathrm{Be}^{3+\gamma}$

Key Concepts

Bohr Model and Energy Quantization

The Bohr model describes the electron in a one?electron system as moving in circular orbits with quantized energies. The energy levels of these electrons are expressed by an equation where the energy is proportional to -Z²/n², with Z being the nuclear charge and n the principal quantum number. This model is fundamental in explaining the discrete ionization energies of hydrogen-like atoms.

Ionization Energy of Hydrogen-like Ions

Ionization energy refers to the minimum energy required to remove an electron completely from an atom or ion. In hydrogen-like species, this energy is determined by the ground state energy given by the Bohr model. The expression for the ionization energy, typically proportional to Z²/n² (with n = 1 for the ground state), is a central concept in understanding the stability and spectral properties of one-electron ions.

Dependence on Nuclear Charge (Z)

In one-electron species, the ionization energy and the energies of the quantized electron orbits scale with the square of the nuclear charge, Z². As Z increases, the attraction between the nucleus and the electron strengthens, leading to deeper (more negative) energy levels and consequently higher ionization energies. This Z² dependency is crucial for comparing species like H, He?, and Li²?.

Photon Energy and Wavelength Relationship

The relationship between a photon’s energy and its wavelength is given by the equation E = hc/?, where h is Planck’s constant and c is the speed of light. This connection is essential when determining the minimum wavelength of light needed to ionize an electron, as the photon must have at least the energy equal to the ionization energy of the electron state from which it is removed.

Transcript

00:01 For this question with multiple parts, we'll start with the general expression for the ionization energy of any one electron species.

00:09 As mentioned in the chapter and in a previous problem, the energy is going to be equal to the constant 2 .18 times 10 to the negative 18, multiplied by the charge of the ion squared, the positive charge, divided by the energy level in which the electron is.

00:32 Found before removal.

00:35 It's then minus 1 over infinity but 1 over infinity is 0 so we don't need to include zero in our denominator.

00:44 We'll then multiply that by avogadro's number because this will be the energy for just one electron to determine the energy for the removal of a mole of electrons we multiply by avogadro's number.

00:59 So to determine the ionization energy for b4 plus it'll be 2 .18 times 10 to the negative 18 joules multiplied by boron has a charge on it in its nucleus of five so it'll be five squared don't let the four plus confuse you there's still one electron that's why the charge on the cat ion is four plus but the actual charge on the nucleus is five plus we divide that by the location of the last electron which which will be in the lowest energy state energy level 1 and multiply it by avogadro's number and we get 3 .28 times 10 to the 7 joules per mole.

01:44 Now to calculate the wavelength associated with the removal of an electron from helium plus in energy level 3, we use the same equation up here because it's helium, z is 2 and because we're removing it from energy level 3, then n is 3.

02:06 This will give us the energy.

02:08 We ultimately need the wavelength, but we need to get the energy first.

02:12 We get 9 .69 times 10 to negative 19 joules...

Please give Ace some feedback