Chapter 7, Problem 7.74P

(a)

Interpretation Introduction

Interpretation:

A general expression for the ionization energy of a one electron species is to be written.

Concept introduction:

Ionization energy is defined as the amount of energy required to remove an electron from an isolated gaseous atom. The energy required to remove an electron from an atom depends on the position of the electron in the atom. The closer the electron is to the nucleus in the atom, the harder it is to pull it out of the atom. As the distance of an electron from the nucleus increases, the magnitude of the forces of attraction between the electron and the nucleus decreases. Thus it becomes easier to remove it from the atom.

The equation to find the difference in the energy between the two levels in hydrogen-like atoms is,

$\Delta E=-2.18\times {10}^{-18}\text{ J}\left(\frac{1}{n_{\text{final}}^2}-\frac{1}{n_{\text{initial}}^2}\right)Z^2$ (1)

(b)

Interpretation Introduction

Interpretation:

The ionization energy of ${\text{B}}^{4+}$ is to be calculated.

Concept introduction:

Ionization energy is defined as the amount of energy required to remove an electron from an isolated gaseous atom. The energy required to remove an electron from an atom depends on the position of the electron in the atom. The closer the electron is to the nucleus in the atom, the harder it is to pull it out of the atom. As the distance of an electron from the nucleus increases, the magnitude of the forces of attraction between the electron and the nucleus decreases. Thus it becomes easier to remove it from the atom.

The general expression for the ionization energy of one mole of a one electron species is $\Delta E=\left(\text{1}\text{.313014}\times {10}^6\right)Z^2\text{ J/mol}$ (2)

(c)

Interpretation Introduction

Interpretation:

The minimum wavelength required to remove the electron from the $n=3$ level of ${\text{He}}^{+}$ is to be determined.

Concept introduction:

Ionization energy is defined as the amount of energy required to remove an electron from an isolated gaseous atom. The energy required to remove an electron from an atom depends on the position of the electron in the atom. The closer the electron is to the nucleus in the atom, the harder it is to pull it out of the atom. As the distance of an electron from the nucleus increases, the magnitude of the forces of attraction between the electron and the nucleus decreases. Thus it becomes easier to remove it from the atom.

The equation that relates to the frequency and wavelength of electromagnetic radiation is as follows:

$\nu =\frac{c}{\lambda }$

Here,

$c$ is the speed of light.

$\lambda$ is the wavelength.

$\nu$ is the frequency of the radiation.

Energy is proportional to the frequency and is expressed by the Plank-Einstein equation as follows:

$\Delta E=h\nu$

Here,

$E$ is the energy.

$h$ is the Plank’s constant.

$\nu$ is the frequency.

The above relation can be modified as follows:

$\Delta E=h\left(\frac{c}{\lambda }\right)$ (3)

(d)

Interpretation Introduction

Interpretation:

The minimum wavelength required to move the electron from $n=2$ level of ${\text{Be}}^{3+}$, is to be determined.

Concept introduction:

The equation to find the difference in the energy between the two levels in hydrogen-like atoms is,

$\Delta E=-2.18\times {10}^{-18}\text{ J}\left(\frac{1}{n_{\text{final}}^2}-\frac{1}{n_{\text{initial}}^2}\right)Z^2$ (1)

The equation that relates to the frequency and wavelength of electromagnetic radiation is as follows:

$\nu =\frac{c}{\lambda }$

Here,

$c$ is the speed of light.

$\lambda$ is the wavelength.

$\nu$ is the frequency of the radiation.

Energy is proportional to the frequency and is expressed by the Plank-Einstein equation as follows:

$\Delta E=h\nu$

Here,

$E$ is the energy.

$h$ is the Plank’s constant.

$\nu$ is the frequency.

The above relation can be modified as follows:

$\Delta E=h\left(\frac{c}{\lambda }\right)$ (3)