Chapter 7, Problem 7.77P

(a)

Interpretation Introduction

Interpretation:

The Bohr radius of an electron in the $n=3$ orbit in the $\text{H}$ atom is to be determined.

Concept introduction:

Bohr’s atomic model had the following postulates:

1) The electrons in atoms revolve around the nucleus in certain circular orbits. The orbits are the energy levels. The orbits closer to the nucleus are of lesser energy than the ones further from the nucleus.

2) The energy of an electron changes sharply upon transition from one energy level to another.

3) The angular momentum of an electron moving around the nucleus is quantized. Thus the angular momentum of an electron is a whole number multiple of $\text{h/}2\pi$.

The Bohr radius equation used to find the radius of an electron’s orbit is as follows:

$r_n=\frac{n^2{h}^2{\in }_0}{\text{π}{m}_e{e}^2}$ (1)

Here,

$n$ is the energy level.

$h$ the Plank’s constant.

${\in }_0$ is a constant related to charge attraction in a vacuum.

$m_e$ is the mass of an electron.

$e$ is the charge on an electron.

(b)

Interpretation Introduction

Interpretation:

The energy of the atom in which the electron is in the $n=3$ orbit is to be determined.

Concept introduction:

Bohr’s atomic model had the following postulates:

1) The electrons in atoms revolve around the nucleus in certain circular orbits. The orbits are the energy levels. The orbits closer to the nucleus are of lesser energy than the ones further from the nucleus.

2) The energy of an electron changes sharply upon transition from one energy level to another.

3) The angular momentum of an electron moving around the nucleus is quantized. Thus the angular momentum of an electron is a whole number multiple of $\text{h/}2\pi$.

(c)

Interpretation Introduction

Interpretation:

The energy of an ${\text{Li}}^{2+}$ ion in which the electron is in the $n=3$ orbit is to be determined.

Concept introduction:

Bohr’s atomic model had the following postulates:

1) The electrons in atoms revolve around the nucleus in certain circular orbits. The orbits are the energy levels. The orbits closer to the nucleus are of lesser energy than the ones further from the nucleus.

2) The energy of an electron changes sharply upon transition from one energy level to another.

3) The angular momentum of an electron moving around the nucleus is quantized. Thus the angular momentum of an electron is a whole number multiple of $\text{h/}2\pi$.

(d)

Interpretation Introduction

Interpretation:

The reason for the difference in the energies of the hydrogen atom and ${\text{Li}}^{2+}$ ion is to be determined.

Concept introduction:

Bohr’s atomic model had the following postulates:

1) The electrons in atoms revolve around the nucleus in certain circular orbits. The orbits are the energy levels. The orbits closer to the nucleus are of lesser energy than the ones further from the nucleus.

2) The energy of an electron changes sharply upon transition from one energy level to another.

3) The angular momentum of an electron moving around the nucleus is quantized. Thus the angular momentum of an electron is a whole number multiple of $\text{h/}2\pi$.