Chapter 28, Problem 88P

(a)

To determine

The ionisation energy required to remove an electron from the ground level.

Answer to Problem 88P

The ionisation energy is $5.1\text{eV}$.

Explanation of Solution

The ground state energy of sodium is $-5.1\text{ eV}$.

The ionisation energy is the energy required to move an electron from the given state to the infinite state and is given by the energy difference between the two states.

Write the relation between $E_n$ and $E_1$ using Bohr model

$E_n=\frac{E_1}{n^2}$                                                               (I)

Here $E_1$ is the ground state energy. $E_n$ is the energy of the $n^{th}$ state and n is the state of the electron.

Write the expression for energy difference between ground state and $n^{th}$ state

$\Delta E=E_n-E_1$                                                         (II)

Here, $\Delta E$ is the energy difference, $E_1$ is the ground state energy. $E_n$ is the energy of the $n^{th}$ state.

Substituting (I) in (II) and substituting $2$ for $n$

$\Delta E=\left(\frac{1}{n^2}-1\right)E_1$                                                          (III)

Substituting $\infty$ for $n$ in (III) to find $\Delta E$ and substituting $-5.1\text{ eV}$ for $E_1$

$\begin{array}{c}\Delta E=\left(\frac{1}{{\infty }^2}-1\right)E_1\\ =-E_1\\ =5.1\text{ eV}\end{array}$

Thus, the ionisation energy is $5.1\text{ eV}$.

(b)

To determine

The wavelength of the emitted radiation during the transition $3d\text{ to }3p$ level.

Answer to Problem 88P

The wavelength of the emitted radiation is $890\text{ nm}$.

Explanation of Solution

The energy of $3d$ level is $-1.6\text{ eV}$ and the energy of $3p$ level is $-3.0\text{ eV}$.

Write the expression for the wavelength of the emitted radiation due to transition.

$\lambda =\frac{hc}{\Delta E}$                                                                    (IV)

Here, $\lambda$ is the wavelength, $h$ is the Planck’s constant, $\Delta E$ is the energy difference and $c$ is the speed of light.

Write the expression for energy difference between $3d\text{ and }3p$ levels

$\Delta E=E_{3d}-E_{3p}$                                                               (V)

Here, $E_{3d}$ is the energy of $3d$ state and $E_{3p}$ is the energy of the $3p$ state.

Substituting (V) in (IV)

$\lambda =\frac{hc}{E_{3d}-E_{3p}}$                                                                   (VI)

Substituting $-1.6\text{ eV}$ for $E_{3d}$ and $-3.0\text{ eV}$ for $E_{3p}$ and $1240\text{ eV}\text{.nm}$ for $hc$ in (VI) to find $\lambda$

$\begin{array}{c}\lambda =\frac{1240\text{ eV}\text{.nm}}{-1.60\text{ eV}-\left(-3.0\text{ eV}\right)}\\ =\frac{1240\text{ eV}\text{.nm}}{3.0\text{ eV}-1.60\text{ eV}}\\ =890\text{ nm}\end{array}$

Thus, the wavelength of the emitted radiation is $890\text{ nm}$.

(c)

To determine

The transition corresponding to characteristic yellow light of sodium.

Answer to Problem 88P

The transition $3p\to 3s$ will emit the characteristic yellow light.

Explanation of Solution

The wavelength of the yellow light is $589\text{nm}$.

The energy of $3s$ level is $-5.1\text{ eV}$ and the energy of $3p$ level is $-3.0\text{ eV}$.

Write the expression for energy

$E=\frac{hc}{\lambda }$                                                                   (VII)

Substituting $1240\text{ eV}\text{.nm}$ for $hc$ and  $589\text{nm}$ for $\lambda$ in (VII) to find $E$

$\begin{array}{c}E=\frac{1240\text{eV}\text{.nm}}{589\text{nm}}\\ =2.11\text{ eV}\end{array}$

Write the expression for energy difference between $3p\text{ and }3s$ level

$\Delta E=E_{3p}-E_{3s}$                                                               (VIII)

Here, $E_{3s}$ is the energy of $3s$ state and $E_{3p}$ is the energy of the $3p$ state.

Substituting $-5.1\text{ eV}$ for $E_{3s}$ and $-3.0\text{ eV}$ for $E_{3p}$ in (VIII) to find $\Delta E$

$\begin{array}{c}\Delta E=-3.0\text{ eV}-\left(-5.1\text{ eV}\right)\\ =2.1\text{ eV}\end{array}$

Since $E$ and $\Delta E$ are approximately the same, the transition $3p\to 3s$ will emit the characteristic yellow light.