Question

Figureis a one-axis graph along which two of the allowed energy values (levels) of an atom are plotted. When the atom is placed in a magnetic field of $\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{}\mathit{T}{\mathit{E}}_{\mathbf{2}}$, the graph changes to that of Figure bbecause of the energy associated with${\overrightarrow{\mathbf{\mu }}}_{\mathbf{o}\mathbf{r}\mathbf{b}}\mathbf{.}\overrightarrow{\mathbf{B}}$. Level ${\mathit{E}}_{\mathbf{1}}$is unchanged, but level splits into a (closely spaced) triplet of levels. What are the allowed values of${\mathit{m}}_{\mathbf{1}}$associated with (a) Energy level ${\mathit{E}}_{\mathbf{1}}$and (b) Energy level ${\mathit{E}}_{\mathbf{2}}$? (c) In joules, what amount of energy is represented by the spacing between the triplet levels?

Short answer

a) Allowed values of$m_l$ associated with energy level$E_1$ is zero.

b)Allowed values of$m_l$ associated with energy level$E_2$asupper line has$m_l=+1$and lower will be$m_l=-1$

c) Amount of energy in joules represented by the spacing between the triplet levels is $4.64\times {10}^{-24}\mathrm{J}$.

Step-by-step solution

Listing the given quantities

Magnetic field,$B_{ext}=0.500\mathrm{T}$

Understanding the concepts of magnetic dipole moment

To calculate the $m_l$values, we can use the formula of energy associated with the orientation of an orbital magnetic dipole moment in an external field. Substituting the values of $∆m_l$, Bohr magnetron, and the magnetic field, we can calculate the spacing between the energy levels.

Formula:

$U=-m_l{\mu }_B{B}_{ext}$

(a) Calculations of the allowed values of ml associated with energy level E1

The energyU associated with the orientation of the orbital magnetic dipole moment in an external magnetic field${\overrightarrow{B}}_{ext}$is

$\begin{array}{rcl}U& =& -{\overrightarrow{\mu }}_{orb}.{\overrightarrow{B}}_{ext}\\ & =& -{\mu }_{orb,z}{B}_{ext}\end{array}$

Since

${\mu }_{orb,z}=-m_l\frac{eh}{4\pi m}$

Also,

$\frac{eh}{4\pi m}={\mu }_B$

Thus,

${\mu }_{orb,z}=-m_l{\mu }_B$

Therefore,

$\begin{array}{rcl}U& =& -{\overrightarrow{\mu }}_{orb}.{\overrightarrow{B}}_{ext}\\ & =& -m_l{\mu }_B{B}_{ext}\end{array}$

From the figure, we can observe that there is no splitting of energy level$E_1$ in the external magnetic field.

Hence its energy does not change by the application of external magnetic field${\overrightarrow{B}}_{ext}$

Hence$U\propto m_l=0$

$\therefore m_l=0$

(b) Calculations of  allowed values of ml associated with energy level E2

The energy level$E_2$ splits into three energy levels, with the middle one unshifted from the original value of energy. $E_2$Hence its$m_l$value is zero.

In other two lines, one line is shifted above the unshifted line and another one is shifted below.

Hence the upper line has$m_l=+1$and lower will be$m_l=-1$

(c) Calculations ofamount of energy in joules represented by the spacing between the triplet levels

The spacing between the energy levels is given by

$\begin{array}{rcl}\Delta U& =& {\mu }_B{B}_{ext}\\ & =& \left|\Delta \left(-m_l{\mu }_B{B}_{ext}\right)\right|\end{array}$

For any pair of adjacent levels in triplet,

$\left|\Delta m_l\right|=1$

Therefore,

where,${\mu }_B$ is Bohr magnetron, and its value is${\mu }_B=9.27\times {10}^{-24}\mathrm{J}/\mathrm{T}$

$\begin{array}{rcl}\Delta U& =& 9.27\times {10}^{-24}\frac{\mathrm{J}}{\mathrm{T}}\times 0.500\mathrm{T}\\ & =& 4.64\times {10}^{-24}\mathrm{J}\end{array}$

Amount of energy in joules represented by the spacing between the triplet levels is $4.64\times {10}^{-24}\mathrm{J}$.