Chapter 2, Problem 24Q

The following is an energy-level diagram for electronic transitions in the Bohr hydrogen atom.

a. Explain why the energy levels get closer together as they increase. Provide mathematical support for this.

b. Verify that the colors given in the diagram are correct. Provide mathematical support.

Interpretation Introduction

Interpretation:

The reason is to be given with mathematical support that energy levels get closer together as they increase. Also  the colours given in the diagram is to be checked.

Concept introduction:

The energy level diagram is simply a spectral line which gives the specific amount of energy for transition of energy levels.

To determine: The explanation with mathematical support that energy levels get closer together as they increase.

Answer to Problem 24Q

Answer

Energy levels get closer together because the negative value of energy will become less with the increase in the value of energy levels.

Explanation of Solution

The Danish physicist Niels Bohr proposed the Bohr Model of hydrogen atom in 1913. This model is also applicable for the atoms having one electron or atoms having hydrogen like characteristics.

The separation in lines in the energy level diagram signifies the separation of energy between the orbits. Hydrogen atom consists of only one electron. Hence, it is present at ground state. The difference in the energy state of orbits reduces for higher levels. This is the main reason that the separation between the lines decreases for higher levels.

The expression of energy of hydrogen given by Bohr is.

$E_{\text{n}}=-2.178\times {10}^{-18}\left(\frac{Z^2}{n^2}\right)\text{J}$

Where,

• $E_{\text{n}}$ is energy of electron.
• $n$ is an integer orbit in which electrons resides.
• $Z$ is nuclear charge.

The integer value of orbit is inversely proportional to negative energy. As the value of integer is $n^2$ this means that negative value of energy will become less with the increase in the value of $n$. Hence, the separation between the lines decreases for higher levels.

Interpretation Introduction

Interpretation:

The reason is to be given with mathematical support that energy levels get closer together as they increase. Also  the colours given in the diagram is to be checked.

Concept introduction:

The energy level diagram is simply a spectral line which gives the specific amount of energy for transition of energy levels.

To determine: The correctness of red color transition given in the diagram with mathematical support.

Answer to Problem 24Q

Answer

The red color transition is correct.

Explanation of Solution

The diagram below shows the transition state from high to low:

Consider the red color transition from state $\text{n=3 to n=2}\text{.}$.

The expression of resultant energy is:

$\Delta E=E_{\text{f}}-E_{\text{i}}$ (1)

Where,

• $\Delta E$ is resultant energy.
• $E_{\text{f}}$ is energy in final state.
• $E_{\text{i}}$ is energy in initial state.

The expression of energy of hydrogen given by Bohr is,

$E_{\text{n}}=-2.178\times {10}^{-18}\left(\frac{Z^2}{n^2}\right)\text{J}$ (2)

The energy of red color transition from state $n=3\text{to}n=2$ using expression (1) and (2) is calculated,

$\begin{array}{c}\Delta E=E_2-E_1\\ =\left[-2.178\times {10}^{-18}\left(\frac{1^2}{2^2}\right)\right]-\left[-2.178\times {10}^{-18}\left(\frac{1^2}{3^2}\right)\right]\\ =-3.03\times {10}^{-19}\text{J}\end{array}$

The negative sign suggests that energy is lost during state transition which means that atom becomes more stable.

The expression of change in energy with respect to wavelength is,

$\begin{array}{c}\Delta E=\frac{hc}{\lambda }\\ \lambda =\frac{hc}{\Delta E}\end{array}$

Where,

• $h$ is Planks constant.
• $c$ is the speed of light.
• $\lambda$ is the wavelength.
• $\Delta E$ is change in energy.

The value of $h$ is $6.626\times {10}^{-34}\text{J}\cdot \text{s}$.

The value of $c$ is $3\times {10}^8\text{m}/\text{s}$.

Substitute the value of $\Delta E$, $h$ and $c$ in above expression.

$\begin{array}{c}\lambda =\frac{6.626\times {10}^{-34}\text{J}\cdot \text{s}\times 3\times {10}^8\text{m}/\text{s}}{-3.03\times {10}^{-19}\text{J}}\\ =6.56\times {10}^{-7}\text{m}\end{array}$

It is clear from the above calculated value of wavelength that red light is in visible spectrum. Hence, color given in this transition is correct.

Interpretation Introduction

Interpretation:

The reason is to be given with mathematical support that energy levels get closer together as they increase. Also  the colours given in the diagram is to be checked.

Concept introduction:

The energy level diagram is simply a spectral line which gives the specific amount of energy for transition of energy levels.

To determine: The correctness of green color transition given in the diagram with mathematical support.

Answer to Problem 24Q

Answer

The green color transition is correct.

Explanation of Solution

The diagram below shows the transition state from high to low:

Consider the green color transition from state $n=4\text{to}n=2$.

The expression of resultant energy is,

$\Delta E=E_{\text{f}}-E_{\text{i}}$ (1)

The expression of energy of hydrogen given by Bohr is,

$E_{\text{n}}=-2.178\times {10}^{-18}\left(\frac{Z^2}{n^2}\right)\text{J}$ (2)

The energy of green color transition from state $n=4\text{to}n=2$ using expression (1) and (2) is calculated,

$\begin{array}{c}\Delta E=E_2-E_1\\ =\left[-2.178\times {10}^{-18}\left(\frac{1^2}{2^2}\right)\right]-\left[-2.178\times {10}^{-18}\left(\frac{1^2}{4^2}\right)\right]\\ =-4.09\times {10}^{-19}\text{J}\end{array}$

The negative sign suggests that energy is lost during state transition which means that atom becomes more stable.

The expression of change in energy with respect to wavelength:

$\begin{array}{c}\Delta E=\frac{hc}{\lambda }\\ \lambda =\frac{hc}{\Delta E}\end{array}$

Substitute the value of $\Delta E$, $h$ and $c$ in above expression.

$\begin{array}{c}\lambda =\frac{6.626\times {10}^{-34}\text{J}\cdot \text{s}\times 3\times {10}^8\text{m}/\text{s}}{4.09\times {10}^{-19}\text{J}}\\ =4.86\times {10}^{-7}\text{m}\end{array}$

It is clear from the above calculated value of wavelength that green light is in visible spectrum. Hence, color given in this transition is correct.

Interpretation Introduction

Interpretation:

The reason is to be given with mathematical support that energy levels get closer together as they increase. Also  the colours given in the diagram is to be checked.

Concept introduction:

The energy level diagram is simply a spectral line which gives the specific amount of energy for transition of energy levels.

To determine: The correctness of blue color transition given in the diagram with mathematical support.

Answer to Problem 24Q

Answer

The blue color transition is correct.

Explanation of Solution

The diagram below shows the transition state from high to low:

Consider the blue color transition from state $n=5\text{to}n=2$.

The expression of resultant energy is:

$\Delta E=E_{\text{f}}-E_{\text{i}}$ (1)

The expression of energy of hydrogen given by Bohr is,

$E_{\text{n}}=-2.178\times {10}^{-18}\left(\frac{Z^2}{n^2}\right)\text{J}$ (2)

The energy of blue color transition from state $n=5\text{to}n=2$ using expression (1) and (2) is calculated,

$\begin{array}{c}\Delta E=E_2-E_1\\ =\left[-2.178\times {10}^{-18}\left(\frac{1^2}{2^2}\right)\right]-\left[-2.178\times {10}^{-18}\left(\frac{1^2}{5^2}\right)\right]\\ =-4.58\times {10}^{-19}\text{J}\end{array}$

The negative sign suggests that energy is lost during state transition which means that atom becomes more stable.

The expression of change in energy with respect to wavelength:

$\begin{array}{c}\Delta E=\frac{hc}{\lambda }\\ \lambda =\frac{hc}{\Delta E}\end{array}$

Substitute the value of $\Delta E$, $h$ and $c$ in above expression.

$\begin{array}{c}\lambda =\frac{6.626\times {10}^{-34}\text{J}\cdot \text{s}\times 3\times {10}^8\text{m}/\text{s}}{4.58\times {10}^{-19}\text{J}}\\ =4.34\times {10}^{-7}\text{m}\end{array}$

It is clear from the above calculated value of wavelength that blue light is in visible spectrum. Hence, color given in this transition is correct.