Chapter 15, Problem 15.6P

Interpretation Introduction

(a)

Interpretation:

The energy of light with a wavelength of $\text{450}\text{nm}$ is to be calculated.

Concept introduction:

The Plank-Einstein equation gives the measure of the energy of an electromagnetic wave. The expression for Plank-Einstein equation is given below.

$E=hc/\lambda$

Answer to Problem 15.6P

The energy of light with a wavelength $\text{450}\text{nm}$ is $2.66\times {\text{10}}^{\text{2}}\text{kJ}{\text{mol}}^{-1}$.

Explanation of Solution

The expression for the energy by Plank-Einstein equation is shown below.

$E=hc/\lambda$…(1)

Where,

• $E$ is the energy of the electromagnetic wave.

• $h$ is the Planks constant having value $6.626\times {10}^{-34}\text{J}\cdot \text{s}$.

• $c$ is the velocity of light having value $3\times {10}^8{\text{ms}}^{-1}$.

• $\lambda$ is the wavelength of light.

The wavelength is given as $\lambda =450\text{nm}$.

The relation between nm and m is given below.

$\text{1}\text{nm}={\text{10}}^{-9}\text{m}$

The conversion of $450\text{nm}$ into m is done as shown below.

$\begin{array}{rll}450\text{nm}& =& \text{450}\text{nm}\times \frac{{\text{10}}^{-9}\text{m}}{\text{1}\text{nm}}\\ & =& 450\times {\text{10}}^{-9}\text{m}\end{array}$

Now, substitute the values of $h$, $c$ and $\lambda$ in the equation (1) and solve for the energy $E$.

$\begin{array}{rll}E& =& \frac{\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)\times \left(3\times {10}^8{\text{ms}}^{-\text{1}}\right)}{\left(450\times {\text{10}}^{-\text{9}}\text{m}\right)}\\ & =& 0.044173\times {10}^{-17}\text{J}\times \frac{{\text{10}}^{-3}\text{kJ}}{1\text{J}}\\ & =& 0.044173\times {10}^{-17}\times {\text{10}}^{-3}\text{kJ}\\ & =& \text{44}\text{.173}\times {10}^{-23}\text{kJ}\end{array}$

The energy in $\text{kJ}{\text{mol}}^{-\text{1}}$, is obtained by multiplying the energy calculated above (in $\text{kJ}$) by the total number of photons present in one mole of photon which is $\text{6}\text{.022}\times {\text{10}}^{\text{23}}{\text{mol}}^{-1}$.

$\begin{array}{rll}E& =& \left(\text{44}\text{.173}\times {10}^{-23}\text{kJ}\right)\times \left(\text{6}\text{.022}\times {\text{10}}^{\text{23}}{\text{mol}}^{-1}\right)\\ & =& 2.66\times {\text{10}}^{\text{2}}\text{kJ}{\text{mol}}^{-1}\end{array}$

Therefore, the energy of light for the given wavelength ($450\text{nm}$) is $2.66\times {\text{10}}^{\text{2}}\text{kJ}{\text{mol}}^{-1}$.

Conclusion

The energy of light is $2.66\times {\text{10}}^{\text{2}}\text{kJ}{\text{mol}}^{-1}$ that has wavelength $\text{450}\text{nm}$.

Interpretation Introduction

(b)

Interpretation:

The energy of light with a wavelength $\text{250}\text{nm}$ is to be stated.

Concept introduction:

The Plank-Einstein equation gives the measure of the energy of an electromagnetic wave. The expression for Plank-Einstein equation is given below.

$E=hc/\lambda$

Answer to Problem 15.6P

The energy of light with a wavelength $\text{250}\text{nm}$ is $4.79\times {\text{10}}^{\text{2}}{\text{kJmol}}^{-1}$.

Explanation of Solution

The expression for the energy by Plank-Einstein equation is shown below.

$E=hc/\lambda$…(1)

Where,

• $E$ is the energy of the electromagnetic wave.

• $h$ is the Planks constant having value $6.626\times {10}^{-34}\text{J}\cdot \text{s}$.

• $c$ is the velocity of light having value $3\times {10}^8{\text{ms}}^{-1}$.

• $\lambda$ is the wavelength of light.

The wavelength is given as $\lambda =250\text{nm}$.

The relation between nm and m is given below.

$\text{1}\text{nm}={\text{10}}^{-9}\text{m}$

The conversion of $250\text{nm}$ into m is done as shown below.

$\begin{array}{rll}250\text{nm}& =& 2\text{50}\text{nm}\times \frac{{\text{10}}^{-9}\text{m}}{\text{1}\text{nm}}\\ & =& 250\times {\text{10}}^{-9}\text{m}\end{array}$

Now, substitute the values of $h$, $c$ and $\lambda$ in the equation (1) and solve for the energy $E$.

$\begin{array}{rll}E& =& \frac{\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)\times \left(3\times {10}^8{\text{ms}}^{-1}\right)}{\left(250\times {\text{10}}^{-9}\text{m}\right)}\\ & =& 0.0795\times {10}^{-17}\text{J}\times \frac{{\text{10}}^{-3}\text{kJ}}{1\text{J}}\\ & =& 0.0795\times {10}^{-17}\times {\text{10}}^{-3}\text{kJ}\\ & =& \text{79}\text{.5}\times {10}^{-23}\text{kJ}\end{array}$

The energy in $\text{kJ}{\text{mol}}^{\text{-1}}$, is obtained by multiplying the energy calculated above (in $\text{kJ}$) by the total number of photons present in one mole of photon which is $\text{6}{\text{.022×10}}^{\text{23}}{\text{mol}}^{\text{-1}}$.

$\begin{array}{rll}E& =& \left(79.5\times {10}^{-23}\text{kJ}\right)\times \left(\text{6}\text{.022}\times {\text{10}}^{\text{23}}{\text{mol}}^{-1}\right)\\ & =& 4.79\times {\text{10}}^{\text{2}}{\text{kJmol}}^{-1}\end{array}$

Therefore, the energy of light for the given wavelength ($250\text{nm}$) is $4.79\times {\text{10}}^{\text{2}}{\text{kJmol}}^{-1}$.

Conclusion

The energy of light is $4.79\times {\text{10}}^{\text{2}}{\text{kJmol}}^{-1}$ that has wavelength $\text{250}\text{nm}$.