Question

How would I calculate the energy (in kilojoules) for $1$ mole of a photon having a wavelength of $623$ nm? If I first calculated the energy of the photon in joule and got $3.20 \times {10^{ - 37}}$ J.

Answer 1

Hint:To determine the answer to this question we should know the energy and wavelength relation, conversion of J to kJ and nm to m. First we will convert the wavelength unit from nm to m. Then by using wavelength and energy relation we will calculate the energy of one photon. Then by using Avogadro number we will calculate the energy of one mole photon.

Formula used: ${\text{E = }}\dfrac{{{\text{hc}}}}{{{\lambda }}}$

Complete step-by-step solution:The relation between energy and wavelength is as follows:
${\text{E = }}\dfrac{{{\text{hc}}}}{{{\lambda }}}$
Where,E is the energy
h is the Planck's constant
c is the speed of light
${{\lambda }}$ is the wavelength
First we will convert the wavelength from nanometre to meter as follows:
\[{10^{ - 9}}{\text{nm}}\,{\text{ = }}\,{\text{1}}\,{\text{m}}\]
\[623\,\,{\text{nm}}\,\,{\text{ = }}\,{\text{623}} \times \,{10^{ - 9}}{\text{m}}\,\]
The value of planck's constant is $6.6 \times {10^{ - 34}}\,{\text{J}}{{\text{s}}^{ - 1}}$ . the value of speed of light is $3 \times {10^{\text{8}}}{\text{m}}{{\text{s}}^{ - 1}}$ .
On substituting $6.6 \times {10^{ - 34}}\,{\text{J}}{{\text{s}}^{ - 1}}$ for h, $3 \times {10^{\text{8}}}{\text{m}}{{\text{s}}^{ - 1}}$ for c, and \[{\text{623}} \times \,{10^{ - 9}}{\text{m}}\] for c,
${\text{E = }}\dfrac{{{\text{hc}}}}{{{\lambda }}}$
$\Rightarrow {\text{E = }}\dfrac{{6.6 \times {{10}^{ - 34}}\,{\text{J}}{{\text{s}}^{ - 1}}\, \times \,3 \times {{10}^{\text{8}}}{\text{m}}{{\text{s}}^{ - 1}}}}{{{\text{623}} \times \,{{10}^{ - 9}}{\text{m}}}}$
$\Rightarrow {\text{E = }}\dfrac{{9.9 \times {{10}^{ - 26}}\,{\text{J}}}}{{{\text{623}} \times \,{{10}^{ - 9}}}}$
$\Rightarrow {\text{E = }}\,{\text{3}}{\text{.2}}\, \times {10^{ - 19}}\,{\text{J}}$
So, the energy of one photon is${\text{3}}{\text{.2}}\, \times {10^{ - 19}}\,{\text{J}}$.
Now, according to Avogadro number one mole of a substance contains $6.02\, \times {10^{23}}$ photons.
Energy of one photon = ${\text{3}}{\text{.2}}\, \times {10^{ - 19}}\,{\text{J}}$
So, energy of one mole photon or $6.02\, \times {10^{23}}$ photons will be,
Energy of one mole photon = ${\text{3}}{\text{.2}}\, \times {10^{ - 19}}\,{\text{J}}\,\, \times \,6.02\, \times {10^{23}}$
${\text{19}}{\text{.3}}\,\, \times {10^4}\,{\text{J}}$
So, the energy of one mole of photon in joule is${\text{19}}{\text{.3}}\,\, \times {10^4}\,{\text{J}}$.
Now, we will convert the energy from joule to kilojoule as follows;
${\text{1000}}\,{\text{J}}\,{\text{ = }}\,{\text{1}}\,{\text{kJ}}$
${\text{19}}{\text{.3}}\,\, \times {10^4}\,\,{\text{J}}\,\,{\text{ = }}\,\,{\text{193}}\,\,{\text{kJ}}$

So, the energy of one mole of photon is ${\text{193}}$ kJ.

Note:Wavelength and energy are inversely proportional. One Joule is equal to ${\text{kg}}{{\text{m}}^{\text{2}}}{{\text{s}}^{ - 2}}$ . The unit of light speed is m/s. we can also take the light speed in cm/s. the unit of wavelength is m. According to Avogadro number one mole of a substance contains $6.02\, \times {10^{23}}$ photons, atoms, ions or molecules. During the solution of this types of problem, units of each term are very important.