Question

What is energy in joules of a mole of photon associated with visible light of wavelength of $486\,nm$ ?

Answer 1

In order to solve this question, we need to understand the definition of energy and dual nature of light. Energy is the work capacity of any body or any material, it can get transferred but it can’t be created nor destroyed. Also the dual nature of light tells us that light is made up of packets of photons each having a certain definite amount of energy like in photoelectric effect, Compton effect and blackbody radiation.

Complete step by step answer:
Before proceeding further we need to identify the definition of $1$ mole which states that it is the amount of material containing $6.022 \times {10^{23}}$ particles. This number is called Avogadro’s number. Also Energy that $1$ photon carries is equal to $h\nu$.
Wavelength of light used is $\lambda = 486\,nm$
So using relation $\nu = \dfrac{c}{\lambda }$
$\nu = \dfrac{{(3 \times {{10}^8})m{{\sec }^{ - 1}}}}{{486 \times {{10}^{ - 9}}m}}$
$\Rightarrow \nu = 0.00617 \times {10^{17}}Hz$
So energy of one photon is $E = h\nu$ where $h = 6.626 \times {10^{ - 34}}Js$
So using energy relation and putting value of frequency we get
$E = (6.626 \times {10^{ - 34}} \times 0.00617 \times {10^{17}})J$
$\Rightarrow E = 0.040 \times {10^{ - 17}}J$
So energy of one mole photon is $E' = {N_A}E$
on putting the values we get,
$E' = (6.022 \times {10^{23}} \times 0.040 \times {10^{ - 17}})J$
$\therefore E' = 0.2408 \times {10^{6.}}J$
we can write $1\,MJ = {10^6}\,J$

Hence the answer is $E' = 0.2408\,MJ$.

Note: It should be remembered that, here Avogadro number denotes number of photons so we multiply one photon energy to total photon number so to get total energy of $1$ mole photon. Also it should be remembered that light has dual nature it can be electromagnetic wave or packets of photons and that is totally depends on different process like in photoelectric effect it act as lumps of photons but in conductor it behaves as electromagnetic wave and that is reason why electric field cannot penetrate inside the conductor.