Question

Lifetimes of the molecules in the excited state are often measured by using pulsed radiation source of duration nearly in the nanosecond range. If the radiation source has the duration of 2 ns and the number of photons emitted during the pulse source is $2.5\times {{10}^{15}}$, calculate the energy of the source.

Answer 1

Hint: There is a relationship between energy, number of photons emitted during the pulse and frequency of the radiation source and it is as follows.
\[E=Nhv\]
E = Energy of the source
N = Number of photons emitted during the pulse
h = Planck’s constant
v = frequency of the pulse

Complete step by step answer:
- In the question it is given that there is a radiation source which has a pulse radiation source of 2 ns with a number of photons of $2.5\times {{10}^{15}}$ . We have to calculate the energy of the source.
- In the question it is given that duration is 2 ns = $\text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}\text{seconds}$.
- In the question it is given that the number of photons released from the source is $2.5\times {{10}^{15}}$ .

 - We know that frequency is inversely proportional to duration of the source.
- Then
\[\begin{align}
 & \text{frequency of the source=}\dfrac{\text{1}}{\text{duration of the source}} \\
 & =\dfrac{1}{2\times {{10}^{-9}}} \\
 & =0.5\times {{10}^{9}} \\
\end{align}\]

- Thus we know the frequency, Planck ’s constant and number of photons emitted then we can calculate the energy of the source as follows.
\[\begin{align}
 & E=Nhv \\
 & =(2.5\times {{10}^{15}})(6.626\times {{10}^{-34}})(0.5\times {{10}^{9}}) \\
 & =8.28\times {{10}^{-10}}J \\
\end{align}\]
- Therefore the energy of the source is $8.28\times {{10}^{-10}}J$ .

Note: As the energy of the source increases then the number of photons released by the source also increases because the energy is directly proportional to the number of photons. We can see this relationship from the above mentioned formula also.