Question

Calculate the number of photons emitted per second by a 10 watt sodium vapour lamp. Assume that 90$\%$ of the consumed energy is converted into light. Wavelength of sodium light is 590 nm, h $= 6.63 \times {10^{ - 34}}Js$.
${\text{A}}{\text{. 2}}{\text{.67}} \times {\text{1}}{{\text{0}}^{17}} \\\ {\text{B}}{\text{. 2}}{\text{.67}} \times {\text{1}}{{\text{0}}^{18}} \\\ {\text{C}}{\text{. 2}}{\text{.67}} \times {\text{1}}{{\text{0}}^{19}} \\\ {\text{D}}{\text{. 2}}{\text{.67}} \times {\text{1}}{{\text{0}}^{20}} \\\$

Answer 1

First of all calculate the total energy emitted by the lamp which is given to be 90$\%$ of the consumed power. Then we need the energy of a single photon emitted by the lamp. Finally, the ratio of total energy emitted to the energy of a single photon gives the required answer.

Formula used:
The energy formula for electromagnetic radiation is given as
$E = h\nu = \dfrac{{hc}}{\lambda }$
Here $\nu$ represents the frequency of the radiation, $\lambda$ is the wavelength of the radiation, c is the speed of light while h represents Planck's constant.

Complete step by step answer:
We are given a sodium lamp which has a power rating of 10 watt. Since, power is equal to the rate of doing work, therefore, the amount of energy emitted by the sodium lamp in one second is 10 joules.
We are also given that only 90$\%$ of the consumed energy is converted into light. This means that only 90$\%$ of work done by the lamp in one second is converted into light.. Therefore the amount of energy used by a lamp per second is $90\%$ of 10 joules which is equal to 9 joules.
Now the energy of a single photon emitted by the sodium lamp can be calculated by energy formula. We are given that the wavelength of light emitted by a sodium lamp has value:
$\lambda = 590nm = 590 \times {10^{ - 9}}m$
Now the energy of a photon emitted by the sodium lamp is calculated in the following way.
$E = \dfrac{{hc}}{\lambda } = \dfrac{{6.63 \times {{10}^{ - 34}} \times 3 \times {{10}^8}}}{{590 \times {{10}^{ - 9}}}} = 0.0337 \times {10^{ - 17}}J$
Now we can easily calculate the number of photons emitted by the sodium lamp by dividing the total energy emitted by the lamp with the energy of one photon emitted by the lamp. This can be done in the following way.
$n = \dfrac{{{\text{Total energy emitted}}}}{{{\text{Energy of one photon}}}} = \dfrac{9}{{0.0337 \times {{10}^{ - 17}}}} \\\ = 267.062 \times {10^{17}} \\\ = 2.67 \times {10^{19}} \\\$
Hence the correct answer is option C.

Note:
It should be noted that we are able to calculate the number of photons by simply dividing the total energy by the energy of a single photon because the electromagnetic radiation is quantized in nature. This implies that the lamp emits light in packets of fixed energy called photons.