dual-nature-of-radiation-and-matter Question 15
Question: Q. 8. The work function $(W)$, of a metal $X$, equals $3 \times 10^{-19} \mathrm{~J}$. Calculate the number $(N)$ of photons, of light of wavelength $26.52 \mathrm{~nm}$, whose total energy equals $W$.
A [Delhi Comptt. 2016]
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Solution:
Ans. Work function,
$$ \begin{aligned} & W=\frac{n h c}{\lambda} \ & n=\frac{W \lambda}{h c} \ &=\frac{3 \times 10^{-19} \times 26.5 \times 10^{-9}}{6.6 \times 10^{-34} \times 3 \times 10^{8}} \ &=4 \times 10^{-2} \ & \text { [CBSE Marking Scheme 2016] } \end{aligned} $$
Detailed Answer :
The work function of a metal is the minimum energy by it to eject an electron. It is given for metal $\mathrm{X}=3 \times 10^{-19} \mathrm{~J}$
Energy of a photon of wavelength $26.52 \mathrm{~nm}=h v$
$$ \begin{aligned} & =\frac{h c}{\lambda} \ & =\frac{6.6 \times 10^{-34} \times 3 \times 10^{8}}{26.52 \times 10^{-9}} \mathrm{~J} \ & =0.754 \times 10^{-17} \mathrm{~J} \end{aligned} $$
If number of photons $=n$; for total energy $3 \times 10^{-19} \mathrm{~J}$. Then,
$$ n \times 0.754 \times 10^{-17}=3 \times 10^{-19} $$
Hence,
$$ \begin{align*} n & =\frac{3 \times 10^{-19}}{0.754 \times 10^{-17}} \ & =4.0 \times 10^{-2} \tag{1} \end{align*} $$
[Students please note here the number of photons is less than 1. It indicates that the energy of a photon is more than work function of the metal and this radiation has required energy for photoemission of electrons.]
