What is the work function of tungsten at $1500\,K$ temperature, when a diode valve with a tungsten filament works at $1500\,K$? Assume the work function of tungsten at $0\,K$ is $4.52\,eV$:
A. $4.71\,eV$
B. $0.39\,eV$
C. $8.86\,eV$
D. $1.25\,eV$

Hint: The work function of a metal is defined as the minimum energy required by an electron to just escape from the metal surface. The work function is measured by a smaller unit of energy called electron volt or $eV$. The energy given to a metal surface is spent in two ways: a part of energy is used in liberating the least tightly bound electron from the metal surface which is equal to the work function of the metal. The rest of the energy is spent as maximum kinetic energy by the emitted electrons.

Complete step by step answer:
It is given that the work function of tungsten at $0\,K$ is ${W_0} = 4.52\,eV$. It is required to find the work function of tungsten at a temperature $T = 1500\,K$. We know the formula for work function
$W = {W_0} + {K_{\max }}$
Where, $W$ is the work function of a metal at a temperature $T$, ${W_0}$ is the work function of the metal at a temperature ${T_0}$ and ${K_{\max }}$ is the maximum kinetic energy of emitted electrons from the surface of the metal at the temperature $T$.

Also, we know that,
${K_{\max }} = \dfrac{3}{2}kT$
Where $k$ is the Boltzmann constant
$k = 8.6 \times {10^{ - 6}}eV{K^{ - 1}}$
Therefore, $W = {W_0} + \dfrac{3}{2}kT$
Substitute all the required values in the above formula.
$W = 4.52 + \dfrac{3}{2} \times 8.6 \times {10^{ - 5}} \times 1500$
Further calculating
$ \Rightarrow W = 4.52 + 0.1935$
$ \Rightarrow W = 4.7135eV$
$\therefore W \approx 4.71eV$

Hence, the correct option is A.

Note: The electrons present below the surface of metal require higher energy than that of the work function of the surface of the metal. The work function is a measure of how tightly a particular metal holds its electrons. The work function is very important in applications where electron emission from metals is involved such as photoelectric devices and cathode-ray tubes.