The work function \(\left(W_0\right)\) of \(\mathrm{Li}, \mathrm{K}, \mathrm{Mg}, \mathrm{Ag}\) and…
The work function \(\left(W_0\right)\) of \(\mathrm{Li}, \mathrm{K}, \mathrm{Mg}, \mathrm{Ag}\) and \(\mathrm{Cu}\) are \(2.42,2.25,3.70,4.30\) and \(4.80 \mathrm{eV}\) respectively. The number of metals which undergo photoelectric effect if a radiation of wavelength \(540 \mathrm{~nm}\) falls on them is \(\left(1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J}\right)\)
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Solution
Key Idea For any metal to show photoelectric effect, \(v > v_0\), where \(v_0\) is threshold frequency, \(v\) is the frequency of light.
Given,
\(\begin{aligned}
W_0 \text { of } \mathrm{Li}=2.42 \mathrm{eV} & =2.42 \times 1.602 \times 10^{-19} \mathrm{~J} \\
& =3.876 \times 10^{-19} \mathrm{~J}
\end{aligned}\)
\(W_0\) of \(\mathrm{K}=2.25 \mathrm{eV}=3.604 \times 10^{-19} \mathrm{~J}\)
\(W_0\) of \(\mathrm{Mg}=3.70 \mathrm{eV}=5.927 \times 10^{-19} \mathrm{~J}\)
\(W_0\) of \(\mathrm{Ag}=4.30 \mathrm{eV}=6.888 \times 10^{-19} \mathrm{~J}\)
\(W_0\) of \(\mathrm{Cu}=4.80 \mathrm{eV}=7.689 \times 10^{-19} \mathrm{~J}\)
Also \(\quad W_0=h v_0\)
\(\because \quad v_0\) of \(\mathrm{Li}=W_0 / h\)
\(=\frac{3.876 \times 10^{-19}}{6.626 \times 10^{-34}}=0.584 \times 10^{15} \mathrm{~s}^{-1}\)
\(\therefore \quad \mathrm{v}_0\) of \(\mathrm{K}=\frac{W_0}{h}=\frac{3.604 \times 10^{-19}}{6.626 \times 10^{-34}}\)
\(=0.54 \times 10^{15} \mathrm{~s}^{-1}\)
\(\therefore \quad v_0\) of \(\mathrm{Mg}=\frac{5.927 \times 10^{-19}}{6.626 \times 10^{-34}}=0.89 \times 10^{15} \mathrm{~s}^{-1}\)
\(\therefore \quad v_0\) of \(\mathrm{Ag}=\frac{6.888 \times 10^{-19}}{6.626 \times 10^{-34}}=1.03 \times 10^{15} \mathrm{~s}^{-1}\)
\(v_0\) of \(\mathrm{Cu}=\frac{7.689 \times 10^{-19}}{6.626 \times 10^{-34}}=1.16 \times 10^{15} \mathrm{~s}^{-1}\)
Frequency of light can be calculated as:
\(\begin{aligned}
v & =\frac{c}{\lambda} \\
& =\frac{3 \times 10^8}{540 \times 10^{-9}}=0.0056 \times 10^{17} \mathrm{~s}^{-1} \\
& \approx 0.56 \times 10^{15} \mathrm{~s}^{-1}
\end{aligned}\)
Thus, only in case of \(K, v > v_0\) thus it will show photoelectric effect.