A Hydrogen-like atom has atomic number Z . Photons emitted in the electronic transitions from level n = 4 to level n = 3 in these atoms are used to perform photoelectric effect experiment on a target

Question

A Hydrogen-like atom has atomic number Z . Photons emitted in the electronic transitions from level n = 4 to level n = 3 in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is 1 . 95 eV . If the photoelectric threshold wavelength for the target metal is 310 nm , the value of Z is _____. [Given h c = 1240 eV - nm and R h c = 13 . 6 eV , where R is the Rydberg constant, h is the Planck’s constant and c is the speed of light in vacuum]

TestHub · Accepted Answer

From Einstein's photoelectric effect, we can write
$E = K E_{m a x} + ϕ \Rightarrow E = K E_{m a x} + \frac{h c}{λ}$
$\Rightarrow ∆ E_{4 to 3} = 1.95 eV + \frac{1240}{310} eV \Rightarrow ∆ E_{4 to 3} = 5.95 eV$
Now, as we know for Hydrogen like atoms, we can write $∆ E = 13.6 Z^{2} (\frac{1}{{(n_{1})}^{2}} - \frac{1}{{(n_{2})}^{2}})$
Therefore, $\Rightarrow 13.6 Z^{2} (\frac{1}{3^{2}} - \frac{1}{4^{2}}) = 5.95$
$\Rightarrow 13.6 Z^{2} \frac{7}{9 \times 16} = 5.95$
$\Rightarrow Z^{2} = \frac{5.95 \times 9 \times 16}{13.6 \times 7} = 9$
$\Rightarrow Z = 3$

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