A stationary $H e^{+}$ ion emits a $(K_{α})$ photon corresponding to the first line of the Lyman series. The photon liberates electron from a stationary hydrogen atom in the ground state. The velocity of the liberated electron is $3.1 \times 10^{x}$ unit. Find $x$ =______(You can make necessary approximations)

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answer is 6.

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Detailed Solution

Energy of a photon emitted for transition n2 to n1 is given by

$Δ E = 13.6 Z^{2} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}) = 13.6 \times 4 \times (\frac{1}{1^{2}} - \frac{1}{2^{2}}) = 40.8 eV$

$\frac{1}{2} m v^{2} = (40.8 - 13.6) eV = 27.2 eV$

$∴ v = 3.1 \times 10^{6} m / \sec$

$∴ x = 6$

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