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The wavelength of emitted radiation when an electron jumps orbits in the Bohr atomic model is given by

1/$\lambda$ = $R_H$ ($\frac{1}{n_f^2}$ - $\frac{1}{n_i^2}$) $Z^2$

But that of X-Ray emission is given by

1/$\lambda$ = $1.1 \times 10^7$ ($\frac{1}{n_f^2}$ - $\frac{1}{n_i^2}$) $(Z-1)^2$

Why is there a difference? Why aren't X-Rays treated like any other emissions?

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The pre-factor is exactly the same thing, look up what $R_H$ is equal to.

The (Z-1) term, is due to the inner electrons shielding part of the nuclear charge such as the electrons involved in the X-ray transitions only see a reduced electric field. It's explained in the link you provided.

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