Chapter 28, Problem 49GP

To determine

The value of $\text{a}$ in expression $\sqrt{\frac{\text{1}}{\text{λ}}}\text{ = a}\left(\text{Z - b}\right)$

Answer to Problem 49GP

The answer is $\text{2869}{\text{m}}^{\text{-}\frac{\text{1}}{\text{2}}}$

Explanation of Solution

Given info: Expression $\sqrt{\frac{\text{1}}{\text{λ}}}\text{ = a}\left(\text{Z - b}\right)$

Formula used:

To find the value of $\text{a}$ , first solve for wavelength, then solve further for transition and effective. After this, use Moseley Plot and find the value of $\text{a}$ .

Calculation:

The wavelength of photon emitted for the transition from $n^{\text{'}}$ to $n$ is:

  $\frac{\text{1}}{\text{λ}}\text{=}\left(\frac{{\text{2π}}^{\text{2}}{\text{Z}}^{\text{2}}{\text{e}}^{\text{4}}{\text{mk}}^{\text{2}}}{{\text{h}}^{\text{3}}\text{c}}\right)\left[\left(\frac{\text{1}}{{\text{n}}^{\text{'2}}}\right)\text{-}\left(\frac{\text{1}}{{\text{n}}^{\text{2}}}\right)\right]{\text{=RZ}}^{\text{2}}\left[\left(\frac{\text{1}}{{\text{n}}^{\text{'2}}}\right)\text{-}\left(\frac{\text{1}}{{\text{n}}^{\text{2}}}\right)\right]$

The ${\text{K}}_{\text{α}}$ line is from the ${\text{n}}^{\text{'}}\text{=2}$ to ${\text{n}}^{\text{'}}\text{=1}$ transition, and the other $\text{n = 1}$ electron shields the nucleus, so the effective $\text{Z}$ is $\text{Z-1}$ :

  $\frac{\text{1}}{\text{λ}}{\text{= RZ}}_{\text{eff}}{}^{\text{2}}\left[\left(\frac{\text{1}}{{\text{n}}^{\text{'2}}}\right)\text{-}\left(\frac{\text{1}}{{\text{n}}^{\text{2}}}\right)\right]\text{= R}{\left(\text{Z-1}\right)}^{\text{2}}\left[\left(\frac{\text{1}}{{\text{2}}^{\text{2}}}\right)\text{-}\left(\frac{\text{1}}{{\text{1}}^{\text{2}}}\right)\right]\text{=}\frac{\text{3}}{\text{4}}\text{R}{\left(\text{Z-1}\right)}^{\text{2}}$

Thus, $\frac{\text{1}}{\sqrt{\text{λ}}}\text{=}{\left(\frac{\text{3}}{\text{4}}\text{R}\right)}^{\frac{\text{1}}{\text{2}}}\left(\text{Z-1}\right)$ which is the equation of a straight line, as in the Moseley plot, with $\text{b =1}$ .

So, the value of $\text{a}$ is

  $\text{a =}{\left(\frac{\text{3}}{\text{4}}\text{R}\right)}^{\frac{\text{1}}{\text{2}}}\text{=}{\left[\frac{\text{3}}{\text{4}}\left(\text{1}{\text{.0974×10}}^{\text{7}}{\text{m}}^{\text{-1}}\right)\right]}^{\frac{\text{1}}{\text{2}}}\text{=2869}{\text{m}}^{\text{-}\frac{\text{1}}{\text{2}}}$

Conclusion:

The required value is $\text{2869}{\text{m}}^{\text{-}\frac{\text{1}}{\text{2}}}$ .