A metal target with atomic number 𝑍 = 46 is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio 𝑟 of the wavelengths of the 𝐾_𝛼-line and the cut-off is found to be 𝑟 = 2. If the same electron beam bombards another metal target with 𝑍 = 41, the value of 𝑟 will be

To find the new ratio $r$ of the wavelengths of the $K_{\alpha }$-line and the cut-off wavelength for a target with atomic number $Z=41$, we can apply Moseley's law and the relation for the Duane-Hunt limit (cut-off wavelength).

First, let the wavelength of the $K_{\alpha }$ X-ray line be denoted by ${\lambda }_K$ and the cut-off wavelength be denoted by ${\lambda }_c$. The ratio is defined as:
$r=\frac{{\lambda }_K}{{\lambda }_c}$

According to Moseley's law for the $K_{\alpha }$ transition, the frequency $\nu$ of the emitted X-ray is given by:
$\nu =Rc{\left(Z-1\right)}^2\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=\frac{3}{4}Rc{\left(Z-1\right)}^2$
where $R$ is the Rydberg constant and $c$ is the speed of light. Since ${\lambda }_K=\frac{c}{\nu }$, we have:
${\lambda }_K=\frac{4}{3R{\left(Z-1\right)}^2}$

The cut-off wavelength ${\lambda }_c$ depends only on the accelerating potential $V$ of the bombarding electron beam:
${\lambda }_c=\frac{hc}{eV}$
Since the same electron beam is used in both cases, the potential $V$, and therefore ${\lambda }_c$, remains constant.

Thus, the ratio $r$ is inversely proportional to ${\left(Z-1\right)}^2$:
$r\propto \frac{1}{{\left(Z-1\right)}^2}$
Using this proportionality, we can write:
$\frac{r_2}{r_1}=\frac{{\left(Z_1-1\right)}^2}{{\left(Z_2-1\right)}^2}$

Given the initial state:
$Z_1=46$ and $r_1=2$
For the second metal target:
$Z_2=41$

Substituting these values into the ratio formula:
$\frac{r_2}{2}=\frac{{\left(46-1\right)}^2}{{\left(41-1\right)}^2}=\frac{{45}^2}{{40}^2}={\left(\frac{9}{8}\right)}^2=\frac{81}{64}$

Now, solving for $r_2$:
$r_2=2\times \frac{81}{64}=\frac{81}{32}=2.53125$
Rounding to two decimal places, we get $r_2\approx 2.53$.

Therefore, the value of $r$ for the new target is 2.53.