Question

The graph between the square root of the frequency of a specific line of characteristic spectrum of X-ray and the atomic number of the target will be
a)

b)

c)

d)

Answer 1

Hint:It has been observed that the frequency of the characteristic spectrum of X-ray and the atomic number of the target is dependent in such a way that the frequency is proportional to the square of the atomic number with an offset value.

Formula used:
Relationship between the frequency of characteristic spectrum of X-ray and the atomic number is given by:
$\nu =a(Z-b{)}^2$...................(1)
Where,
$\nu$ is the frequency of characteristic spectrum of X-ray,
Z denotes the atomic number of the target atom,
a is the proportionality constant and a is positive,
b is offset constant and it’s positive also.

Complete step by step answer:
From eq.(1) we get the relationship between$\sqrt{\nu }$and Z as:
$$\begin{gathered} \nu =a(Z-b{)}^2 \\ \therefore \sqrt{\nu }=\sqrt{a}Z-\sqrt{a}b \end{gathered}$$.............................(2)

Now, compare this eq.(2) with the equation of a straight line i.e. $y=mx+c$. You will see that$\sqrt{\nu }$and Z are linearly related to each other with slope$\sqrt{a}$with an intercept $-\sqrt{a}b$. So, the line will cut x-axis at +b point. Hence, the graph of $\sqrt{\nu }$ vs Z will look like:

Where, square roots of frequency values are given at y axis and the atomic number Z is given in the x-axis.

Correct answer:
The correct graph is given in option (b).

Note: From the given relation between frequency of characteristic spectrum and atomic number in eq.(1) a student can easily determine that the relationship between$\sqrt{\nu }$and Z will be linear. But they might get confused when choosing one between the first 3 straight line options. So, you need to focus on eq.(2). You will see that the line must have a negative intercept with y-axis. But in option a and option c the intercepts are positive and zero respectively. So, you’ll get the correct answer as graph b.