Question

It is observed that characteristic $X$-ray spectra of elements show regularity When frequency to the power " $n$ " ie $v^n$ of $X$-rays emitted is plotted against atomic number " $Z$ ", following graph is obtained

• A. 2
• B. 3
• C. 1
• D. $\frac{1}{2}$

Answer 1

The Correct Option is D

The question is related to the characteristic X-ray spectra of elements, which show regular patterns when the frequency of X-rays is plotted against the atomic number \( Z \). According to Moseley's law, the frequency \( v \) of the X-rays is related to the atomic number by the formula:

\(v = a(Z - b)^2\)

where:

• \(a\) is a proportionality constant.
• \(b\) is another constant depending on the series of lines considered (for instance, K or L series).

This can be rearranged as:

\(v^2 \propto Z\)

Thus, when \( v^n \) is plotted against \( Z \), if a linear graph is obtained, then \( n \) corresponds to the power required to make the relationship linear.

Since \( v^2 \propto Z \), the correct value of \( n \) should be \( 1/2 \). This is because of the following reasoning:

Let's consider the options provided:

• When \( n = 2 \), then \( v^2 \propto Z \), which is correct.
• When \( n = 1 \), then \( v \propto Z^{1/2} \), which does not fit the original relationship.
• When \( n = 3 \) or \( n = \frac{1}{2} \), the linear relationship only holds with \( n = \frac{1}{2} \).

Therefore, the correct value of \( n \) is \( \frac{1}{2} \) as per the explanation and the standard equation of Moseley's law.