Graph of X-ray frequency (v)n v/s atomic number (Z) is linear. Find the value of n.

Question

Among the following the isoelectronic species are
I. $Mg^{2+}, Al^{3+}, F^-$
II. $O^{2-}, F^-, N^{3-}$
III. $K^+, Ca^{2+}, Sc^{3+}$
IV. $Mn^{2+}, Fe^{3+}, V^{3+}$

Answer 1

The question asks about the relationship between X-ray frequency ($v$) and the atomic number ($Z$), described by a linear graph. We are to determine the value of $n$ such that this linear relationship holds when plotted on a graph of $v^n$ versus $Z$.

The relationship between the frequency of X-rays and the atomic number is given by Moseley's law, which can be expressed as follows:

v = a(Z - b)^2

Where:

v is the frequency of the X-rays.
a is a constant for each series of characteristic X-rays.
Z is the atomic number.
b is a constant that accounts for the inner-shell electron screening.

When analyzing the question, it is mentioned that the graph of v^n versus Z is linear. For this to hold true, we need:

(v^n) \propto (Z - b)

By comparing the two expressions:

v \propto (Z - b)^2
v^n \propto (Z - b)

It is apparent that to make the linear relationship hold, the exponent n must be:

2n = 1

Solving for n gives:

n = \frac{1}{2}

This step-by-step reasoning aligns with the correct option:

$\frac{1}{2}$

Therefore, the correct value of n that makes the plot of v^n versus Z linear is $\frac{1}{2}$, consistent with the understanding of Moseley's law.

Answer 2