Question 16

MATRIX MATCHHARD

List-I shows various functional dependencies of energy ( $E$ ) on the atomic number ( $Z$ ). Energies associated with certain phenomena are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

List - I

$E \propto Z^2$

$E \propto (Z - 1)^2$

$E \propto Z(Z - 1)$

$E$ is practically independent of $Z$

List-II

energy of characteristic x-rays

electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170

energy of continuous x-rays

average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170

energy of radiation due to electronic transitions from hydrogen-like atoms

Correct Match:

P → 5

Q → 1

R → 2

S → 4

Detailed Solution

(P) According to Bohr's model, the energy of electronic transitions in hydrogen-like atoms is given by $E = 13.6 Z^2 (\frac{1}{n_1^2} - \frac{1}{n_2^2})$ , so $E \propto Z^2$ . Thus, P $\rightarrow$ 5.

(Q) According to Moseley's Law for characteristic X-rays, the frequency $\nu$ is related to $Z$ as $\sqrt{\nu} \propto (Z - b)$ . For $K_\alpha$ lines, $b \approx 1$ . Since $E = h\nu$ , $E \propto (Z - 1)^2$ . Thus, Q $\rightarrow$ 1.

(R) The electrostatic (Coulomb) potential energy of a nucleus with $Z$ protons is proportional to the number of proton pairs, which is $\frac{Z(Z-1)}{2}$ . Thus, the electrostatic part of binding energy $E \propto Z(Z-1)$ . Thus, R $\rightarrow$ 2.

(S) For stable nuclei with mass numbers $30 < A < 170$ , the binding energy per nucleon ( $BE/A$ ) is nearly constant (approximately $8$ MeV/nucleon). Since $A$ is roughly proportional to $Z$ for stable nuclei, $E_{avg}$ is practically independent of $Z$ . Thus, S $\rightarrow$ 4.

Matching these results: P-5, Q-1, R-2, S-4. Correct Option is C.

Question 16

List - I

List-II

Correct Match:

Detailed Solution

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