Q.16
List-I shows various functional dependencies of energy (E) on the atomic number (Z). Energies associated with certain phenomena are given in List-II.

Task: Match the correct energy dependency (List-I) to the corresponding phenomenon (List-II).

List-I (Energy Dependencies):
(P) $E\propto Z^2$
(Q) $E\propto (Z-1{)}^2$
(R) $E\propto Z(Z-1)$
(S) $E$ is practically independent of $Z$

List-II (Phenomena):
(1) Energy of characteristic x-rays
(2) Electrostatic part of the nuclear binding energy for stable nuclei (mass numbers 30–170)
(3) Energy of continuous x-rays
(4) Average nuclear binding energy per nucleon for stable nuclei (mass numbers 30–170)
(5) Energy of radiation due to electronic transitions in hydrogen-like atoms

Answer: C

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Solution:

1. Key Concepts

• Energy of Hydrogen-like Atoms (Option 5):
  The energy levels of hydrogen-like atoms are given by:

  $${\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\propto </span>\frac{{\mathrm{<span\; style="background-color:\; white;">Z</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

  Thus, for transitions, $E\propto Z^2$.

• Characteristic X-rays (Option 1):
  The energy of characteristic x-rays (e.g., Kα line) is proportional to $(Z-1{)}^2$ due to shielding by one electron.

• Nuclear Binding Energy (Options 2 & 4):

  • The electrostatic part of nuclear binding energy (Coulomb repulsion) scales as $Z(Z-1)\mathrm{/}{A}^{1\mathrm{/}3}$. For fixed $A$, $E\propto Z(Z-1)$.

  • The average binding energy per nucleon is roughly constant (independent of $Z$) for stable nuclei ($A=30\text{–}170$).

• Continuous X-rays (Option 3):
  The maximum energy of bremsstrahlung (continuous x-rays) depends on the accelerating voltage, not $Z$. However, the intensity depends on $Z$, but the problem refers to energy, which is independent of $Z$.

2. Matching List-I to List-II

• (P) $E\propto Z^2$: Matches hydrogen-like atom transitions (5).

• (Q) $E\propto (Z-1{)}^2$: Matches characteristic x-rays (1).

• (R) $E\propto Z(Z-1)$: Matches the electrostatic part of nuclear binding energy (2).

• (S) $E$ independent of $Z$: Matches average binding energy per nucleon (4).

3. Correct Option

The correct matching is:

• P→5, Q→1, R→2, S→4, which corresponds to Option C.

4. Final Answer

The correct match is:

\boxed{C}