Imagine you are a detective in the quantum realm. You are handed four mysterious energy dependencies and asked to match them with their physical origins. This is not just a matching exercise; it is a tour through the greatest hits of modern physics! Let's break them down one by one.
Unlocking the First Clue
Bohr's Model
Our first clue is the dependency E∝Z2. Where have we seen this before?
Recall Bohr's model for hydrogen-like atoms. The energy of an electron in the n-th orbit is given by the famous equation:
When an electron jumps between energy levels, the energy of the emitted radiation is simply the difference between these levels. Because both levels scale with Z2, the emitted radiation energy also scales with Z2.
Thus, this dependency perfectly describes the energy of radiation due to electronic transitions from hydrogen-like atoms.
The Second Clue
Moseley's Law
Next, we encounter E∝(Z−1)2. This specific (Z−1) term is a massive hint.
It points directly to the shielding effect in multi-electron atoms. When we study characteristic X-rays, we rely on Moseley's Law, which states:
For K-series X-rays, an electron drops into the innermost shell. The remaining electron in that shell shields the nucleus, making the effective nuclear charge approximately Z−1. This means the screening constant b≈1.
Since the energy of a photon is $E = h
u$, squaring Moseley's equation reveals that the energy of characteristic X-rays is proportional to (Z−1)2.
The Third Clue
Inside the Nucleus
Our third dependency is E∝Z(Z−1). This looks like a combinatorics problem, and that is exactly what it is!
Inside a nucleus, there are Z positively charged protons. Every single proton repels every other proton via the Coulomb force.
To find the total electrostatic repulsion, we must count the number of interacting proton pairs. The number of ways to choose 2 protons from a total of Z is given by:
Therefore, the electrostatic part of the nuclear binding energy scales directly with Z(Z−1).
The Final Clue
The Plateau of Stability
Finally, we have an energy E that is practically independent of Z.
Think about the curve of binding energy per nucleon (BE/A) plotted against the mass number A. For very light and very heavy nuclei, this value changes significantly.
However, for stable nuclei with mass numbers in the range of 30 to 170, the curve flattens out into a plateau. In this region, the strong nuclear force saturates, and the average nuclear binding energy per nucleon remains roughly constant at about 8 to 8.5 MeV.
It does not depend strongly on Z or A, perfectly matching our final clue.
Bringing It All Together
We have successfully decoded all the clues.
The Z2 dependency belongs to hydrogen-like atoms. The (Z−1)2 dependency is the signature of characteristic X-rays. The Z(Z−1) term arises from proton pair repulsion in the nucleus. And the constant energy represents the stable plateau of binding energy per nucleon.
Final Answer: The correct matching is P→5, Q→1, R→2, and S→4.
