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Chapter 10: Problem 12

The ethyne molecule \((\mathrm{HC} \equiv \mathrm{CH})\) consists of two fermions \(\left(^{1} \mathrm{H}\right)\) and two bosons \(\left(^{12} \mathrm{C}\right) .\) What are the implications for the statistical weights of the levels of various \(J\) ? What are the implications of replacing (a) one \(^{12} \mathrm{C}\) by \(^{13} \mathrm{C},\) (b) both \(^{12} \mathrm{Cby}^{13} \mathrm{C}\). (The \(^{13} \mathrm{C}\) nucleus is a fermion, \(\left.I=\frac{1}{2} .\right)\)

Short Answer

Expert verified
The replacement of \(\mathrm{C}\) bosons with \(\mathrm{C}\) fermions in the ethyne molecule will decrease the density of states as fermions cannot occupy the same quantum state. The decrease will be more significant when both bosons are replaced.

Step by step solution

01

APPREHENSION OF ORIGINAL MOLECULE

Ethyne molecule (\(\mathrm{HC} \equiv \mathrm{CH}\)) is composed of two fermions (\(^{1}\mathrm{H}\)) and two bosons (\(^{12}\mathrm{C}\)). Since bosons can occupy the same quantum state and fermions cannot, each separate quantum state (J) could be occupied by multiple bosons but only one fermion.
02

CONSEQUENCE OF REPLACMENT: PART a

For part a, when one \(^{12}\mathrm{C}\) boson is replaced by a \(^{13}\mathrm{C}\) fermion, the molecule now contains three fermions and one boson. This will somewhat decrease the density of states. Each level of J can still be occupied by the boson, but the three fermions have to occupy three different quantum states.
03

CONSEQUENCE OF REPLACMENT: PART b

For part b when both \(^{12}\mathrm{C}\) bosons are replaced by \(^{13}\mathrm{C}\) fermions, the molecule becomes entirely composed of fermions. This will further decrease the density of states as now, each level of J is to be occupied by only a single fermion.

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