Understanding ESR Splitting in $[Mo(CN)_8]^{3-}$

Electron Spin Resonance (ESR) spectroscopy is a technique used to study molecules or ions containing unpaired electrons. The ESR spectrum can provide detailed information about the electronic structure and the surrounding environment. One important phenomenon observed in ESR is hyperfine splitting, which occurs due to the interaction between the unpaired electron's magnetic moment and the magnetic moments of nearby atomic nuclei.

Hyperfine Coupling and Spectral Lines

This interaction causes a single ESR line to split into a pattern of multiple lines, known as a multiplet. The number of lines observed ($N$) depends directly on the nuclear spin quantum number ($I$) of the interacting nucleus and the number of equivalent nuclei ($n$) involved in the interaction. The relationship is defined by the formula:

$ N = 2nI + 1 $

In this equation:

• $N$ represents the total number of spectral lines observed.
• $n$ is the count of magnetically equivalent nuclei interacting with the unpaired electron.
• $I$ is the nuclear spin quantum number of one such nucleus.

Analyzing the $[Mo(CN)_8]^{3-}$ Spectrum

The question provides specific details about the ESR spectrum of the complex ion $[Mo(CN)_8]^{3-}$:

• In its natural isotopic abundance (primarily containing $^{12}C$, where $I=0$), the spectrum exhibits a single line. This implies minimal or no hyperfine splitting from relevant nuclei.
• However, when the sample is specifically enriched with the isotope $^{13}C$, the ESR spectrum changes significantly, showing nine lines.

The appearance of these nine lines specifically upon $^{13}C$ enrichment indicates that the unpaired electron interacts with the nuclei of the $^{13}C$ atoms, causing the observed hyperfine splitting.

Determining the Nuclear Spin (I) of $^{13}C$

We observe $N=9$ lines due to the interaction with $^{13}C$. The nuclear spin quantum number ($I$) for the $^{13}C$ isotope is a known fundamental property. We can use the observed number of lines ($N=9$) and the splitting formula to determine or confirm the value of $I$.

The standard nuclear spin for $^{13}C$ is $I = 1/2$. Let's use this known value in the formula $N = 2nI + 1$ to see how many equivalent $^{13}C$ nuclei ($n$) would be required to produce 9 lines.

$ 9 = 2n(1/2) + 1 $

Simplifying the equation:

$ 9 = n + 1 $

Solving for $n$ gives:

$ n = 9 - 1 = 8 $

The result $n=8$ indicates that the unpaired electron interacts with 8 equivalent $^{13}C$ nuclei. This number aligns perfectly with the structure of the $[Mo(CN)_8]^{3-}$ complex, which contains eight cyanide ($CN$) ligands. Each cyanide ligand has one carbon atom. Therefore, the observation of nine lines in the ESR spectrum upon $^{13}C$ enrichment is a direct consequence of the unpaired electron interacting with the 8 carbon atoms, confirming that the nuclear spin of $^{13}C$ is $I=1/2$.