Step 1: Understanding the Question:
The question asks for the nuclear spin quantum number of the carbon-13 isotope used in NMR spectroscopy.
Step 2: Key Concept:
A nucleus is NMR active only when it has a non-zero spin, and spin arises from unpaired protons and neutrons in the nucleus; nuclei with an odd mass number always carry a spin that is a half-integer.
Step 3: Detailed Explanation:
Carbon-13 has 6 protons and 7 neutrons, an odd total nucleon count, giving it one unpaired nucleon. This gives carbon-13 a spin quantum number of \(\frac{1}{2}\), the simplest possible non-zero spin, which is why it produces a clean, easily interpreted NMR signal, unlike carbon-12, which has an even nucleon count and no spin at all.
Step 4: Final Answer:
\(\frac{1}{2}\).