Discussion
The 13C NMR spectrum of acetone-d6 has a signal at 30 ppm as a septet in the intensity ratio
Answer: B
The septet has intensities 1 : 3 : 6 : 7 : 6 : 3 : 1.
The CD3 carbon couples to three deuterium nuclei, each with spin I = 1. The number of lines is \(2nI + 1 = 2 \times 3 \times 1 + 1 = 7\). Each deuterium has three equally likely spin states (-1, 0, +1), so the intensities are the coefficients of \((1 + x + x^{2})^{3}\):
\[ (1 + x + x^{2})^{3} \to 1, 3, 6, 7, 6, 3, 1 \]
This is the familiar septet of the solvent peak of acetone-d6. 1 : 6 : 15 : 20 : 15 : 6 : 1 is the binomial pattern for six spin-1/2 nuclei, as in the CH of an isopropyl group in 1H NMR, not three deuteriums. The other two sequences do not come from any set of equivalent nuclei.
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