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The above is the correct $ ^{13}C $ NMR spectrum of butanone.
In the $ ^{13}C $ NMR spectrum of butanone, I figure that the peak locations of the first and third carbons should be switched (27.3, 35.2). Shouldn't the third carbon be shifted upfield by the terminal methyl group and therefore have a "smaller ppm" than the first carbon?
No, substitution by an alkyl group produces a small downfield shift in both proton and carbon nmr. Therefore, in a typical hydrocarbon a secondary carbon will be downfield of a primary carbon. The assignments in your figure are correct.
A possible explanation for the downfield shift of a carbon atom when a proton is replaced with another carbon atom is as follows. Hydrogen is more electropositive than carbon, therefore, in a C-H bond a significant amount of the electron density in the bond resides around the carbon. When we replace the electropositive hydrogen with a relatively more electronegative carbon, then there is less electron density around the central carbon atom. This reduction in electron density deshields the carbon nucleus and shifts it downfield. Every time another hydrogen is replaced by carbon a further downfield shift occurs.
