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The above is the correct $\ce{^{13}C}$-$\mathrm{NMR}$ spectrum of butanone.
In the $\ce{^{13}C}$-$\mathrm{NMR}$ spectrum of butanone, I figure that the peak locations of the first $(\pu{27.3 ppm})$ and third $(\pu{35.2 ppm})$ carbons should be switched. Shouldn't the third carbon be shifted upfield by the terminal methyl group and therefore have a "smaller $(\pu{ppm})$" 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.
I agree with the Ron's notation that a substitution by an alkyl group produces a small downfield shift in both proton and carbon NMR. My intention here to educate OP that he/her can learn some mathematical calculations, which can predict expected chemical shifts of most of simple compounds such as 2-butanone. If OP is studing spectroscopy, my recommendation is Ref.1, which gave pretty good amount of mathematical calculations. For instance, following is the predicted $\ce{^{13}C}$ chemical shifts for 2-butanone:
The difference between butane and 2-butanone is the carbonyl group at second carbon. According to Ref.1 (page 269):
Replacement of $\ce{CH2}$ of alkanes by $\ce{C=O}$ causes a downfield shift at $\alpha$-$\ce{C}$ $(\pu{10-14 ppm})$ and an upfield shift at the $\beta$-$\ce{C}$ (several $\pu{ppm}$ in acyclic compounds).
Following diagram would show the actual $\ce{^{13}C}$ chemical shifts butane (Table II; page 260, Ref.1) and calculated $\ce{^{13}C}$ chemical shifts of 2-butanone accordingly:
According to the calculations, $\ce{^{13}C}$ chemical shift of $\ce{C}$1 of 2-butanone (a carbon $\alpha$ to carbonyl carbon) is approximately $13.4 + 14 = \pu{27.4 ppm}$, actual value of which is $\pu{27.3 ppm}$. Similarly, the calculated $\ce{^{13}C}$ chemical shift of $\ce{C}$3 of 2-butanone, which is also $\alpha$ to the carbonyl group, is approximately $25.2 + 14 = \pu{34.2 ppm}$. The actual value given by OP is $\pu{35.2 ppm}$. This prove that the original assignment is not by mistake. they are the actual values of 2-butanone.
References:
