I am currently struggling with the following NMR spectrum for this compound:
I know for sure that we should multiply the integrals of the hydrogen atoms by four, and that the compound contains at least one ethoxy group. Furthermore, the molar mass of the compound is estimated to be $\pu{258g/mol}$. We tried our best, but could not find the solution. We also figured that we should multiply the $\ce{^{13}C}$-$\mathrm{NMR}$ by two, but aren't sure about it. Any tips would be helpful.
I appreciate Karsten Theis for his effort to solve this problem, but I have to say that I completely disagree with the structure he has come with as the answer (as a matter of fact, I came up with the same structure but it didn't fit the given $\ce{^1H}$-$\mathrm{NMR}$ spectrum; vide infra). Followings are my reasons:
As correctly pointed out by Karsten Theis, the resonance at about $\pu{174 ppm}$ of the $\ce{^{13}C}$-$\mathrm{NMR}$ spectrum indicates a presence of an ester carbonyl carbon. The quartet at about $\pu{4.05 ppm}$ and triplet at about $\pu{1.15 ppm}$ in $\ce{^{1}H}$-$\mathrm{NMR}$ spectrum indicates a presence of $\ce{-O-CH2-CH3}$ group, as hinted by OP's teacher. The presence of $\ce{-O-CH2\!-}$ fragment is confirmed by the resonance at about $\pu{60.0 ppm}$ in $\ce{^{13}C}$-$\mathrm{NMR}$ spectrum.
The second triplet at about $\pu{2.20 ppm}$ in $\ce{^{1}H}$-$\mathrm{NMR}$ spectrum also indicates a presence of $\ce{-CH2-C(=O)-}$ fragment. Thus, Karsten Theis' suggestion of the presence of $\ce{-CH2-CO-O-CH2-CH3}$ fragment is correct. But here is the problem. The proton signal of $\ce{-CH2-C(=O)-}$ fragment displays a triplet, not a doublet as Karsten Theis' suggested structure would have given in reality. Consequently, most possible fragment should have been $\ce{-CH2-CH2-CO-O-CH2-CH3}$ (total of 5 carbons).
Since the $\ce{^{13}C}$-$\mathrm{NMR}$ spectrum shows only 6 carbon resonances and the molar mass of the compound is suggested to be about $\pu{258 g mol-1}$, the compound should have at least one plane of symmetry as suggested by OP. It is also noteworthy that the intensity of the third carbon from the right in $\ce{^{13}C}$-$\mathrm{NMR}$ spectrum is almost double compared to other possible $\ce{-CH2 \!-}$ carbon signals. Therefore, arguably, the compound should have at least two $\ce{-CH2-CH2-CO-O-CH2-CH3}$ fragments plus two $\ce{-CH2 \!-}$ parts, giving that $\ce{C14H26O4}$ molecular formula (molar mass of which come up as $\pu{258.36 g mol-1}$ that agrees with the given value). Thus the structure should be:
The molecule is diethyl decanedioate, with molecular formula of $\ce{C14H26O4}$ and molar mass of $\pu{258.36 g mol-1}$. To confirm the structure, let's look at the integrations given in $\ce{^{1}H}$-$\mathrm{NMR}$. The proton signal of $\ce{-CH2-C(=O)-}$ at around $\pu{2.2 ppm}$ is integrated as $1.00$, meaning one proton is equivalent to $0.25$ integration value. The integration value of the triplet and the multiplet coincided with it at the range of $\pu{1.1-1.3 ppm}$ is marked as $3.5$, thus is equivalent to 14 protons $(\frac{3.5}{0.25} = 14)$. Also, the integrations of multiplet at $\pu{1.55 ppm}$ and quartet at $\pu{4.05 ppm}$ are marked as $\approx 1$, and hence each of them is equivalent to 4 protons. Thus, the total proton count is $3 \times 4 + 14 = 26$ protons, which agrees with the suggested molecular formula.
Further, the ChemDraw stimulated $\ce{^{1}H}$- and $\ce{^{13}C}$-$\mathrm{NMR}$ spectra for the suggested molecule are shown below:
Both spectra are in good agreement with actual spectra, specifically see the overlapped carbon signals at around $\pu{29 ppm}$. However, the triplet at around $\pu{1.07 ppm}$ has moved downfield to $\pu{1.17 ppm}$ in real time spectrum, showing more deshielded nature, probably due to the presence of oxygen at $\beta$-position.
Edit:
As Buttonwood has suggested, it is always better to compare your data with experimentally recorded data. The SDBS Database, which provides the spectral data of compound in interest under SDBS 2666 has given following $\ce{^{1}H}$- and $\ce{^{13}C}$-$\mathrm{NMR}$ spectra to compare:
Both spectra match the given spectra of the compound.
Taking the hint from your instructor and the 13-C signal at 175 ppm fits nicely to a $\ce{-CH2-CO-O-CH2-CH3}$ fragment, i.e an ester. The 13-C spectrum shows only 6 peaks and the molar mass is about 258 g/mol, so there is probably some symmetry in the molecule.
There is something that puzzles me. The signals around 1.2 ppm in the proton NMR integrate to 3.5, so after doubling (to explain the other peaks as $\ce{CH2}$ groups), we get 7. There is a $\ce{CH3}$ group already accounted for (from the ethoxy group), so there would be another two $\ce{CH2}$ groups (or even more $\ce{XH}$ groups). That means there are at least 6 different signals in the proton NMR, but just 6 different peaks in the 13-C NMR. With one coming from the carbonyl, that's a mismatch and we need another carbon. One of the peaks in the 13-C NMR is about twice as high, so maybe that's the solution to this puzzle.
A possible molecular formula would be $\ce{C14H26O4}$. That would mean all the signals in the proton NMR are "doubled up" because of symmetry, and all of the ones in the 13-C NMR as well, except for one that corresponds to four carbons.
I did not find a structure that satisfies all of these constraints. The closest I could find is this:
This has the correct number of 13-C peaks with pretty good matching of chemical shifts, and explains much of the proton NMR as well. Unfortunately, the molar mass is off by 2 (degree of unsaturation is 3 instead of 2), and one of the signals has an integral that is too low, and the signal at 2.2 ppm in the proton NMR would be a doublet instead of the observed triplet. This incorrect structure, however, does illustrate how having different degree of symmetry in different parts of the molecule can explain the number of signals found in the 13-C spectrum.
