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Taken from OCR A Chemistry Unified Chemistry Sample Paper.

Information about a monobasic organic acid D is shown below.

  • D reacts by both electrophilic substitution and electrophilic addition.
  • The molecular formula of D is $\ce{C_xH_yO2}$.
  • The mass spectrum of D has a molecular ion peak at $m/z = 148$.
  • The $\ce{^13C}$ NMR spectrum of D contains seven peaks.

Determine and draw a possible structure for D. Explain your reasoning from the evidence provided.

I understand that D must contain a benzene ring (electrophilic substitution) and a double carbon bond (electrophilic addition). However, I'm at a loss about how to deduce the molecular formula. Taking away $M_r(\ce{O2})$ from $\pu{148 g mol^-1}$, gives $M_r(\ce{C_xH_y})= \pu{116 g mol^-1}$. The mark scheme gives the molecular formula as $\ce{C9H8O2}$. I can't see how the values of $x$ and $y$ were deduced. All you know is that D contains a $\ce{C=C}$, benzene ring and most likely a $\ce{-COOH}$ group (due to monobasic nature).

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    $\begingroup$ Divide 116 by 12 (with integer then modulo division) which gives 9 with a remainder of 8. You know the molecule contains only C, H and O, so this gives your answer. $\endgroup$ – porphyrin Mar 5 at 13:33
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    $\begingroup$ If you were a mathematician, you would say why $\ce{C9H8}$ and not $\ce{C8H20}$ or even $\ce{H116}$. Studying chemistry, you can exclude those "solutions" because even in an alkane, you never have more than 2n+2 hydrogen atoms in a hydrocarbon with n carbon atoms. $\endgroup$ – Karsten Theis Mar 5 at 13:56
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    $\begingroup$ If you were an engineer, you might say why $\ce{C9H8}$ and not $\mathrm{C_{9.5}H_2}$. Studying chemistry, you can exclude a formula with non-integers if you want to describe a molecule with distinct structure. $\endgroup$ – Karsten Theis Mar 5 at 13:59
  • $\begingroup$ It's nothing other than Cinnamic acid. $\endgroup$ – Soumik Das Mar 5 at 16:28
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    $\begingroup$ @Soumik Das You might be right, but that was not the question. The OP is asking how to deduce the chemical formula from the molar mass. $\endgroup$ – Karsten Theis Mar 5 at 18:19
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There is no contradiction on that question. Because it is an A-Level question (assuming you are high school student), you should be able to solve with your basic knowledge:

You have correctly suggested that D contains a $\ce{C=C}$, a benzene ring, and most likely a $\ce{-COOH}$ group, based on the data given (at least $4+1+1=6$ number of unsaturations).

The molecular formula of D is $\ce{C_xH_yO2}$ and molecular ion peak of its mass spectrum is at $148\:m/z$, thus, $12\times x + 1\times y + 16\times 2 = 148$ or $12 x + y = 148-32= 116$.

Since $\ce{^13C}$ NMR contains 7 peaks, the molecule must have at least 7 $\ce{C}$ atoms so you can deduce that amount from your $\ce{C_xH_y}$ portion. That leaves you with: $\pu{(116-12\times7) g/mol} = \pu{32 g/mol}$ to resove ($x-7\lt3$), which can be either $\ce{CH20}$ ($= \pu{32 g/mol}$) or $\ce{C2H8}$ ($= \pu{32 g/mol}$). That leave you with the molecular formula of your compound been either $\ce{C8H20O2}$ ($\mathrm{FM}= \pu{148 g/mol}$) or $\ce{C9H8O2}$ ($\mathrm{FM}= \pu{148 g/mol}$). Since you know that compound has at least $6$ unsaturations, molecular formula $\ce{C8H20O2}$ can be eliminated. Therefore the molecular formula of your compound should be $\ce{C9H8O2}$.

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I'm at a loss about how to deduce the molecular formula.

Compounds containing only C, H, and O have a certain number of hydrogens (h) compared to the number of carbons (c).

If there are no double bonds and no rings, h = 2c + 2. This is true for linear chain alkanes. Adding oxygens to those (hydroxyl groups, ethers) does not change the number of hydrogens. Turning the linear chain to a branched structure does not change the number of hydrogens.

For every double bond you add, you lose two hydrogens (think of it as turning sigma bonds to hydrogen into pi bonds between the heavier atoms). For every ring you close, you lose two hydrogens (one additional C-C bond instead of each carbon bound to a hydrogen, for example).

So for molecules with double bonds (d) and rings (r), you get:

h = 2c + 2 - 2d - 2r

All you know is that D contains a C=C, benzene ring and most likely a -COOH group

So the benzene ring accounts for three double bonds and a ring, the C=C for a double bond and the -COOH for a double bond (C=O). At this point, we don't know if there are even more double bonds or rings, so we have d >= 5 and r >= 1 and the following inequality

h = 2c + 2 - 2d - 2r < 2c - 10

So the molecule will not have the maximum possible number of hydrogen atoms (2c + 2) but much less.

$M_r(\ce{C_xH_y})= \pu{116 g mol^{-1}}$

Just like porphyrin said in his comment, there is only one solution to x and y that makes chemical sense: 9 carbons and 8 hydrogens. You don't even have to think about the rings and double bonds like I discussed above, but it is a nice check. For a hydrocarbon with 9 carbon atoms we would expect 20 hydrogen atoms in the absence of rings and double bonds. We have 12 less than that, so the number of rings and double bonds in the molecules is 6. That fits with the expectation of a benzene ring (accounting for 4), another C=C double bond (1 more) and a carboxylate group (1 more).

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