$^1H$ NMR spectrum of a mixture containing $CH_3Br$ ($x$ mol) and $(CH_3)_3CBr$ ($y$ mol) shows two singlets at 2.7 ppm and 1.8 ppm, with the relative ratio of 3:1 (integration value), respectively. The value of $x/y$ is ____________
(rounded off to the nearest integer)
NMR Spectrum Interpretation
The $^1H$ NMR spectrum provides information about the number and environment of hydrogen atoms in a molecule. The given spectrum shows two signals, indicating two distinct types of proton environments in the mixture.
- Signal 1: Observed at 2.7 ppm with a relative integration of 3.
- Signal 2: Observed at 1.8 ppm with a relative integration of 1.
Identifying Proton Sources
We need to correlate these signals to the protons in the two compounds: $CH_3Br$ (moles = $x$) and $(CH_3)_3CBr$ (moles = $y$).
- $CH_3Br$: Contains a methyl group ($CH_3$) with 3 protons. These protons are expected to resonate further downfield due to the electronegativity of the adjacent bromine atom.
- $(CH_3)_3CBr$: Contains a tert-butyl group ($(CH_3)_3C$), which has three equivalent methyl groups. This results in a total of 9 equivalent protons. These protons typically resonate at a higher field (lower ppm value) compared to $CH_3Br$.
Therefore:
- The signal at 2.7 ppm corresponds to the 3 protons of $CH_3Br$.
- The signal at 1.8 ppm corresponds to the 9 protons of $(CH_3)_3CBr$.
Calculating Molar Ratio ($x/y$)
The integration value of an NMR signal is directly proportional to the number of protons generating that signal.
- The total number of protons contributing to the 2.7 ppm signal is $3 \times x$ (3 protons per molecule times $x$ moles).
- The total number of protons contributing to the 1.8 ppm signal is $9 \times y$ (9 protons per molecule times $y$ moles).
The observed relative integration ratio is 3:1 for the 2.7 ppm signal ($CH_3Br$) and the 1.8 ppm signal ($(CH_3)_3CBr$), respectively. This ratio must equal the ratio of the total number of protons for each compound.
Setting up the ratio:
$ \frac{\text{Integration of Signal 1}}{\text{Integration of Signal 2}} = \frac{\text{Total Protons from } CH_3Br}{\text{Total Protons from } (CH_3)_3CBr} $
$ \frac{3}{1} = \frac{3x}{9y} $
Simplify the equation:
$ \frac{3}{1} = \frac{x}{3y} $
Now, solve for the molar ratio $x/y$:
$ 3 \times (3y) = 1 \times x $
$ 9y = x $
$ \frac{x}{y} = 9 $
Final Answer
The calculated value of $x/y$ is 9. Rounded to the nearest integer, the value remains 9.
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