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In mass spectrometry, 1-pentanol can be observed to have no peak at $m/z = 59$, whilst 3-pentanol can be observed to have a peak at $m/z = 59$.

Clearly the 3-pentanol peak at $m/z = 59$ is due to the cleavage creating $\ce{CH3CH2CHOH | CH2CH3}$

Why is this the case, as isn't 1-pentanol able to form the fragment $\ce{CH2CH2CH2OH}$ by cleaving at the third carbon?

i.e. fragmenting into $\ce{CH3CH2}$ and $\ce{CH2CH2CH2OH}$ (this should have $m/z = 59$?).

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    $\begingroup$ RE: "... Why is this the case, as isn't 1-pentanol able to form the fragment CH2CH2CH2OH by cleaving at the third carbon?..." Of course 1-pentanol can cleave in such a manner. It is just that such cleavage is highly improbable. $\endgroup$ – MaxW Dec 19 '19 at 0:34
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In mass spectrometry, organic compounds can be ionized by different methods. The compounds ionized by electron impact ionization display particular fragmentation patterns. When subjected to mass spectroscopy, cleavage of the $\ce{C-C}$ bond next to the oxygen or nitrogen can usually occur, when the compound is ionized by electron impact ionization.

Let’s look at what happens to 3-pentanol: $$\ce{CH3CH2CH(OH)CH2CH3 ->[e- beam] CH3CH2CH(O^.+H)CH2CH3 }$$

$$\ce{CH3CH2CH(O^.+H)CH2CH3 -> CH3CH2CH=O^+H + ^.CH2CH3}$$

Now, base peak (most probably), would be $\ce{[CH3CH2CH=OH]+}$, which should be at $ 59 \ m/z$. Note that in general, the molecular ion of an alcohol (here it is $ 88 \ m/z$) is either trace or non-existent, probably loosing $\ce{H2O^.}$.

Now, we'll look at what happens to 1-pentanol: $$\ce{CH3CH2CH2CH2CH2-OH ->[e- beam] CH3CH2CH2CH2CH2-O^.+H }$$

$$\ce{CH3CH2CH2CH2CH2-O^.+H -> CH2=O^+H + ^.CH2CH2CH2CH3}$$

Now, base peak (again, most probably), would be $\ce{[CH2=OH]+}$, which should be at $ 31 \ m/z$ ($88-31=57$ mass loss). Note that here again, the molecular ion is either trace or non-existent, probably loosing $\ce{H2O^.}$, and possibly $[M-18]$ ion should be significant here, since there is equal chance of cleaving $\ce{C_1-O}$ bond (giving peak at $ 70 \ m/z$) or $\ce{C_2-C_3}$ bond (giving peak at $ 31 \ m/z$):

The Mass Spectrum of 1-pentanol

However, according to the mass spectrum given, $\ce{C_5}$-carbon chain has undergone McLafferty type rearrangement (which is common in MS of aldehydes and ketones) to give the base peak at $ 42 \ m/z$ by loosing a water and ethylene simultaneously from parent ion (see the insert in above diagram).

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