Difference between a Primary, Secondary, Tertiary, Quaternary Carbon and Nitrogen

A primary carbon/nitrogen is denoted by numeral 1, secondary with numeral 2, tertiary with numeral 3 and quaternary with numeral 4 with a degree (o) sign in the superscript. Example, primary (1o), secondary (2o), tertiary (3o), quaternary (4o).
 
The nomenclature is used for an sp3 hybridized carbon of an alkane and an alkyl group carrying halide or the hydroxy functional group. Amines and amides can also be classified under this nomenclature. 
 
A carbon is called primary if it has only one neighboring alkyl carbon atom, secondary if it has two and tertiary for three linked alkyl carbon atoms. For an alkane shown below, the terminal methyls will be primary (1o), the penultimate ones will be secondary (2o), and the central one will be the tertiary (3o) carbon. 
primary, secondary, tertiary carbon in an alkane
 
An alkane can also have a quaternary (4o) carbon , a carbon that is linked to four other alkyl carbon atoms. Example, neo-pentane. 
primary, secondary, tertiary carbon neopentane
 
For assigning the nomenclature for the alkyl halides and alcohols, the functional group is not counted when adding the number of the connected substituents. Only carbon partners are taken into account. Therefore, a terminal methyl carbon in an alkane is a primary carbon, but in an alkyl halide, it is not as it has no carbon neighbor. 
number of primary, secondary, tertiary carbons in alkyl halide and alcohol
 
For an amine or an amide, the number of carbons attached to the nitrogen becomes an important parameter. A primary amine has one carbon linked to the nitrogen (CH3-CH2-NH2), a secondary amine has two (CH3-CH2-NH-CH2-CH3), and a tertiary amine has three (CH3-CH2-)3-N. 
primary amine, secondary amine, tertiary amine nomenclature
 
The same logic applies to the amides. A simple 1o amide would have an -NH2 group linked to the carbonyl carbon. Example, CH3-CO-NH2. A secondary amide would be two carbon linkages extending from the Nitrogen (Example, CH3-CO-NH-CH3), and a tertiary amide would have three. Example, CH3-CO-N(CH3)2
primary amide, secondary amide, tertiary amide nomenclature
 
In amines, the Nitrogen lone pair sometimes participates in bonding, forming a quaternary ammonium derivate (CH3-CH2-)4N+. It is called as ammonium due to the positive charge on the Nitrogen. It would mostly exist as salt with a negative counterion (Cl-, HSO4- etc.) also being present. Example, tetraethylammonium bromide.
quaternary amide example
 
For an amide, a quaternary salt is not stable and therefore, not found.
 
The aim of the primary to tertiary nomenclature is to understand a substrate's reactivity and stability under certain reaction conditions- For Example,
 
The order of reactivity of alkyl halides towards SN1 is-  -CH3 < 1o < 20 < 30 
 
The order of the acidic strength of alcohol is - 1o>  2o > 3o 
 
The stability of the carbocation follows the order-  3o>  2o>  1o
 
The basicity of the amines in the gaseous phase is -  3o>  2o>  1o
 
But, can the nomenclature be used if the alkyl sp3 carbon is next to an sp2 hybridized carbon of an alkene or a carbonyl?
 
For an alkene, the alkyl carbon can be called as primary, secondary or tertiary but the alkene carbon is called as monosubstituted or disubstituted. This again has to do with the reactivity and the stability of the mono versus disubstituted alkene. The nomenclature of the neighboring alkyl carbon has the least importance. It gains importance only when an electrophilic substitution reaction happens on an alkene, and the product formation depends on the stability of the carbocation. 
primary and secondary carbocation nomenclature
 
But if the sp3 carbon carries a halide or a hydroxy functional group, it is called as an allylic halide or allylic alcohol.
allylic halide and allylic alcohol example
 
Similarly, the nomenclature of an sp3 carbon next to an sp2 carbonyl carbon is denoted using Greek alphabets of alpha (α), beta (β), gamma (γ), delta (δ). The alpha carbon being the closest and the delta farthest to the carbonyl functional group. Primary, secondary, tertiary nomenclature is not used in such cases.
Alpha, Beta, Gamma and Delta Nomenclature of Carbon next to carbonyl
 
The 1o to 3o nomenclature also extends to the hydrogens. The hydrogens attached to a 1o, 2o, 3o carbon are called 1o, 2o, and 3o hydrogens. 
Primary Hydrogens, Secondary Hydrogens and Tertiary Hydrogens
 
 

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