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What would be the best reaction to increase the carbon chain length in an alkane? Preferably, it would be by one at a time; however, if that is not possible, just generally increasing the total length of the carbon chain (such as producing pentane from propane) would be fine.
Moreover, is there a way that I can add a methyl group to a compound in the form $\ce{R-CH3}$?
Homologation methods exist, but typically not for unbranched alkanes. A typical example is the Arndt-Eistert homologation of carboxylic acids.
For simple unbranched alkanes, stepwise chain elongation is not a technical relevant process. On the contrary, petrochemical processes reach for the opposite (formation of smaller alkanes) by thermal or catalytical cracking.
Formation of long chain alkanes, although not via subsequent step up reactions, is possible by catalytic oxidative dimerization of terminal alkynes (Glaser, Hay or Eglinton coupling) and subsequent hydrogenation or by Kolbe dimerization of carboxylic acids.
Branched alkanes, such as triptane (2,2,3-trimethylbutane), which are interesting as 'anti-knocking' fuel additives, are another league. Here, the source of the methyl group to be transfered is methanol, catalysts are either zeolites or indium(III)-iodide ($\ce{InI3}$) (DOI).
One method could be metathesis of alkanes. There is a nice paper about it published by J. M. Basset et al., SCIENCE 1997, 276, 99-102, "Metathesis of Alkanes Catalyzed by Silica-Supported Transition Metal Hydrides" and for sure more recent ones have been published since then.
Abstract The silica-supported transition metal hydrides $\ce{(≡Si-O-Si≡)(≡Si-O-)2Ta-H}$ and $\ce{(≡Si-O-)xM-H (M, chromium or tungsten)}$ catalyze the metathesis reaction of linear or branched alkanes into the next higher and lower alkanes at moderate temperature ($\pu{25^\circ}$ to $\pu{200^\circ C}$). With $\ce{(≡Si-O-Si≡)(≡Si-O-)2Ta-H}$, ethane was transformed at room temperature into an equimolar mixture of propane and methane. Higher and lower homologs were obtained from propane, butane, and pentane as well as from branched alkanes such as isobutane and isopentane. The mechanism of the step leading to carbon-carbon bond cleavage and formation likely involves a four-centered transition state between a tantalum-alkyl intermediate and a carbon-carbon bond of a second molecule of alkane.
Carbenes, like $\ce{CH2}$, are known to insert into everything, including, in case of absence of better substrate, carbon-hydrogen bonds. However, the process is not clean, as carbene will insert into any suitable position it can find, with no selectivity. Carbenes are usually produced by decomposition of precursors, usually ketene ($\ce{CH2=C=O}$) or diazomethane ($\ce{CH2=N=N}$).
In case there is some group in the compounds, it is often possible to selectively substitute hydrogen, activated by the group. For example, many electronegative groups, like carbonyl group, activate nearby hydrogenes, making them quite acidic by organic chemistry standards. Produced anion can react with equivalent of carbocation. Carbonyl group may be later eliminated by Wolff–Kishner reduction (heating with hydrazine and alkaly metal hydroxide)
However, there is no general way to increase chain of alkanes cleanly and selectively.
