Well I was studying organic chemistry, and I came across this reaction for increasing the length of the carbon chain. Upon asking my teacher about the following reaction, she said she was unsure about it. Is this a valid reaction? And if it is, can you please provide the name of the reagent and the actual mechanism?
Answer
I think the reagent you're talking about is ketene (R=H in the figure below). It is a linear molecule (the central carbon is $\ce{sp}$ hybridized) and very reactive.
When mixed with an alkane and heated it dimerizes to form diketene rather than react with the alkane.
However, when irradiated with light, ketene will produce the reactive carbene "methlylene" ($\ce{CH2:}$). When this photolysis is performed in the presence of an alkane, the methylene will insert itself into the various $\ce{C-H}$ bonds like you've drawn. However the reaction is not synthetically useful for a number of reasons. For example, the $\ce{C-H}$ insertion reaction is very indiscriminate and a variety of products will be formed if different $\ce{C-H}$ bonds exist in the molecule.
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