What is the chemical equation for the decarboxylation of butyric acid with soda lime?

Decarboxylation is the removal of carbon dioxide from a molecule.

For a carboxylic acid, the reaction would be #"R-COO-H" → "R-H" + "CO"_2#

However, only the sodium or potassium salt undergoes ready decarboxylation.

Soda lime is essentially a mixture of sodium hydroxide, calcium oxide, and calcium hydroxide.

The first step is the reaction of butyric acid with soda lime to form sodium butyrate.

#"CH"_3"CH"_2"CH"_2"COOH" + "NaOH" → "CH"_3"CH"_2"CH"_2"COONa" + color(red)(cancel(color(black)("H"_2"O")))#
#"CaO" + color(red)(cancel(color(black)("H"_2"O"))) → "Ca(OH)"_2#
#stackrel(————————————————————)("CH"_3"CH"_2"CH"_2"COOH" + "NaOH" + "CaO" → "CH"_3"CH"_2"CH"_2"COONa" + "Ca(OH)"_2)#

Second is the decarboxylation of the sodium salt.

#"CH"_3"CH"_2"CH"_2"COONa" + "NaOH" → "CH"_3"CH"_2"CH"_3 + "Na"_2"CO"_3#

The overall equation is the sum of the two equations.

#"CH"_3"CH"_2"CH"_2"COOH" + "NaOH" + "CaO" → color(red)(cancel(color(black)("CH"_3"CH"_2"CH"_2"COONa"))) + "Ca(OH)"_2#
#color(red)(cancel(color(black)("CH"_3"CH"_2"CH"_2"COONa"))) + "NaOH" → "CH"_3"CH"_2"CH"_3 + "Na"_2"CO"_3#
#stackrel(————————————————————————)("CH"_3"CH"_2"CH"_2"COOH" + "2NaOH" + "CaO" → "CH"_3"CH"_2"CH"_3 + "Na"_2"CO"_3+ "Ca(OH)"_2)#