Explanation:
\({\rm{C}}{{\rm{H}}_{\rm{3}}}{\rm{COONa + NaOH}}\)\(\xrightarrow[\Delta ]{{{\text{CaO}}}}\)\({\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \mathop {{\rm{C}}{{\rm{H}}_{\rm{4}}}}\limits_{({\rm{X}})} {\rm{ + Na2C}}{{\rm{O}}_{\rm{3}}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ({\rm{decarboxylation}})\)
\({\text{2C}}{{\text{H}}_{\text{3}}}{\text{COONa + 2}}{{\text{H}}_{\text{2}}}{\text{O}}\xrightarrow{\Delta }\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {{{\text{C}}_{\text{2}}}{{\text{H}}_{\text{6}}}}\limits_{{\text{(Y)}}} {\text{ + 2NaOH + 2C}}{{\text{O}}_{\text{2}}}{\text{ + }}{{\text{H}}_{\text{2}}}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(Kolbe's}}\,\,{\text{electrolysis)}}\)
\({\mathrm{\therefore \mathrm{X}=\mathrm{CH}_{4} \&}}\)
\({\mathrm{\mathrm{Y}=\mathrm{C}_{2} \mathrm{H}_{6}}}\)