We can verify the statements by looking at the carbon count and reaction mechanisms for the synthesis of methane.
Examination of Statement I:
Methane ($CH_4$) is the simplest alkane with one carbon.
- Decarboxylation of sodium ethanoate removes a $CO_2$ group from a 2-carbon chain, leaving 1 carbon ($CH_4$). This works.
- Kolbe's electrolysis involves the dimerization of alkyl groups. For sodium acetate, the methyl groups ($CH_3$) pair up to form ethane ($CH_3-CH_3$), which has 2 carbons. Thus, methane is not the product of this reaction.
- Grignard reagents like $CH_3MgBr$ react with any active hydrogen source (like water) to give the corresponding alkane. Here, $CH_3$ picks up $H$ from water to form $CH_4$. This works.
Since the second method produces ethane instead of methane, Statement I is incorrect.
Examination of Statement II:
- Unsaturated hydrocarbons like ethene ($C_2H_4$) or ethyne ($C_2H_2$) have a minimum of two carbon atoms. Reducing them via hydrogenation results in ethane ($C_2H_6$). You cannot get a 1-carbon alkane from a 2-carbon (or more) unsaturated chain using these methods.
- The Wurtz reaction is a coupling reaction where two alkyl groups join together. Even with the smallest methyl groups, the resulting alkane must have at least 2 carbons ($CH_3-CH_3$). Therefore, methane cannot be produced.
Both claims in Statement II are true.
In summary, Statement I is false because Kolbe's electrolysis yields ethane, and Statement II is true because neither unsaturated hydrocarbons nor the Wurtz reaction can produce a single-carbon alkane.