Concept:
Grignard reagents react with carbon dioxide to form carboxylic acids after acidic hydrolysis.
\[\begin{aligned}
RMgX + CO_2
\rightarrow
RCOOMgX
\xrightarrow{H^+}
RCOOH
\end{aligned}\]
Carboxylic acids undergo Hell-Volhard-Zelinsky (HVZ) reaction with \(Cl_2/red\;P\) to form \(\alpha\)-halo acids.
Soda-lime decarboxylation gives an alkane having one carbon atom less than the parent acid.
Step 1: Identify compound \(P\).
Ethyl magnesium bromide reacts with carbon dioxide as
\[\begin{aligned}
C_2H_5MgBr + CO_2
\rightarrow
C_2H_5COOMgBr
\end{aligned}\]
On acidic hydrolysis,
\[\begin{aligned}
C_2H_5COOMgBr
\xrightarrow{H^+}
C_2H_5COOH
\end{aligned}\]
Thus,
\[\begin{aligned}
P=CO_2,\qquad Q=C_2H_5COOH
\end{aligned}\]
Step 2: Identify compound \(R\).
Propanoic acid undergoes HVZ reaction.
\[\begin{aligned}
CH_3CH_2COOH
\xrightarrow{Cl_2/red\,P}
CH_3CHClCOOH
\end{aligned}\]
Therefore,
\[\begin{aligned}
R=CH_3CHClCOOH
\end{aligned}\]
Step 3: Identify compound \(S\).
Soda-lime decarboxylation:
\[\begin{aligned}
CH_3CH_2COOH
\xrightarrow{\text{soda lime}}
C_2H_6
\end{aligned}\]
Thus,
\[\begin{aligned}
S=C_2H_6
\end{aligned}\]
Step 4: Summarize the compounds.
\[
\begin{aligned}
P &:\quad CO_2 \\
Q &:\quad C_2H_5COOH \\
R &:\quad CH_3CHClCOOH \\
S &:\quad C_2H_6
\end{aligned}
\]
\[\begin{aligned}
\boxed{
CO_2,\;
C_2H_5COOH,\;
CH_3CHClCOOH,\;
C_2H_6
}
\end{aligned}\]
Hence, option \(\mathbf{(C)}\) is correct.