Step 1: Understanding the reaction.
The reaction that explains the acidic property of carboxylic acids is their ability to react with bases like sodium to form salts (sodium carboxylates) and release hydrogen gas. This is a characteristic reaction of acids, where they neutralize with bases to form salts.
Step 2: Analyzing the options.
(A) \( 2 \, R-C-OH + 2 \, Na \xrightarrow{\Delta} 2 \, R-C-ONa + H_2 \uparrow \): Correct — This is the typical reaction of a carboxylic acid with a metal (sodium), explaining its acidic nature.
(B) \( 2 \, R-C-OH \xrightarrow{P_2O_5} (R-CO)_2O + H_2O \): Incorrect. This reaction is the formation of an ester, not related to acidity.
(C) \( R-C-OH + R-OH \xrightarrow{\text{conc.} H_2SO_4} R-C-OR + H_2O \): Incorrect. This is the esterification reaction, not an explanation of acidity.
(D) \( R-C-OH + SOCl_2 \xrightarrow{\Delta} R-C-Cl + SO_2 \uparrow + HCl \uparrow \): Incorrect. This is a reaction to form an acyl chloride, not related to acidity.
Step 3: Conclusion.
The correct answer is (A) \( 2 \, R-C-OH + 2 \, Na \xrightarrow{\Delta} 2 \, R-C-ONa + H_2 \uparrow \), which explains the acidic property of carboxylic acids.