Step 1: Understanding the Concept:
The sign of the enthalpy change indicates whether the reaction is endothermic or exothermic. Le Chatelier's principle states that increasing temperature favors the endothermic direction.
Step 2: Detailed Explanation:
The reaction is given as \(2\text{NO}_2(g) \rightleftharpoons \text{N}_2\text{O}_4(g) + 14.6 \text{ kcal}\). The "+14.6 kcal" on the right side means the forward reaction (formation of N\(_2\)O\(_4\)) is exothermic. The reverse reaction (decomposition of N\(_2\)O\(_4\)) is endothermic.
According to Le Chatelier's principle, an increase in temperature favors the endothermic direction, i.e., the decomposition of N\(_2\)O\(_4\).
Step 3: Final Answer:
An increase in temperature favors the decomposition of N\(_2\)O\(_4\), which corresponds to option (B).