Step 1: Understanding the Concept:
The effect of temperature on the solubility of a solid in a liquid is governed by Le Chatelier's Principle.
If the dissolution process is endothermic (\( \Delta H>0 \), heat is absorbed), increasing the temperature shifts the equilibrium towards more dissolution. Hence, solubility increases.
If the dissolution process is exothermic (\( \Delta H<0 \), heat is released), increasing the temperature shifts the equilibrium towards crystallization. Hence, solubility decreases.
Step 2: Key Formula or Approach:
Evaluate the enthalpy of solution for the given salts.
Endothermic: \( \text{Solute} + \text{Solvent} + \text{Heat} \rightleftharpoons \text{Solution} \) (Solubility \( \propto \) T)
Exothermic: \( \text{Solute} + \text{Solvent} \rightleftharpoons \text{Solution} + \text{Heat} \) (Solubility \( \propto 1/T \))
Step 3: Detailed Explanation:
Let's analyze the given salts:
Salts like \( \text{NaBr} \), \( \text{NaCl} \), and \( \text{NaNO}_3 \) have endothermic enthalpies of solution. Therefore, their solubility generally increases with an increase in temperature (though the increase for NaCl is very slight).
Sodium sulfate (\( \text{Na}_2\text{SO}_4 \)) exhibits unique, anomalous solubility behavior due to a phase transition.
Below \( 32.4^\circ\text{C} \), the stable solid phase is the decahydrate, \( \text{Na}_2\text{SO}_4 \cdot 10\text{H}_2\text{O} \) (Glauber's salt). The dissolution of the decahydrate is endothermic, so its solubility increases as the temperature rises up to \( 32.4^\circ\text{C} \).
At \( 32.4^\circ\text{C} \) (the transition temperature), it loses its water of crystallization and converts to anhydrous \( \text{Na}_2\text{SO}_4 \).
Above \( 32.4^\circ\text{C} \), the stable solid phase is anhydrous \( \text{Na}_2\text{SO}_4 \). The dissolution of anhydrous \( \text{Na}_2\text{SO}_4 \) is an exothermic process.
Therefore, according to Le Chatelier's principle, the solubility of anhydrous \( \text{Na}_2\text{SO}_4 \) decreases with a further increase in temperature.
Step 4: Final Answer:
The salt whose solubility decreases with an increase in temperature is \( \text{Na}_2\text{SO}_4 \).