91Ó°ÊÓ

The equilibrium constant, denoted as \( K_p \) in the context of reactions involving gases, provides valuable information about the ratio of products to reactants at equilibrium. In the given reaction, \( \mathrm{CuSO}_{4} \cdot 3 \mathrm{H}_{2} \mathrm{O}(\mathrm{s}) \rightleftharpoons \mathrm{CuSO}_{4}(\mathrm{s})+3 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \), the equilibrium constant determines the balance between hydrated copper sulfate and its gaseous components at equilibrium.

\( K_p \) values signify the pressure of the gaseous water in equilibrium with the solid reactants:

• At different temperatures, \( K_p \) varies; for instance, here \( K_p \) is \( 10^{-6} \ \mathrm{atm}^3 \) at 298 K and \( 10^{-4} \ \mathrm{atm}^3 \) at 323 K.
• The change indicates that the equilibrium shifts towards producing more gaseous water as temperature increases.

Understanding \( K_p \) helps predict how changes in conditions affect the reaction's position, crucial for controlling industrial processes.

Temperature is a critical factor that influences chemical equilibrium. The Van't Hoff equation specifically provides a mathematical approach to understand this dependence.

This equation shows that as temperature varies, the equilibrium constant \( K_p \) changes, reflecting a shift in the equilibrium position.

• For endothermic reactions like the dissociation of \( \mathrm{CuSO}_{4} \cdot 3 \mathrm{H}_{2} \mathrm{O} \), an increase in temperature often results in a higher \( K_p \), favoring the formation of products.
• In contrast, for exothermic reactions, increasing temperature usually decreases \( K_p \), pushing the equilibrium towards reactants.

By using the Van't Hoff equation, \( \ln \left( \frac{K_{2}}{K_{1}} \right) = -\frac{\Delta_{r} H^{\circ}}{R} \left( \frac{1}{T_{2}} - \frac{1}{T_{1}} \right) \), we can calculate the change in \( K_p \) with temperature and explore the nature of the reaction.

The enthalpy change, \( \Delta_{r} H^{\circ} \), reveals whether a reaction absorbs or releases heat. For any given chemical reaction, understanding \( \Delta_{r} H^{\circ} \) offers insights into the thermal characteristics of the process:

• In the provided exercise, the positive value of \( \Delta_{r} H^{\circ} = 147.41 \ \mathrm{kJ/mol} \) indicates the reaction is endothermic, meaning it requires heat to proceed.
• This is reflected in the shift of equilibrium toward more product formation as temperature increases, consistent with Le Chatelier's Principle.
• Solving the Van't Hoff equation gives this value, reinforcing the relationship between temperature, \( K_p \), and enthalpy.
• It's also key in designing industrial processes, like optimizing conditions for desired product yields.

Overall, knowing \( \Delta_{r} H^{\circ} \) not only aids in understanding reaction spontaneity but also assists in predicting temperature effects on equilibrium.