Question

For the reaction $$\mathrm{A}(\mathrm{s})\rightleftharpoons \mathrm{B}(\mathrm{s})+2\mathrm{C}(\mathrm{g})+\frac{1}{2}\mathrm{D}(\mathrm{g})\mathrm{\Delta }{H}^{\circ }=0$$ (a) Will $K_p$ increase, decrease, or remain constant with temperature? Explain. (b) If a constant-volume mixture at equilibrium at 298 K is heated to 400 K and equilibrium re-established, will the number of moles of $\mathrm{D}(\mathrm{g})$ increase, decrease, or remain constant? Explain.

Short answer

(a) $K_p$ will remain constant with changes in temperature as the $\mathrm{\Delta }{H}^{\circ }$ for the reaction is 0. (b) The number of moles of $D_{(g)}$ will increase when the system is heated from 298 K to 400 K. This is due to the shift in equilibrium towards the products side as per Le Chatelier's principle.

Step-by-step solution

Understanding the Equilibrium Reaction

The given equation represents an equilibrium reaction where reactants and products are in different phases. Note that the reaction yields an increase of gaseous particles, thus an increase in entropy. The enthalpy change $\mathrm{\Delta }{H}^{\circ }$ for the reaction is 0.

Evaluate the effect of temperature on the equilibrium constant (Kp)

The Van't Hoff equation describes how the equilibrium constant $K_p$ changes with temperature. When $\mathrm{\Delta }{H}^{\circ }$ is zero as in this case, the equilibrium constant $K_p$ doesn't depend on temperature. Therefore, $K_p$ will remain constant irrespective of the temperature changes.

Analyze the effect of temperature on equilibrium position

Le Chatelier's principle states that if a system at equilibrium is subjected to a change, the system will adjust itself in a way that minimizes that change. In this case, since the reaction involves the production of more gaseous particles, an increase in temperature will shift the reaction to the product side. This is due to the positive entropy change. Therefore, the number of moles of $D_{(g)}$ will increase when the system is heated from 298 K to 400 K.

Key concepts

Van't Hoff equation

The Van't Hoff equation is an essential concept in chemical equilibrium. It helps us understand how the equilibrium constant, denoted as $K_p$ for gas-phase equilibria, varies with temperature. The equation can be expressed as:

• $\ln K=-\frac{\mathrm{\Delta }{H}^{\circ }}{R}(\frac{1}{T})+C$

Here, $\mathrm{\Delta }{H}^{\circ }$ is the standard enthalpy change, $R$ is the universal gas constant, and $T$ is the temperature in Kelvin. The constant $C$ refers to some constant related to the integration process.

In cases where the enthalpy change ($\mathrm{\Delta }{H}^{\circ }$) is zero, like in our original exercise, the equation simplifies the analysis. Since there is no heat absorbed or released, the factor $-\frac{\mathrm{\Delta }{H}^{\circ }}{R}=0$, leaving $K_p$ unaffected by temperature changes.
Thus, even with temperature variations, $K_p$ remains constant, making the Van't Hoff equation a powerful tool for predicting temperature-impact on chemical equilibria.

Le Chatelier's principle

Le Chatelier's principle is a cornerstone in chemical equilibrium that explains how a system at equilibrium responds to external changes. When a change is applied, the equilibrium will shift in a direction to counteract the effect of the change.

In the context of the given reaction:

• Since the production of gaseous particles increases the entropy (disorder) of the system, a temperature increase would favor the formation of products according to Le Chatelier's principle.

This principle asserts that the reaction will shift towards the side with more gaseous products when heated, because more gas molecules result in higher entropy, which the system tries to reach under higher temperatures.

For a reaction having zero enthalpy change, the shift is primarily driven by entropy changes, leading to an increased number of moles of the gaseous product at higher temperatures. This explains the increase in moles of $D_{(g)}$ as the system moves from 298 K to 400 K.

Gaseous equilibria

Gaseous equilibria involve reactions where gases are part of the reactants or products, affecting the equilibrium in a unique way. For such reactions, the equilibrium constant is typically expressed in terms of partial pressures as $K_p$.

The behavior of gaseous equilibria can be heavily influenced by changes in pressure and temperature.

• An increase in the total number of gas molecules usually indicates a greater disorder or entropy, leading reactions in the gas phase to favor the side with more gas molecules under constant pressure conditions.
• If we consider a constant volume system, changes in temperature can cause shifts in equilibrium positions as per Le Chatelier's principle.

Reactions in the gas phase, like the one in our exercise, demonstrate that when the system is heated, the increase in gaseous products corresponds to an increase in entropy.
Understanding gaseous equilibria allows us to predict how changes in conditions such as temperature and pressure will impact the equilibrium state, crucial for controlling and optimizing industrial chemical processes.