Question

Consider the following system at equilibrium at \(25^{\circ} \mathrm{C}\) : $$\mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{PCl}_{5}(g) \quad \Delta G^{\circ}=-92.50 \mathrm{~kJ}$$ What will happen to the ratio of partial pressure of \(\mathrm{PCl}_{5}\) to partial pressure of \(\mathrm{PCl}_{3}\) if the temperature is raised? Explain completely.

Short answer

In conclusion, raising the temperature in the given equilibrium system will decrease the ratio of partial pressure of \(\mathrm{PCl}_{5}\) to the partial pressure of \(\mathrm{PCl}_{3}\), as the system will shift towards the reverse reaction according to Le Chatelier's principle, favoring the production of reactants over the product.

Step-by-step solution

Write down the given reaction and ΔG°

We are given the following reaction at equilibrium: \(\mathrm{PCl}_3(g) + \mathrm{Cl}_2(g) \rightleftharpoons \mathrm{PCl}_5(g)\) and \(\Delta G^{\circ} = -92.50 \,\mathrm{kJ/mol}\)

Calculate the equilibrium constant Kp from ΔG°

Using the equation for the relationship between the standard Gibbs free energy change and the equilibrium constant Kp: \(\Delta G^{\circ} = -RT\ln(K_p)\) We can solve for Kp by rearranging the formula: \(K_p = e^{-\frac{\Delta G^{\circ}}{RT}}\) Given the temperature T=298K (25°C) and universal gas constant R=8.314 J/molK, we can plug in the values: \(K_p = e^{\frac{92.50\times10^3}{8.314\times298}}\) \(K_p \approx 2.77\times10^4\)

Consider the effect of the temperature change

From the Van't Hoff equation, we can conclude that an increase in temperature will favor the endothermic direction of the reaction. Now, let's consider the relationship between ΔG and the temperature. Recall the expression for the Gibbs free energy change in terms of enthalpy change (ΔH) and entropy change (ΔS): \(\Delta G = \Delta H - T\Delta S\) Since the reaction has a negative ΔG, it's spontaneous at 25°C, and it implies that the reaction is exothermic (ΔH is negative). Therefore, when the temperature is raised, according to Le Chatelier's principle, the system will counteract the change by shifting the equilibrium toward the endothermic direction, which, in this case, is the reverse reaction.

Predict the effect on the ratio of partial pressures

As stated in the previous step, when the temperature is raised, the system will shift toward the reverse reaction, which implies that the equilibrium will favor the production of reactants, PCl3 and Cl2, rather than the product, PCl5. This means that the ratio of the partial pressures of PCl5 to PCl3 will decrease, as the concentration of PCl3 will increase and the concentration of PCl5 will decrease. In conclusion, when the temperature is raised, the ratio of partial pressure of PCl5 to the partial pressure of PCl3 will decrease due to the shift in the equilibrium position, favoring the reverse (endothermic) reaction as per Le Chatelier's principle.

Key concepts

Le Chatelier's Principle

Le Chatelier's principle is a fundamental concept in chemical equilibrium that helps us predict how a system at equilibrium responds to external changes. This principle states that if an external stress is applied to a system at equilibrium, the system will adjust in such a way as to counteract that stress.

External stresses can include changes in concentration, pressure, volume, or temperature. For instance, if the temperature of a system is increased, the system will favor the reaction that absorbs heat to try to lower the temperature back to its original state. Similarly, decreasing the temperature would shift the equilibrium to favor the reaction that releases heat.

In the context of the exercise, when the temperature is raised, the equilibrium shifts in the direction that absorbs the added thermal energy, which is the endothermic direction. As a result, more reactants (PCl₃ and Cl₂) are produced, which decreases the ratio of the partial pressure of PCl₅ to PCl₃. This shift aligns with Le Chatelier's principle, ensuring that the system dynamically adjusts to the new equilibrium state.

Equilibrium Constant

The equilibrium constant, represented as K, is a numerical value that expresses the ratio of the concentrations of the products to the reactants at equilibrium, raised to the power of their respective stoichiometric coefficients. It's essential for understanding the extent to which a reaction proceeds under a given set of conditions.

The value of the equilibrium constant is fixed for a given reaction at a specific temperature. An increase or decrease in the value of K indicates the shift of the equilibrium position towards products or reactants, respectively. For gaseous reactions, the equilibrium constant in terms of partial pressures is represented by K_p.

Using the Gibbs free energy change (abla G) in the given exercise, we calculated K_p and found it to be a large number, suggesting a strong tendency toward product formation at 25°C. However, a change in temperature will alter K_p, leading to an adjustment in the ratio of products to reactants at the new equilibrium state.

Gibbs Free Energy

Gibbs free energy (G) is a thermodynamic quantity that serves as a criterion for spontaneity at constant temperature and pressure. A negative value of Gibbs free energy (abla G) indicates that a reaction is spontaneous in the direction written, while a positive value suggests non-spontaneity.

The relationship between Gibbs free energy and the equilibrium constant is given by the equation abla G° = -RTln(K), where R is the gas constant, T is the temperature in Kelvin, and K is the equilibrium constant. This equation allows us to calculate the equilibrium constant using the standard Gibbs free energy change for a reaction.

In our example, the negative abla G° suggests that the formation of PCl₅ is spontaneous at 25°C. However, as temperature increases, abla G will become less negative due to the term -Tabla S becoming larger, assuming the reaction has a positive entropy change (abla S). Consequently, the spontaneous nature of the forward reaction decreases, and the system is no longer as favorable towards product formation as before, which is consistent with the behavior predicted by Le Chatelier's principle.