Question

Consider the reaction: $$\mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{PCl}_{5}(g)$$ At \(25^{\circ} \mathrm{C}, \Delta G^{\circ}=-92.50 \mathrm{kJ}\) Which of the following statements is (are) true? a. This is an endothermic reaction. b. \(\Delta S^{\circ}\) for this reaction is negative. c. If the temperature is increased, the ratio \(\frac{\mathrm{PCl}_{5}}{\mathrm{PCl}_{3}}\) will increase. d. \(\Delta G^{\circ}\) for this reaction has to be negative at all temperatures. e. When \(\Delta G^{\circ}\) for this reaction is negative, then \(K\) is greater than 1.00 .

Short answer

The short answer based on the step-by-step solution is: a. False b. False (Cannot be determined) c. False (Cannot be determined) d. False e. True

Step-by-step solution

(Determine if the reaction is endothermic or exothermic)

The given reaction has a ΔG° = -92.50 kJ at 25°C. An endothermic reaction is one in which the standard enthalpy change (ΔH°) is positive, and an exothermic reaction is one in which ΔH° is negative. Keep in mind that there isn't enough information to directly evaluate ΔH°. However, since the question refers to ΔG, and we know that ΔG = ΔH - TΔS (ΔS being the entropy), we can infer that a negative ΔG value indicates a spontaneous reaction. Answer: a. False #Step 2: Determine the sign of ΔS° (entropy change)#

(Determine the sign of ΔS°)

Given that ΔG = ΔH - TΔS, with T > 0, a negative ΔG° suggests that the term (ΔH - TΔS) is negative. We can't uniquely determine the signs of ΔH or ΔS from this reaction, but the following statement is not necessarily true. Answer: b. False (Cannot be determined) #Step 3: Effect of temperature on the reaction equilibrium#

(Effect of temperature on the reaction equilibrium)

Le Chatelier's principle states that if an external stress (e.g., temperature, pressure, or concentration) is applied to a system in equilibrium, the system will adjust to counteract the stress and restore equilibrium. In this case, we consider the effect of increasing temperature. However, we do not have enough information available currently to determine how the temperature will impact the equilibrium concentration ratio. Answer: c. False (Cannot be determined) #Step 4: ΔG° at different temperatures#

(ΔG° at different temperatures)

ΔG° is not constant at all temperatures - its value depends on the standard enthalpy change (ΔH°) and the standard entropy change (ΔS°) according to the equation ΔG = ΔH - TΔS. We don't have enough information about these variables to determine if ΔG° has to be negative at all temperatures. Answer: d. False #Step 5: Relate ΔG° to K (equilibrium constant)#

(Relate ΔG° to K)

There is a relationship between ΔG° and the equilibrium constant K, according to the equation \(ΔG° = -RT \ln{K}\) (where R is the gas constant and T is the temperature). When ΔG° is negative, it implies that the Natural logarithm (ln) of K is positive (because -RT is negative), meaning K > 1.00. Answer: e. True

Key concepts

Gibbs free energy

Gibbs free energy (ΔG) is a thermodynamic potential that helps predict whether a process will occur spontaneously at constant pressure and temperature. If ΔG is negative, the process is spontaneous; if positive, it is non-spontaneous.
Understanding Gibbs free energy can be distilled into knowing its role in chemical reactions:

• Spontaneous Reaction: ΔG < 0.
• Non-Spontaneous Reaction: ΔG > 0.
• Equilibrium: The system is at equilibrium when ΔG = 0.

Gibbs free energy connects with other thermodynamic properties through the equation:\[ΔG = ΔH - TΔS\]Where ΔH is enthalpy, T is temperature, and ΔS is entropy. In our exercise, with ΔG^{°}=-92.50 ext{kJ}, it indicates the reaction is spontaneous at 25°C.

enthalpy

Enthalpy ( ΔH ) is the measure of the total heat content in a thermodynamic system and reflects the energy required to form the system. It signifies whether a reaction absorbs heat (endothermic, ΔH > 0 ) or releases heat (exothermic, ΔH < 0 ).
When considering a reaction's spontaneity using Gibbs free energy, ΔH is a key component:

• Endothermic reactions: absorb heat; they often need energy input to proceed.
• Exothermic reactions: release heat; they can proceed spontaneously under proper conditions.

In our exercise, the Gibbs free energy is used to infer reaction characteristics, but without explicit ΔH values, determining if the reaction is endothermic or exothermic requires more information. However, the negative ΔG suggests a tendency towards exothermic properties.

entropy

Entropy ( ΔS ) measures the level of disorder or randomness in a system. It is a fundamental concept in determining the feasibility and direction of chemical processes.
A process leading to more disorder has a positive ΔS , while one leading to less disorder has a negative ΔS .

• Positive ΔS : increases randomness.
• Negative ΔS : decreases randomness.

In the equation ΔG = ΔH - TΔS , ΔS contributes to the free energy change, and it can heavily influence the reaction outcome, especially at varying temperatures.
For the reaction given, without direct calculation of ΔS , it's challenging to definitively state its positivity or negativity. However, generally, gas reactions tend to increase ΔS due to the dispersal of molecules.

equilibrium constant

The equilibrium constant (K) is a crucial part of chemical thermodynamics. It quantifies the ratio of product concentration to reactant concentration when a reaction is at equilibrium.
The equation linking Gibbs free energy and equilibrium constant is:\[ΔG^{°} = -RT \, \ln{K}\]Here, R is the gas constant, T the temperature, and K indicates reaction favorability:

• K > 1: favoring products, spontaneous reaction.
• K < 1: favoring reactants, non-spontaneous reaction.

In our exercise, since ΔG^{°} is negative, K must be greater than 1, showing that the products are favored at equilibrium. This directly ties how spontaneous reactions are favorably inclined towards product formation, dictated by ΔG's relationship with K.