Question

Discuss the effect of temperature change on the spontaneity of the following reactions at 1 atm. (a) \(\mathrm{Al}_{2} \mathrm{O}_{3}(s)+2 \mathrm{Fe}(s) \longrightarrow 2 \mathrm{Al}(s)+\mathrm{Fe}_{2} \mathrm{O}_{3}(s)\) $$ \Delta H^{\circ}=+851.5 \mathrm{~kJ} ; \Delta S^{\circ}=+38.5 \mathrm{~J} / \mathrm{K} $$ (b) \(\mathrm{N}_{2} \mathrm{H}_{4}(l) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2}(g)\) $$ \Delta H^{\circ}=-50.6 \mathrm{~kJ} ; \Delta S^{\circ}=0.3315 \mathrm{~kJ} / \mathrm{K} $$ (c) \(\mathrm{SO}_{3}(g) \longrightarrow \mathrm{SO}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)\) $$ \Delta H^{\circ}=98.9 \mathrm{~kJ} ; \Delta S^{\circ}=+0.0939 \mathrm{~kJ} / \mathrm{K} $$

Short answer

Based on the analysis of Gibbs free energy change for the given reactions, the effect of temperature change on the spontaneity of the reactions can be summarized as follows: - Reaction (a): The temperature needs to be higher than 22117 K for the reaction to be spontaneous. - Reaction (b): This reaction is spontaneous at all temperatures. - Reaction (c): The temperature needs to be higher than 1053 K for the reaction to be spontaneous.

Step-by-step solution

Reaction (a) - Analyzing Gibbs free energy change

For the reaction (a), we have the following values of enthalpy and entropy change: \(\Delta H^{\circ} = +851.5 \mathrm{~kJ}\) and \(\Delta S^{\circ} = +38.5 \mathrm{~J} / \mathrm{K}\) Both enthalpy and entropy change are positive. Let's plug these values into the Gibbs free energy change formula and calculate the critical temperature at which \(\Delta G\) becomes negative: \(\Delta G = 851.5 - T(0.0385)\) For the reaction to be spontaneous, \(\Delta G < 0\). Therefore: \(851.5 - T(0.0385) < 0\) Now, find the critical temperature: \(T > \frac{851.5}{0.0385} \approx 22117 \mathrm{K}\) So, for reaction (a) to be spontaneous, the temperature needs to be greater than 22117 K.

Reaction (b) - Analyzing Gibbs free energy change

For the reaction (b), we have the following values of enthalpy and entropy change: \(\Delta H^{\circ} = -50.6 \mathrm{~kJ}\) and \(\Delta S^{\circ} = +0.3315 \mathrm{~kJ} / \mathrm{K}\) In this case, enthalpy change is negative and entropy change is positive. Let's plug these values into the Gibbs free energy change formula: \(\Delta G = -50.6 - T(0.3315)\) For the reaction to be spontaneous, \(\Delta G < 0\). Therefore: \(-50.6 - T(0.3315) < 0\) In this case, no matter the value of temperature, the \(\Delta G\) will always be negative, making the reaction spontaneous at all temperatures.

Reaction (c) - Analyzing Gibbs free energy change

For the reaction (c), we have the following values of enthalpy and entropy change: \(\Delta H^{\circ} = +98.9 \mathrm{~kJ}\) and \(\Delta S^{\circ} = +0.0939 \mathrm{~kJ} / \mathrm{K}\) Both enthalpy and entropy change are positive. Let's plug these values into the Gibbs free energy change formula and calculate the critical temperature at which \(\Delta G\) becomes negative: \(\Delta G = 98.9 - T(0.0939)\) For the reaction to be spontaneous, \(\Delta G < 0\). Therefore: \(98.9 - T(0.0939) < 0\) Now, find the critical temperature: \(T > \frac{98.9}{0.0939} \approx 1053 \mathrm{K}\) So, for reaction (c) to be spontaneous, the temperature needs to be greater than 1053 K. In conclusion, the temperature conditions for each reaction to be spontaneous are: - Reaction (a): spontaneous at T > 22117 K - Reaction (b): spontaneous at all temperatures - Reaction (c): spontaneous at T > 1053 K

Key concepts

Spontaneity of Reactions

When we talk about the spontaneity of reactions, we are referring to whether or not a reaction occurs without outside intervention. One of the main tools we use to evaluate this is the Gibbs Free Energy, denoted as \( \Delta G \). Spontaneous reactions tend to occur naturally and result in a decrease in free energy, meaning \( \Delta G \) is less than zero.
To calculate \( \Delta G \), we use the equation: \[ \Delta G = \Delta H - T \Delta S \] where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy. A negative \( \Delta G \) value indicates a spontaneous process. So, the sign and magnitude of \( \Delta G \) help predict whether a process will occur naturally. In general, if the enthalpy change \( \Delta H \) is negative and the entropy change \( \Delta S \) is positive, the reaction tends to be spontaneous at any temperature, as we have seen in reaction (b). However, reactions like (a) and (c), where both \( \Delta H \) and \( \Delta S \) are positive, are dependent on temperature for spontaneity. As temperature increases, the entropy's role becomes more significant because of the \( T \Delta S \) term. Thus, reactions can shift from non-spontaneous to spontaneous depending on the temperature.

Effect of Temperature on Reactions

The temperature of a system plays a crucial role in determining the spontaneity of a reaction. Temperature affects the \( T \Delta S \) component of the Gibbs Free Energy equation. As temperature increases, the entropy change \( \Delta S \) can have a larger effect.
This influence can define whether a reaction becomes spontaneous or not. For instance, in reaction (c), increasing the temperature can make the reaction spontaneous because the positive \( \Delta S \) becomes significant enough with a higher temperature, lowering the \( \Delta G \) to a negative value.

• For reactions where \( \Delta H > 0 \) and \( \Delta S > 0 \), like (a) and (c), higher temperatures can favor spontaneity, as shown by the calculated critical temperatures for each reaction.
• In contrast, when \( \Delta H < 0 \) and \( \Delta S < 0 \), higher temperatures can make reactions less spontaneous.

This highlights that temperature can be a determining factor for spontaneity and must be considered when analyzing reactions.

Enthalpy and Entropy Changes

Enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)) are two fundamental thermodynamic quantities that play key roles in the Gibbs Free Energy equation. Each of them provides insight into different aspects of reactions.

Enthalpy (\( \Delta H \))

Enthalpy change refers to the heat absorbed or released in a reaction at constant pressure.

• A negative \( \Delta H \) indicates an exothermic reaction where heat is released, often contributing to reaction spontaneity at low temperatures.
• A positive \( \Delta H \) signifies an endothermic reaction where heat is absorbed, which may require increased temperature for the reaction to become spontaneous, as seen in reactions (a) and (c).

Entropy (\( \Delta S \))

Entropy change reflects the disorder or randomness of a system.

• Positive \( \Delta S \) indicates an increase in disorder, which is often beneficial for spontaneity, especially at high temperatures, as is the case in reactions (a) and (c).
• Negative \( \Delta S \) shows a decrease in disorder, which can hinder spontaneity if not compensated by a large \( \Delta H \).

Understanding both \( \Delta H \) and \( \Delta S \) is essential as they collectively dictate the direction and spontaneity of chemical reactions through their impact on \( \Delta G \). This is why they are critical tools for chemists and students when analyzing reactivity and stability of a system.