Question

Discuss the effect of temperature change on the spontaneity of the following reactions at 1 atm: $$ \text { (a) } \begin{aligned} \mathrm{Al}_{2} \mathrm{O}_{3}(s)+2 \mathrm{Fe}(s) \longrightarrow 2 \mathrm{Al}(s)+\mathrm{Fe}_{2} \mathrm{O}_{3}(s) \\ \Delta H^{\circ}=851.4 \mathrm{~kJ} ; & \Delta S^{\circ}=38.5 \mathrm{~J} / \mathrm{K} \end{aligned} $$ (b) \(\mathrm{N}_{2} \mathrm{H}_{4}(l) \longrightarrow \mathrm{N}_{2}+2 \mathrm{H}_{2}(g)\) $$ \Delta H^{\circ}=-50.6 \mathrm{~kJ} ; \quad \Delta S^{\circ}=0.3315 \mathrm{~kJ} / \mathrm{K} $$ (c) \(\mathrm{SO}_{3}(g) \longrightarrow \mathrm{SO}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)\) $$ \Delta H^{\circ}=98.9 \mathrm{~kJ} ; \quad \Delta S^{\circ}=0.0939 \mathrm{~kJ} / \mathrm{K} $$

Short answer

Answer: (a) Increasing the temperature increases the spontaneity of the reaction. (b) Increasing the temperature decreases the spontaneity of the reaction. (c) At temperatures lower than 1052 K, the reaction is non-spontaneous, and at higher temperatures, it is spontaneous.

Step-by-step solution

Calculating ΔG for each reaction at various temperatures

ΔG is calculated using the formula: $$ \Delta G = \Delta H - T\Delta S $$ We will do this calculation for each reaction with different temperatures - below are examples of the reactions at 298 K: (a) $$\Delta G = 851.4\,\text{kJ} - (298\,\text{K})(0.0385\,\text{kJ/K}) = 738.67\,\text{kJ}$$ (b) $$\Delta G = -50.6\,\text{kJ} - (298\,\text{K})(-0.3315\,\text{kJ/K}) = -100.372\,\text{kJ}$$ (c) $$\Delta G = 98.9\,\text{kJ} - (298\,\text{K})(-0.0939\,\text{kJ/K}) = 115.71\,\text{kJ}$$

Analyzing the effect of temperature change

To analyze the effect of temperature change, we will consider the relationships between ΔG, ΔH, and ΔS for each reaction: (a) ΔH > 0 and ΔS > 0: As the temperature increases, TΔS increases and ΔG decreases. Therefore, increasing the temperature increases the spontaneity of the reaction. (b) ΔH < 0 and ΔS < 0: As the temperature increases, TΔS decreases and ΔG increases. Therefore, increasing the temperature decreases the spontaneity of the reaction. (c) ΔH > 0 and ΔS < 0: Since both ΔH and ΔS have the same signs, this system will have a critical temperature (\(T_c\)) where the reaction goes from non-spontaneous to spontaneous, or vice versa. We can find this temperature by letting ΔG = 0 and using the formula $$ T_c = \frac{\Delta H}{\Delta S} $$ For the reaction (c), we get: $$T_c = \frac{98.9\,\text{kJ}}{0.0939\,\text{kJ/K}} = 1052\,\text{K}$$ So, at temperatures lower than 1052 K, reaction (c) will be non-spontaneous, and at temperatures higher than 1052 K, it will be spontaneous. In summary: (a) Increasing the temperature increases the spontaneity of the reaction. (b) Increasing the temperature decreases the spontaneity of the reaction. (c) At temperatures lower than 1052 K, the reaction is non-spontaneous, and at higher temperatures, it is spontaneous.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat and temperature and their relation to energy and work. It tells us how thermal energy is converted to other forms of energy and how it affects matter. The laws of thermodynamics govern the principles of energy conversion, enabling us to understand the workings of engines, refrigerators, and even cells in our bodies.

Within this context, chemical reactions often involve changes in thermal energy, and understanding these changes is crucial in predicting whether a reaction will occur spontaneously. A spontaneous reaction is one that occurs naturally under a given set of conditions without needing to be driven by an external energy source. However, a reaction's spontaneity doesn't necessarily relate to the speed of the reaction—it tells us only about the feasibility of the reaction occurring on its own.

Gibbs Free Energy

Gibbs free energy, denoted as \( G \), is a thermodynamic quantity that's extremely useful in predicting the direction of a chemical reaction. The change in Gibbs free energy \( \Delta G \) during a process indicates the amount of energy available to do work. When \( \Delta G \) is negative, the process is spontaneous, and when it's positive, the process is non-spontaneous.

The relationship between \( \Delta G \) and the enthalpy \( \Delta H \) and entropy \( \Delta S \) changes of a reaction at constant temperature and pressure is given by the formula \( \Delta G = \Delta H - T\Delta S \). This equation is pivotal because it connects the heat exchange of a system (\( \Delta H \) represents the total heat content) with the temperature (\( T \) in Kelvin) and the system's disorder (\( \Delta S \) for entropy change). By analyzing these variables, one can determine the spontaneity of a chemical reaction under given conditions.

Enthalpy

Enthalpy, represented by \( H \), is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure. The change in enthalpy, \( \Delta H \) during a reaction, reflects the heat absorbed or released at constant pressure.

A positive \( \Delta H \) indicates that a reaction absorbs heat from its surroundings (endothermic), while a negative \( \Delta H \) means that a reaction releases heat to its surroundings (exothermic). In a textbook exercise, understanding if the reaction is exothermic or endothermic helps us predict how changes in temperature can influence \( \Delta G \) and, thus, the spontaneity of the reaction.

Entropy

Entropy is denoted by \( S \) and is a measure of disorder or randomness in a system. Thermodynamically, it's a key factor because, at a microscopic level, nature tends towards disorder. The second law of thermodynamics states that the entropy of an isolated system never decreases; it can either increase or remain constant.

In terms of chemical reactions, an increase in entropy (\( \Delta S > 0 \) ) signifies that the products are more disordered than the reactants, often associated with the formation of gases from solids or liquids. Conversely, a decrease in entropy (\( \Delta S < 0 \) ) implies a transition towards a more ordered system. The temperature at which a reaction switches from non-spontaneous to spontaneous can be identified as the critical temperature (\( T_c \) ), calculated by setting \( \Delta G = 0 \) and determining the temperature via the equation \( T_c = \frac{\Delta H}{\Delta S} \).