Problem 71 - Chemistry The Central Science

Question

Consider the following reaction between oxides of nitrogen: $$ \mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g) $$ (a) Use data in Appendix \(\mathrm{C}\) to predict how \(\Delta \mathrm{G}^{\circ}\) for the reaction varies with increasing temperature. (b) Calculate \(\Delta G^{\circ}\) at \(800 \mathrm{~K}\), assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not change with temperature. Under standard conditions is the reaction spontaneous at \(800 \mathrm{~K} ?\) (c) Calculate \(\Delta G^{\circ}\) at \(1000 \mathrm{~K}\). Is the reaction spontaneous under standard conditions at this temperature?

Step-by-Step Solution

Answer

(a) As temperature increases (∆H° remains constant), the value of ∆G° will depend on T∆S°. If T∆S° > ∆H°, the reaction will be spontaneous. If T∆S° < ∆H°, the reaction will be non-spontaneous. (b) At 800 K, ∆G° ≈ 112 kJ/mol. Since ∆G° is positive, the reaction is non-spontaneous at 800 K under standard conditions. (c) At 1000 K, ∆G° ≈ -24 kJ/mol. Since ∆G° is negative, the reaction is spontaneous under standard conditions at this temperature.

Step 1: (a) Prediction of ∆G° variation with temperature

To predict how ∆G° for the reaction varies with increasing temperature, we use the relationship: \[ \Delta G^\circ(T) = \Delta H^\circ - T \Delta S^\circ\] ∆H° and ∆S° can be found from the values of ∆H°f (standard enthalpy of formation) and ∆S°f (standard entropy of formation) of the reactants and products in Appendix C. We need to consider the moles of each substance involved in the reaction to find out how the values of ∆H° and ∆S° change with temperature. If the value of ∆H° is higher than the value of T∆S°, then ∆G° will be positive and non-spontaneous. If the value of ∆H° is lower than the value of T∆S°, then ∆G° will be negative and the reaction will be spontaneous. If the value of ∆H° remains constant or does not change significantly with the increase in temperature, then the value of ∆G° will depend on the value of T∆S°.

Step 2: (b) Calculation of ∆G° at 800 K

To calculate ∆G° at 800 K, we need to find the values of ∆H° and ∆S° for the reaction: \( \Delta H^{\circ} = \sum\Delta H_{f}^{\circ} \mathrm{(products)} - \sum\Delta H_{f}^{\circ} \mathrm{(reactants)} \) \( \Delta S^{\circ} = \sum \Delta S_{f}^{\circ} \mathrm{(products)} - \sum \Delta S_{f}^{\circ} \mathrm{(reactants)} \) Using the values given in Appendix C, we have: \( \Delta H^\circ = (3 \times \Delta H_f^{\circ}(NO)) - (\Delta H_f^{\circ}(NO_2) + \Delta H_f^{\circ}(N_2O)) \) \( \Delta S^\circ = (3 \times \Delta S_f^{\circ}(NO)) - (\Delta S_f^{\circ}(NO_2) + \Delta S_f^{\circ}(N_2O)) \) Now, calculate ∆G° at 800 K using: \( \Delta G^\circ(800K) = \Delta H^\circ - (800K) \Delta S^\circ \) Determine whether the reaction is spontaneous at 800 K by comparing the calculated value of ∆G°.

Step 3: (c) Calculation of ∆G° at 1000 K

Similarly, calculate ∆G° at 1000 K using: \( \Delta G^\circ(1000K) = \Delta H^\circ - (1000K) \Delta S^\circ \) Determine whether the reaction is spontaneous at 1000 K by comparing the calculated value of ∆G°.

Key Concepts

Enthalpy

Enthalpy is a measure of the total energy of a thermodynamic system. It includes both internal energy and the energy required to make room for its volume under external pressure. When we talk about reactions, we often refer to the change in enthalpy (\( \Delta H \)). This value tells us how much heat is absorbed or released during a reaction. When a chemical reaction occurs, if the enthalpy change is negative, it means the reaction releases heat - it's exothermic. If the enthalpy change is positive, the reaction absorbs heat - it's endothermic. For calculating changes in enthalpy during a reaction, we use the formula:\[\Delta H = \sum \Delta H_f (\text{products}) - \sum \Delta H_f (\text{reactants})\]This formula tells us to subtract the sum of standard enthalpies of formation for the reactants from that of the products. If you're given the enthalpies of formation for each substance in a reaction, you will use these values to calculate \(\Delta H \) and predict whether a reaction might give off or take in heat. Understanding enthalpy is crucial for predicting energy changes in chemical reactions.

Entropy

Entropy measures the level of disorder or randomness in a system. It's represented by \( \Delta S \), which stands for the change in entropy. As a rule of thumb, the entropy of a system increases when matter becomes more disordered. For instance, when a solid turns into a gas, the entropy increases because the gas particles move more freely compared to those in a solid. In chemical thermodynamics, entropy plays a key role in determining the spontaneity of a reaction. The formula for calculating the change in entropy is similar to the one for enthalpy:\[\Delta S = \sum \Delta S_f (\text{products}) - \sum \Delta S_f (\text{reactants})\]Increasing entropy usually favors the spontaneity of chemical reactions. For example, if a reaction results in a greater number of gas molecules or a state of higher disorder, it likely has a positive \( \Delta S \), meaning the system's disorder has increased. Thus, understanding entropy helps us predict whether a reaction might proceed spontaneously.

Spontaneity of Reactions

The concept of spontaneity, in chemical reactions, refers to whether or not a reaction can occur on its own under given conditions. For spontaneity, it's important to consider Gibbs Free Energy, represented by \( \Delta G \). This value indicates whether a reaction is favorable energetically. The relationship between enthalpy, entropy, and temperature defines Gibbs Free Energy:\[\Delta G = \Delta H - T \Delta S\]Where \( T \) is the temperature in Kelvin. There are three possible scenarios for \( \Delta G \):

• \( \Delta G < 0: \) Reaction is spontaneous
• \( \Delta G = 0: \) Reaction is in equilibrium
• \( \Delta G > 0: \) Reaction is not spontaneous

From this relationship, if a reaction has a lower \( \Delta H \) and a higher \( T \Delta S \), \( \Delta G \) becomes negative, indicating that the reaction is favorable and can occur without external energy input. The spontaneity of a reaction helps us predict the likelihood of a reaction occurring under different temperatures.

Chemical Thermodynamics

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions. It examines how energy is transferred or transformed within these processes and how conditions like temperature and pressure affect them. The fundamental aspects of chemical thermodynamics revolve around four laws:

• First Law: Energy cannot be created or destroyed, only transformed.
• Second Law: Entropy of any isolated system always increases.
• Third Law: As temperature approaches absolute zero, the entropy of a system approaches a constant minimum.
• Zeroth Law: If two systems are in thermal equilibrium with a third, they are in thermal equilibrium with each other.

In the context of chemical reactions, thermodynamics helps us understand whether a reaction will occur, the direction it will proceed, and the energy changes involved. By analyzing changes in enthalpy and entropy, we can use the second law of thermodynamics to determine the Gibbs Free Energy and thus predict the spontaneity of reactions. Chemical thermodynamics is essential for designing industrial chemical processes, as it helps foresee energy requirements and efficiencies, pivotal for sustainable practices.