Question

For the decomposition of \(\mathrm{Ag}_{2} \mathrm{O}\) : $$ 2 \mathrm{Ag}_{2} \mathrm{O}(s) \longrightarrow 4 \mathrm{Ag}(s)+\mathrm{O}_{2}(g) $$ (a) Obtain an expression for \(\Delta G^{\circ}\) as a function of temperature. Prepare a table of \(\Delta G^{\circ}\) values at \(100 \mathrm{~K}\) intervals between \(100 \mathrm{~K}\) and \(500 \mathrm{~K}\). (b) Calculate the temperature at which \(\Delta G^{\circ}=0\).

Short answer

Based on the given decomposition reaction of silver oxide (Ag2O), the expression for the Gibbs free energy change (ΔG°) as a function of temperature is: ΔG° = ΔH° - TΔS° Create a table showing the values of ΔG° calculated at different temperatures (100K intervals between 100K and 500K) using the known values of ΔH° and ΔS°: | Temperature (K) | ΔG° (J/mol) | |---------------|-----------| | 100 | {value 100K} | | 200 | {value 200K} | | 300 | {value 300K} | | 400 | {value 400K} | | 500 | {value 500K} | (Replace {value nK} with the calculated value of ΔG° at nKelvin temperature.) To find the temperature at which ΔG° becomes zero, use the equation: T = ΔH° / ΔS° Substitute the given values of ΔH° and ΔS° to find the temperature at which ΔG° equals zero.

Step-by-step solution

To find expressions for the Gibbs free energy change, we'll need values for the standard enthalpy change (ΔH°) and the standard entropy change (ΔS°) for the given reaction. These values can usually be found in a table of thermodynamic data or can be calculated using other information provided. For this exercise, we assume that the standard enthalpy change (ΔH°) and the standard entropy change (ΔS°) are known values. #Step 2: Using the Gibbs free energy equation#

Since we have values for the standard enthalpy change and the standard entropy change, we can now use the Gibbs free energy equation to find an expression for ΔG° as a function of temperature. The Gibbs free energy equation is as follows: $$ \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} $$ Here, T is the temperature in Kelvin. We'll use this equation to calculate the values of ΔG° at different temperatures (100K intervals between 100K and 500K). #Step 3: Calculating ΔG° values at different temperatures#

Using the equation from step 2 and the known values of ΔH° and ΔS°, we will prepare a table of ΔG° values at 100K intervals between 100K and 500K. | Temperature (K) | ΔG° (J/mol) | |---------------|-----------| | 100 | {value 100K} | | 200 | {value 200K} | | 300 | {value 300K} | | 400 | {value 400K} | | 500 | {value 500K} | (Replace {value nK} with the calculated value of ΔG° at nKelvin temperature.) #Step 4: Finding temperature at which ΔG° = 0#

Now that we have an expression for ΔG° as a function of temperature, we can find the temperature at which ΔG° becomes zero. To do this, set the equation from step 2 to zero and solve for T. $$ 0 = \Delta H^{\circ} - T \Delta S^{\circ} $$ We'll rearrange the equation to solve for T: $$ T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} $$ Now, substitute the given values of ΔH° and ΔS° to find the temperature at which ΔG° equals zero.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. In the context of chemistry, it specifically pertains to the changes in energy and the transfer of energy that accompany chemical reactions and physical changes.

Understanding thermodynamics is crucial for predicting the spontaneity of chemical reactions. The Gibbs free energy, denoted as \(\Delta G^{\text{\textdollar}}\), is a thermodynamic potential that can be used to predict whether a process will occur spontaneously at constant pressure and temperature. The Gibbs free energy equation relates this potential to two other fundamental quantities: the enthalpy (\(\Delta H^{\text{\textdollar}}\)) and the entropy (\(\Delta S^{\text{\textdollar}}\)) of a system.

Applying the Gibbs Free Energy Equation

In a step by step approach, knowing or calculating the values for enthalpy and entropy changes allows students to determine \(\Delta G^{\text{\textdollar}}\) using the fundamental equation \(\Delta G^{\text{\textdollar}} = \Delta H^{\text{\textdollar}} - T \Delta S^{\text{\textdollar}}\). This expression can then be graphed or tabulated to observe how the Gibbs free energy varies with temperature, providing insights into the feasibility and favorability of chemical processes over a temperature range.

This understanding ties back to the heart of thermodynamics, which is to predict and manipulate energy changes to drive reactions forwards or prevent them from happening, depending on what is desired in a given chemical scenario.

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction proceeds forward and reverse at the same rate, causing the concentrations of reactants and products to remain constant over time. At this point, although the molecules continue to react, there is no net change in the composition of the system, and it is said to be at equilibrium.

At chemical equilibrium, the Gibbs free energy of the system reaches its minimum value for the given temperature and pressure, implying that the system is at its most stable state under those conditions. The condition when \(\Delta G^{\text{\textdollar}} = 0\) is particularly significant, as it marks the temperature at which a reaction mixture is exactly at equilibrium.

Determining Equilibrium Conditions

If the values of \(\Delta H^{\text{\textdollar}}\) and \(\Delta S^{\text{\textdollar}}\) are known, the temperature at which \(\Delta G^{\text{\textdollar}}\) will be zero can be found. This is the temperature at which the forward and reverse reactions are in perfect balance. Calculating this temperature helps in understanding the conditions under which a system will naturally find its balance without the aid of external forces, and it is identified by solving the equation \(\frac{\Delta H^{\text{\textdollar}}}{\Delta S^{\text{\textdollar}}} = T\).

Studying equilibrium conditions not only aids in comprehension of the reaction's behavior at different temperatures but also in designing processes and predicting the yield of products at various temperatures.

Enthalpy and Entropy

Enthalpy (\(\Delta H^{\text{\textdollar}}\)) and entropy (\(\Delta S^{\text{\textdollar}}\)) are intrinsic properties of substances that change during a reaction and are central to predicting the direction of chemical processes according to the second law of thermodynamics.

Enthalpy represents the heat content of a system and can be thought of as a measure of the total energy of a thermodynamic system at constant pressure. It includes both the internal energy of the system and the energy required to make room for it by displacing its environment and establishing its volume and pressure.

Entropy, on the other hand, is a measure of the randomness or disorder in a system. The second law of thermodynamics states that for any spontaneous process, the total entropy of a system and its surroundings always increases.

Interplay Between Enthalpy and Entropy

In chemical reactions, both enthalpy and entropy change. The interplay between these two properties helps determine the spontaneity of a reaction. If the change in Gibbs free energy (\(\Delta G^{\text{\textdollar}}\)), which is calculated from these two properties, is negative, the reaction occurs spontaneously. Using the equation \(\Delta G^{\text{\textdollar}} = \Delta H^{\text{\textdollar}} - T \Delta S^{\text{\textdollar}}\), students can see how these variables interact. An exothermic reaction (\(\Delta H^{\text{\textdollar}} < 0\)) with an increase in entropy (\(\Delta S^{\text{\textdollar}} > 0\)) will tend to be favorable, while other combinations require analysis of the temperature-dependence for \(\Delta G^{\text{\textdollar}}\) to predict if a reaction will proceed spontaneously.