Question

What must be the temperature if the following reaction has \(\Delta G^{\circ}=-45.5 \mathrm{kJ}, \Delta H^{\circ}=-24.8 \mathrm{kJ},\) and \(\Delta S^{\circ}=15.2 \mathrm{JK}^{-1} ?\) $$\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})+3 \mathrm{CO}(\mathrm{g}) \longrightarrow 2 \mathrm{Fe}(\mathrm{s})+3 \mathrm{CO}_{2}(\mathrm{g})$$

Short answer

The temperature at which this reaction would occur is approximately 1365 K.

Step-by-step solution

Assign given variables

Assign the given variables to the correct symbols. Here, we have \(\Delta G^{\circ} = -45500 \, J\), \(\Delta H^{\circ} = -24800 \, J\), and \(\Delta S^{\circ} = 15.2 \, J/K\). Note that we've converted the energy units from kJ to J for consistency.

Apply Gibbs-Helmholtz equation

The Gibbs-Helmholtz equation is \( \Delta G = \Delta H - T \Delta S \). We can rearrange this to solve for temperature, getting \( T = (\Delta H - \Delta G) / \Delta S \).

Substitute values and calculate temperature

Substitute the given values into the rearranged Gibbs-Helmholtz equation to find the temperature. So, \( T = ((-24800 \, J) - (-45500 \, J)) / 15.2 \, J/K \), which simplifies down to about 1365 K.

Key concepts

Gibbs-Helmholtz Equation

The Gibbs-Helmholtz Equation is a fundamental formula that relates the change in Gibbs Free Energy of a system (\(\triangle G\)) with its enthalpy change (\(\Delta H\)) and entropy change (\(\Delta S\)) at constant temperature (\(T\)). This equation is expressed as \(\Delta G = \Delta H - T\Delta S\).
The concept is crucial in determining whether a process will occur spontaneously under constant temperature and pressure. A negative \(\Delta G\) indicates a spontaneous process, while a positive \(\Delta G\) suggests that the process is non-spontaneous and will require input of energy to proceed.
In the context of the textbook exercise, the Gibbs-Helmholtz Equation helps us find the temperature at which the chemical reaction is in equilibrium—meaning that \(\Delta G\) is zero. By rearranging the equation, we can solve for \(T\) as follows: \(T = (\Delta H - \Delta G) / \Delta S\). It's important to keep units consistent, hence in the step-by-step solution, all energy units were converted from kJ to J to ensure accurate temperature calculation. This equation is vital in thermochemistry and offers students a valuable tool for understanding energy changes during reactions.

Thermodynamic Temperature Calculation

Thermodynamic temperature calculation is a process used to quantify the absolute temperature of a system. The absolute temperature, measured in kelvin (K), is a key component in numerous thermodynamic equations, including the Gibbs-Helmholtz Equation discussed earlier.
To understand the temperature at which a reaction is at equilibrium, we utilize the rearranged Gibbs-Helmholtz Equation, as shown in the solution steps. Temperature directly influences the spontaneity of chemical processes; higher temperatures can increase the entropy (\(\Delta S\)), potentially making an otherwise non-spontaneous reaction spontaneous due to the \(T\Delta S\) term becoming larger than the enthalpy change (\(\Delta H\)).
In the given exercise, the calculation shows that when the variables from the reaction are plugged into the equation, the thermodynamic temperature of equilibrium is approximately 1365 K. This type of calculation is practical when predicting reaction behavior under different temperature conditions and is critical for scientists and engineers designing processes in various fields.

Chemical Reaction Equilibrium

Chemical reaction equilibrium refers to the state in which the rate of the forward reaction equals the rate of the reverse reaction. At equilibrium, the concentrations of reactants and products remain constant over time. The position of the equilibrium can be influenced by a variety of factors, including temperature, which we've explored through thermodynamic temperature calculation.
In equilibrium considerations, Gibbs Free Energy is a valuable indicator. A \(\Delta G = 0\) corresponds to a system at equilibrium. This state of equilibrium can be understood through Le Chatelier's principle, which tells us that a system at equilibrium will adjust concentrations, pressure, or temperature to counteract any changes imposed upon it.
Therefore, knowing the temperature at which a chemical reaction reaches equilibrium, as determined by the Gibbs-Helmholtz Equation, gives us the ability to predict how the system will behave under different conditions and to control the reaction to favor the formation of either reactants or products. Such insights are essential in industrial chemistry where yield optimization is necessary for cost-effective production.