If the enthalpy and entropy changes are not affected by the change in temperature calculate the temperature at which the system will attain equilibrium.

1. Introduction to Gibbs Free Energy

Gibbs free energy (G) is a thermodynamic potential that helps predict the direction of chemical processes and reactions. The change in Gibbs free energy (\(\Delta G\)) indicates the spontaneity of a process: if \(\Delta G\) is negative, the process is spontaneous; if positive, it is non-spontaneous. The equation governing Gibbs free energy is:

\[
\Delta G = \Delta H – T\Delta S
\]

Here, \(\Delta H\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S\) is the change in entropy. This formula is crucial in determining the energy changes under constant pressure and temperature conditions.

2. Understanding Enthalpy (\(\Delta H\)) and Entropy (\(\Delta S\))

Enthalpy (\(\Delta H\)) measures the total energy of a thermodynamic system, incorporating both the internal energy and the energy due to pressure and volume, expressed in joules per mole (J/mol). A positive \(\Delta H\) signifies heat absorption by the system (endothermic process), while a negative \(\Delta H\) signifies heat release (exothermic process).

Entropy (\(\Delta S\)), on the other hand, is a measure of the system’s disorder or randomness, expressed in joules per mole per Kelvin (J/K/mol). An increase in entropy (\(\Delta S > 0\)) suggests a transition to more disorder, whereas a decrease (\(\Delta S < 0\)) indicates a transition to less disorder.

3. Spontaneity and Temperature Dependence

The spontaneity of a process is determined by \(\Delta G\). At constant temperature and pressure, if \(\Delta G < 0\), the process is spontaneous. The influence of \(\Delta G\) is dependent on \(\Delta H\), \(\Delta S\), and the temperature \(T\). Temperature significantly affects spontaneity by altering the contribution of entropy to the Gibbs free energy.

If \(\Delta H\) and \(\Delta S\) are constant and unaffected by temperature changes, the equation \(\Delta G = \Delta H – T\Delta S\) can be directly used to predict how temperature influences the spontaneity of the process.

4. Calculation of Equilibrium Temperature

To find the temperature at which the system is at equilibrium (\(\Delta G = 0\)), we can rearrange the Gibbs free energy equation:

\[
0 = \Delta H – T_{eq}\Delta S
\]

Solving for \(T_{eq}\) (equilibrium temperature):

\[
T_{eq} = \frac{\Delta H}{\Delta S}
\]

Given that \(\Delta H = 52 \text{ kJ mol}^{-1}\) and \(\Delta S = 165 \text{ JK}^{-1} \text{ mol}^{-1}\), we convert \(\Delta H\) to joules:

\[
\Delta H = 52000 \text{ J mol}^{-1}
\]

Substituting these values into the equation for \(T_{eq}\):

\[
T_{eq} = \frac{52000 \text{ J mol}^{-1}}{165 \text{ JK}^{-1} \text{ mol}^{-1}}
\]

\[
T_{eq} = 315.15 \text{ K}
\]

This calculation implies that at a temperature of 315.15 K, the system reaches a state of equilibrium where the process is neither spontaneous nor non-spontaneous.

Conclusion

The analysis of Gibbs free energy provides critical insights into the temperature dependence of chemical reactions and processes. By evaluating changes in enthalpy and entropy, one can determine not only the spontaneity of a process at a given temperature but also predict the temperature at which the system will achieve equilibrium. For the given changes in enthalpy and entropy, the calculated equilibrium temperature is 315.15 K. At this temperature, the changes in enthalpy and entropy balance each other out, resulting in zero change in Gibbs free energy, indicating a state of equilibrium. This understanding is crucial in chemical thermodynamics for designing processes that require precise control over temperature to achieve desired outcomes.