The entropy change of a cell reaction can be determined from the change of the cell potential with temperature. (a) Show that \(\Delta S^{\circ}=n F\left(E_{\text {cell }, 2^{\circ}}-E_{\text {cell }, 1^{\circ}}\right) /\left\langle T_{2}-T_{1}\right) .\) Assume that \(\Delta S^{\circ}\) and \(\Delta H^{\circ}\) are constant over the temperature range considered. (b) Calculate \(\Delta S^{\circ}\) and \(\Delta H^{\circ}\) for the cell reaction \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}(\mathrm{~s})+\) \(\mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{Hg}(\mathrm{l})+2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{Cl}^{-}(\mathrm{aq})\), given that \(E^{\circ}=+0.2699 \mathrm{~V}\) at \(293 \mathrm{~K}\) and \(+0.2669 \mathrm{~V}\) at \(303 \mathrm{~K}\).

The Nernst Equation is a fundamental relation in electrochemistry, allowing us to calculate the cell potential at any given temperature, for a specific chemical reaction, once we know the standard cell potential and the concentrations of the reactants and products. This equation is expressed as:

\( E = E^\circ - \frac{RT}{nF} \ln Q \)

where \( E \) is the cell potential, \( E^\circ \) is the standard cell potential, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons transferred, \( F \) is Faraday's constant, and \( Q \) is the reaction quotient which gives us the ratio of the concentrations of products over reactants. In the context of an entropy change calculation, the Nernst Equation helps us understand the relationship between the entropy of a system and how it influences the cell potential.

The standard cell potential, denoted as \( E^\circ \), is a measure of the electromotive force of an electrochemical cell under standard conditions: a temperature of 298 K, a 1 M concentration for each aqueous species, and a 1 atm pressure for any gases involved. It is a vital component in the Nernst Equation and plays a crucial role in determining the spontaneity and equilibrium of a given reaction.\( E^\circ \) can be positive or negative, with a positive \( E^\circ \) indicating that the reaction can proceed spontaneously under standard conditions.

Understanding the standard cell potential is key not just in calculating current and potential at various states, but also in evaluating the free energy changes associated with the reactions in electrochemical cells.

Gibbs free energy, denoted by \( \Delta G \), is arguably the most important thermodynamic function for understanding chemical reactions. It indicates the maximum amount of work that can be obtained from a process at constant temperature and pressure. A negative \( \Delta G \) means the reaction is spontaneous, while a positive value indicates non-spontaneity. The relationship between Gibbs free energy and cell potential under standard conditions is given by the equation:

\( \Delta G^\circ = -nFE^\circ \)

Here, \( \Delta G^\circ \) is the change in Gibbs free energy under standard conditions, \( n \) is the number of moles of electrons transferred during the reaction, \( F \) is Faraday’s constant, and \( E^\circ \) is the standard cell potential. The connection between free energy and electrochemical cell potential is a cornerstone of understanding how energy is transformed in batteries and other electrochemical systems.

Thermochemistry in electrochemical cells involves the study of heat energy and its interplay with chemical reactions within the cells. Two primary thermodynamic quantities of interest are the entropy change (\( \Delta S \)) and the enthalpy change (\( \Delta H \)). Entropy is a measure of the disorder or randomness in a system, and its change during a reaction can affect the voltage output of the cell. As illustrated in the original exercise, the entropy change can be related to the change in cell potential with temperature.

The enthalpy change, on the other hand, represents the heat absorbed or released. When combined with the entropy change, it forms the basis for the Gibbs free energy equation:\( \Delta G = \Delta H - T\Delta S \), where \( T \) is temperature. By calculating \( \Delta G \) through the cell potential, one can deduce \( \Delta H \), offering insights into the thermal characteristics of electrochemical reactions and processes.