A voltaic cell has one half-cell with a Cu bar in a \(1.00 M\) \(\mathrm{Cu}^{2+}\) salt, and the other half-cell with a Cd bar in the same volume of a \(1.00 \mathrm{M} \mathrm{Cd}^{2+}\) salt. (a) Find \(E_{\mathrm{cell}}^{\circ}, \Delta G^{\circ},\) and \(K\) (b) As the cell operates, \(\left[\mathrm{Cd}^{2+}\right]\) increases; find \(E_{\text {cell }}\) and \(\Delta G\) when \(\left[\mathrm{Cd}^{2+}\right]\) is \(1.95 M\) (c) Find \(E_{\text {cell }}, \Delta G,\) and \(\left[\mathrm{Cu}^{2+}\right]\) at equilibrium.

Standard electrode potential is a measure of the individual potential of a reversible electrode at standard state (1.00 M concentration, 1 atm pressure, and 298 K temperature). It represents the tendency of a chemical species to be reduced, measured in volts (V) relative to the standard hydrogen electrode, which is assigned a potential of 0.00 V.

For instance, in the given exercise, the standard electrode potential for Copper \((Cu^{2+}/Cu)\) is +0.34 V, and for Cadmium \((Cd^{2+}/Cd)\) is -0.40 V. These values indicate that Copper is more readily reduced compared to Cadmium, which helps in determining the direction of electron flow in a voltaic cell.

The Nernst equation provides a way to calculate the cell potential under non-standard conditions by accounting for the ion concentrations in the solution. It’s given by:

\[ E_{\text{cell}} = E_{\text{cell}}^\text{0} - \frac{0.0592}{n} \text{log} \frac{[\text{products}]}{[\text{reactants}]} \]

In this equation, \( E_{\text{cell}}^\text{0} \) is the standard cell potential, \( n \) is the number of moles of electrons transferred, and the term inside the log function represents the reaction quotient. For the given problem, when the concentration of \(\text{Cd}^{2+}\) changes from 1.00 M to 1.95 M, the Nernst equation helps us to calculate the new cell potential, ensuring we account for the variation in ion concentration.

Gibbs Free Energy Change (\(\triangle G\)) is crucial in predicting the spontaneity of a reaction. A negative \(\triangle G\) indicates a spontaneous process, while a positive one signals non-spontaneity. The relationship between Gibbs free energy and cell potential is given by:

\[ \triangle G = -nFE_{\text{cell}} \]

Here, \(\triangle G^\text{0}\) represents the change in standard conditions, \(\triangle G\) denotes non-standard conditions, \(\text{n}\) is the number of electrons transferred in the reaction, and \(\text{F}\) is the Faraday constant (96485 C/mol). For our voltaic cell, we use this equation to calculate \(\triangle G^\text{0}\) as -142.16 kJ/mol and \(\triangle G\) for varied concentrations of \(\text{Cd}^{2+}\).

The equilibrium constant (K) quantifies the ratio of product concentrations to reactant concentrations at equilibrium, providing insight into the reaction's favorability. The relationship between Gibbs free energy change and equilibrium constant is expressed as:

\[ \triangle G^\text{0} = -RT \text{ln}(K) \]

Rearranging gives us \[ K = e^{-\triangle G^\text{0}/RT} \] where \(\text{R}\) is the gas constant (8.314 J/(mol·K)) and \(\text{T}\) is the temperature in Kelvin (298 K for standard conditions). From the given problem's data, we calculate K to be approximately \[7.17 \times 10^{24}\] indicating that, at equilibrium, product formation is highly favored.

The cell potential (\(\text{E}_{\text{cell}}\)) is the measure of the voltage difference between two electrodes of a voltaic cell. It predicts the direction of electron flow and is determined by the standard electrode potentials of the cathode and anode. The cell potential under standard conditions (\(\text{E}_{\text{cell}}^\text{0}\)) is calculated using:

\[ E_{\text{cell}}^\text{0} = E^\text{0}_{\text{cathode}} - E^\text{0}_{\text{anode}} \]

In our voltaic cell, Copper (Cu) is the cathode and Cadmium (Cd) is the anode, resulting in a standard cell potential of 0.74 V. When the concentrations change, like \([\text{Cd}^{2+}] = 1.95 M\), the Nernst equation helps update \(\text{E}_{\text{cell}}\), adjusting it to 0.723 V, reflecting the shift from standard to non-standard conditions.