The nature of curve of \(E_{\text {cell }}^{\circ}\) vs. \(\log K_{c}\) is : (a) straight line (b) parabola (c) hyperbola (d) elliptical curve

When delving into electrochemistry, one of the fundamental concepts to grasp is the standard cell potential, denoted by \(E_{\text{cell}}^{\circ}\). It's a measure of how much voltage, or 'driving force', a cell can produce under standard conditions, which typically means at a temperature of 298 K (25 °C), a pressure of 1 atmosphere, and all solutes at a concentration of 1 M.

In essence, the standard cell potential tells us the potential difference between the two electrodes of a galvanic cell when no current is flowing. It's a critical factor determining whether a redox reaction will occur spontaneously. If \(E_{\text{cell}}^{\circ}\) is positive, it predicts a spontaneous reaction under standard conditions, while a negative value suggests the reaction is non-spontaneous.

To calculate \(E_{\text{cell}}^{\circ}\), we use standard reduction potentials for the half-reactions taking place at the anode and cathode, which are readily available in electrochemical tables. The standard cell potential thus provides a quick way to predict the feasibility of reactions and helps to rank substances according to their oxidizing and reducing strengths.

Another key topic is the equilibrium constant, represented as \(K_c\). It is a dimensionless value that gives us an idea about the position of equilibrium for a particular reaction. The equilibrium constant is determined by the concentrations of the reactants and products at equilibrium.

The relationship between the equilibrium constant and standard cell potential is deeply rooted in thermodynamics. As discussed in the step by step solution, using the Nernst equation we can deduce that\[E_{\text{cell}}^{\circ} = \frac{RT}{nF} \ln K_c\]where \(R\) is the gas constant, \(T\) is temperature in kelvin, \(n\) is the number of moles of electrons transferred, and \(F\) is the Faraday's constant. This formula implies that for a given temperature and number of electrons involved in the reaction, the standard cell potential is directly related to the natural logarithm of the equilibrium constant.

This interconnectedness is crucial for students to understand because it shows how the thermodynamics of a cell is reflective of its equilibrium properties. It also forms the basis for the prediction of the spontaneous direction of a reaction and the extent to which a reaction will proceed before reaching equilibrium.

In the study of electrochemistry, the linear relationship is a simple yet powerful mathematical concept that characterizes the direct proportionality between two quantities. In the context of our exercise, when plotting the standard cell potential \(E_{\text{cell}}^{\circ}\) against the logarithm of the equilibrium constant \(\log K_{c}\), we obtain a straight line.

This linearity is derived from the modified Nernst equation where it is shown that:\[E_{\text{cell}}^{\circ} = \frac{RT}{nF \ln{10}} \log K_c\]The slope of the straight line in this context is proportional to \(\frac{RT}{nF \ln{10}}\) while the intercept on the \(E_{\text{cell}}^{\circ}\)-axis is 0 when \(\log K_c\) is 1 (since \(\log 1 = 0\)). Understanding this linear relationship is critical because it simplifies complex electrochemical concepts into straightforward, predictable patterns.

Therefore, not only does this relationship allow for the easy calculation of the standard cell potential for various reactions, but it also visually reinforces the concept that as the equilibrium constant increases, indicating a reaction more likely to proceed to completion, the standard cell potential also increases.