Question

Which graph correctly correlates \(\dot{E}_{\text {cell }}\) as' a function of concentrations for the cell $$ \mathrm{Zn}(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+2 \mathrm{Ag}(s), \quad E_{\text {cell }}^{\circ}=1.56 \mathrm{~V} $$

Short answer

The correct graph will show a nonlinear, downward-sloping curve depicting \(E_{\text{cell}}\) as it decreases with increasing \(\text{Zn}^{2+}\) or decreasing \(\text{Ag}^{+}\) concentration.

Step-by-step solution

Understanding the Nernst Equation

In order to relate the cell potential \(E_{\text{cell}}\) to the concentrations of reactants and products, we use the Nernst Equation, which is given by:\[ E_{\text{cell}} = E_{\text{cell}}^\circ - \frac{0.0592}{n} \log Q \]where \(E_{\text{cell}}^\circ\) is the standard cell potential, \(n\) is the number of moles of electrons transferred in the reaction, and \(Q\) is the reaction quotient.

Determining Standard Cell Potential and Number of Electrons Transferred

For the given reaction, the standard cell potential \(E_{\text{cell}}^\circ\) is already provided as 1.56 V. Two moles of electrons are transferred, as indicated by the stoichiometry of the reaction. So, \(n = 2\).

Writing the Reaction Quotient (Q)

The reaction quotient \(Q\) is calculated using the concentrations of the products raised to their stoichiometric coefficients divided by the reactants raised to their stoichiometric coefficients, ignoring solids as they do not affect the quotient. For this reaction:\[ Q = \frac{[\text{Zn}^{2+}]^{1}}{[\text{Ag}^{+}]^{2}} \]

Graphing the Cell Potential

Because the relationship described by the Nernst equation is logarithmic, a graph of cell potential \(E_{\text{cell}}\) as a function of concentration will be nonlinear. \(E_{\text{cell}}\) will decrease as the concentration of \(\text{Zn}^{2+}\) increases or as the concentration of \(\text{Ag}^{+}\) decreases. The graph will be a downward curve, reflecting the negative sign in the Nernst equation.

Key concepts

Electrochemical Cell Potential

Understanding electrochemical cell potential is integral to grasping the workings of batteries and electrochemical reactions. Simply put, it is the voltage or electrical potential difference between the two electrodes of an electrochemical cell. It's what drives the electrical current through an external circuit.

When a metal like zinc reacts with a silver ion solution, as in our exercise, electrons are transferred from zinc to the silver ions, which creates an electrical potential. This potential depends not only on the types of materials involved but also on their concentrations. As reactions proceed and concentrations change, so does the cell potential. Knowing how to predict this change is key in fields such as material science and energy storage technology.

In real-world applications, the potential of an electrochemical cell can determine the amount of power a battery can provide, the efficiency of fuel cells, or even the rate of corrosion in metals. It's a cornerstone concept that links chemistry to the generation and use of electrical energy.

Reaction Quotient

Let's talk about the reaction quotient (Q), which is a snapshot of a reaction at any point in time. It considers the concentrations of the products and reactants at a specific moment, not just when the system is at equilibrium. The concept is crucial for predicting the direction of chemical reactions.

In the exercise, the reaction quotient for the reaction of zinc and silver ions is written as a ratio of the product concentration, in this case the zinc ions, to the power of their stoichiometric coefficient, and the reactant concentration, the silver ions, each raised to the power of their stoichiometric coefficient as well.

Understanding Q is useful beyond just academic exercises. It has implications in industrial chemical processes where the yield of a reaction must be maximized, or in environmental sciences, where it can help predict the movement of pollutants through different mediums.

Standard Cell Potential

The standard cell potential, denoted as \(E_{\text{cell}}^\circ\), is another critical factor in the Nernst Equation. It represents the potential difference between two half-cells under standard conditions, which include solutes at 1 M concentration, gases at 1 atm pressure, and a temperature of 298 K (25°C).

For the given zinc and silver ion reaction, the standard cell potential is a positive 1.56 V. This indicates a spontaneous reaction under standard conditions. The higher the standard cell potential, the greater the ability of the cell to do electrical work. It's pivotal in predicting whether a reaction can occur spontaneously and is used to calculate the maximum possible electrical energy a cell can generate.

In industries like metal plating or the manufacturing of batteries, knowing the standard cell potential is vital for designing processes and products. It helps engineers and scientists choose the right materials and conditions for optimal performance.

Logarithmic Relationships in Chemistry

Finally, let's delve into the logarithmic relationships in chemistry, especially as they relate to the Nernst Equation. The logarithmic term in the Nernst Equation (\(-\frac{0.0592}{n} \log Q\)) illustrates how the cell potential changes with the reaction quotient. Because the log function is involved, the relationship between the cell potential and concentration of reactants and products is nonlinear.

Logarithms help in understanding concepts like pH in acid-base chemistry, reaction kinetics, and equilibrium. They convert multiplicative relationships into additive ones, making it easier to work with the large range of values that chemical concentrations can have. It's mainly because of these logarithmic relationships that small changes in reactant or product concentration can result in significant shifts in cell potential.

Furthermore, the use of logarithms simplifies complex calculations, which is especially helpful in higher levels of chemistry where reactions can become very intricate. Comprehending log relationships is thus essential for students and professionals working with chemical processes, as it affects everything from the speed of a reaction to the energy it can produce.