Question

Consider a Daniell cell operating under non-standardstate conditions. Suppose that the cell's reaction is multiplied by 2 . What effect does this have on each of the following quantities in the Nernst equation: (a) \(E\) (b) \(E^{\circ},(\mathrm{c}) Q\) (d) \(\ln Q\), (e) \(n\) ?

Short answer

Only \(n\) is affected, it becomes twice, while \(E\), \(Q\), and \(\ln Q\) remain the same; \(E^{\circ}\) is unaffected.

Step-by-step solution

Understanding Nernst Equation

The Nernst equation relates the cell potential, concentration of ions, and temperature in a Daniell cell. It is given by \(E = E^{\circ} - \frac{RT}{nF} \ln Q\), where \(E\) is the cell potential, \(E^{\circ}\) is the standard cell potential, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient.

Effect of Reaction Multiplication on E

The cell potential \(E\) is affected by the reaction quotient \(Q\) and \(n\). Multiplying the reaction by 2 changes \(Q\), but \(E^{\circ}\) and \(n\) must be determined first to find the effect on \(E\).

Effect on Standard Potential \(E^{\circ}\)

Multiplying the reaction coefficients by 2 does not change the standard potential \(E^{\circ}\) since it in the standard state; therefore, \(E^{\circ}\) remains unchanged.

Effect on Reaction Quotient \(Q\)

When the entire reaction is multiplied by 2, the reaction quotient \(Q\), which is the ratio of product concentrations to reactant concentrations, raised to the power of their stoichiometric coefficients, is affected. \(Q\) itself is calculated with concentrations that have not changed in magnitude due to simple reaction scaling and remains the same numerically.

Effect on \(\ln Q\)

Since \(Q\) does not numerically change by multiplying the reaction, \(\ln Q\) also remains unchanged. Although the exponent values in \(Q\) in the Nernst calculation would double, this is counteracted by the doubling in \(n\).

Effect on Number of Electrons \(n\)

Multiplying the reaction by 2 doubles the stoichiometric coefficients, which includes the number of moles of electrons \(n\). Thus, \(n\) is also doubled.

Key concepts

Cell Potential

The cell potential, denoted as \(E\), represents the voltage or electrical potential difference between the two electrodes in an electrochemical cell such as a Daniell cell. It is a measurable value and gives insight into the cell's ability to perform work by driving an electric current.
As described by the Nernst equation, the cell potential under non-standard conditions can vary according to the concentration of reactants and products. It is calculated as:

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

Here, \(R\) represents the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient. The cell potential \(E\) provides a dynamic picture of the cell's operation beyond the simple standard conditions, taking into account the real-world chemical activities occurring within the cell.

Standard Cell Potential

The standard cell potential, \(E^{\circ}\), is a key reference value in electrochemistry which indicates the voltage of a cell under standard state conditions: 1 M concentration for solutions, a pressure of 1 atm for gases, and pure solids or liquids. This potential is derived from the difference in the standard reduction potentials of the two half-cells making up the electrochemical cell.

• It remains constant, regardless of the scaling of the chemical reaction or the stoichiometric coefficients, as it reflects the inherent properties of the reactants under standard conditions.

An unaltered standard cell potential means that even if the reaction in a Daniell cell is multiplied by any factor, \(E^{\circ}\) does not change. This stability makes \(E^{\circ}\) vital for comparing different electrochemical cells and predicting the feasibility and direction of reactions under standard conditions.

Reaction Quotient

The reaction quotient \(Q\) is a vital concept for understanding the progress of a reaction in a chemical cell. It is calculated similarly to the equilibrium constant \(K\) but uses the initial concentrations or partial pressures of the reactants and products.

• In a Daniell cell, \(Q\) is determined by the expression formed from the ratio of the activities (or concentrations) of the products raised to their coefficients, divided by the activities of the reactants raised to theirs.

When the overall stoichiometry of the reaction doubles, theoretically \(Q\) remains unchanged because it depends solely on the mass action expression given actual concentrations or pressures are unchanged. Thus, multiplying the reaction by a number does not directly affect the value of \(Q\) calculated with real concentrations.

Daniell Cell

The Daniell cell is a classic type of galvanic cell involving a spontaneous redox reaction between zinc and copper. It consists of two different metal electrodes submerged in electrolyte solutions and connected by a salt bridge that facilitates ionic flow.

• In the Daniell cell, zinc acts as the anode, oxidizing by losing electrons, and copper acts as the cathode, reducing by gaining those electrons.
• This setup generates an electric current proportional to the standard and actual cell potentials.

Understanding the operation of a Daniell cell is crucial for interpreting the effects of any changes in reaction conditions on parameters like cell potential and electron flow, as discussed in relation to the Nernst equation, particularly under non-standard conditions.

Number of Moles of Electrons

The number of moles of electrons \(n\) is a crucial component in the electrochemical cell's Nernst equation, as it relates to the full charge transfer between the electrodes during the electrochemical reactions. In an equation such as:

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

\(n\) directly influences the magnitude of the term \(\frac{RT}{nF}\), which is subtracted from \(E^{\circ}\). When the coefficients of a reaction double, \(n\) also doubles, since more electrons are involved in the redox processes. This doubling balances changes in \(Q\) caused by scaling the reaction, keeping \(E\) stable unless concentration conditions alter. Understanding \(n\) is critical for calculating the precise potential and operation of real-world cells.