Problem 14 - General Chemistry: Principles and Modern Applications

Question

Consider the reaction \(\operatorname{Co}(\mathrm{s})+\mathrm{Ni}^{2+}(\mathrm{aq}) \longrightarrow\) \(\mathrm{Co}^{2+}(\mathrm{aq})+\mathrm{Ni}(\mathrm{s}), \quad\) with \(\quad E_{\mathrm{cell}}^{\circ}=0.02 \mathrm{V} . \quad\) If \(\quad \mathrm{Co}(\mathrm{s}) \quad\) is added to a solution with \(\left[\mathrm{Ni}^{2+}\right]=1 \mathrm{M},\) should the reaction go to completion? Explain.

Step-by-Step Solution

Answer

Yes. Since the calculated cell potential is positive, the reaction is spontaneous and will go to completion.

Step 1: Determining the Nernst Equation

The Nernst equation is used to calculate cell potentials under non-standard conditions. It is given by \(E = E° - (RT/nF) * lnQ\) where E is the cell potential, E° is the standard cell potential, R is the gas constant (8.314 J/mol.K), T is the temperature in Kelvin (assuming it to be 298K as it is not given), n is the number of electron transfers in the reaction (here it is 2 as observed from the balanced reaction), F is Faraday's constant (96485 C/mol) and Q is the reaction quotient.

Step 2: Determination of the Reaction Quotient (Q)

Q is the ratio of the concentrations of the products divided by the concentrations of the reactants, each raised to the power of their stoichiometric coefficients. In this case, as the reactant Co(solid) is in solid form and the product Ni(solid) is also in solid form their concentrations are omitted from the reaction quotient. Also, Co²⁺ is produced in the reaction. However, since its concentration is not given, we assume Q to be 1 by default.

Step 3: Substituting values into the Nernst Equation

Substituting the values into the Nernst equation, we obtain \( E = 0.02 - (8.314*298/2*96485)*ln(1)\).

Step 4: Calculation and interpretation of results

Since ln(1) = 0, the equation simplifies to \( E=0.02V \). The cell potential is positive, indicating that the reaction is spontaneous and would go to completion.

Key Concepts

Nernst Equation

The Nernst equation is a fundamental formula used in electrochemistry to determine the cell potential of an electrochemical cell under non-standard conditions. This equation is essential because it allows us to predict how changes in concentration can affect the cell potential. The equation is expressed as:

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

Here, \(E\) is the cell potential that we are trying to calculate. \(E^\circ\) is the standard cell potential, which is a constant value at standard conditions of 1M concentration, pressure at 1atm, and temperature at 298K. \(R\) is the gas constant with a value of 8.314 J/mol.K. \(T\) represents the temperature in Kelvin. \(n\) is the number of moles of electrons transferred in the reaction, and for our example, \( n = 2 \).
\(F\) is Faraday’s constant (96485 C/mol), which connects electric charge to moles of electrons. Lastly, \(Q\) is the reaction quotient, determining the ratio of the product concentrations to reactants. Thus, Nernst Equation helps to calculate when cell potential veers from standard assumptions.

Cell Potential

The cell potential, also known as electromotive force (emf), is a measure of the voltage or electrical potential difference of an electrochemical cell. It is the driving force behind the electrochemical reaction.The standard cell potential, \(E^\circ\), is measured under standard conditions and denotes the inherent potential difference when all reactants and products are at 1M concentration and 298K.
However, in real-world scenarios, conditions often vary, requiring use of the Nernst equation to calculate actual cell potential. Cell potential can either be positive or negative, which gives insights into whether a reaction is spontaneous or non-spontaneous:

• Positive cell potential: Suggests a spontaneous reaction.
• Negative cell potential: Indicates a non-spontaneous reaction.

In our example, after using the Nernst equation, the cell potential under applied conditions remains the same as the standard potential, \(E = 0.02V\), confirming a spontaneous reaction.

Reaction Quotient

The reaction quotient, denoted by \(Q\), is a dimensionless quantity that reflects the relative concentrations of products and reactants at any point in a reaction. It is calculated similarly to the equilibrium constant, but without assuming equilibrium conditions.For a general reaction \(aA + bB \rightarrow cC + dD\), the reaction quotient is:

• \( Q = \frac{[C]^c[D]^d}{[A]^a[B]^b} \)

In electrochemical reactions like the one in our exercise, pure solids and liquids are excluded from \(Q\). Consequently, only aqueous concentrations are used. Here, since Co and Ni are solids, they do not contribute to \(Q\), simplifying it to be 1, assuming equilibrium-like condition before the reaction majorly proceeds.
Understanding how \(Q\) works is crucial since it aids in using the Nernst equation to determine if the reaction naturally proceeds forward when actual conditions deviate from standard.

Spontaneous Reaction

A spontaneous reaction is one that proceeds on its own under given conditions, without being driven by external energy sources. In electrochemistry, spontaneity is often determined by the sign of the cell potential, \(E\).When the cell potential is positive, the reaction is favorable and spontaneous under the given conditions. This happens because a positive \(E\) implies that the cell can produce an electric current when connected with a conductor. In our exercise, since the calculated cell potential is \(0.02V\), which is positive, we predict that the reaction naturally proceeds to convert the reactants into products without needing external intervention.Understanding spontaneity aids in assessing whether adding reagents like Co to a Ni system will spontaneously shift to form more products, just as seen when the reaction quotient and cell potential are interpreted accurately using Nernst equation.