Problem 34 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Zinc granules are added in excess to a \(500 \mathrm{ml}\) of \(1.0 \mathrm{M}\) nickel nitrate solution at \(25^{\circ} \mathrm{C}\) until the equilibrium is reached. If the standard reduction potential of \(\mathrm{Zn}^{2+} \mid \mathrm{Zn}\) and \(\mathrm{Ni}^{2+} \mid \mathrm{Ni}\) are \(-0.75 \mathrm{~V}\) and \(-0.24 \mathrm{~V}\), respectively, the concentration of \(\mathrm{Ni}^{2+}\) in solution at equilibrium is \((2.303 R T / F=0.06)\) (a) \(1.0 \times 10^{-17} \mathrm{M}\) (b) \(1.0 \times 10^{17} \mathrm{M}\) (c) \(5 \times 10^{-17} \mathrm{M}\) (d) \(2 \times 10^{-17} \mathrm{M}\)

Step-by-Step Solution

Answer

The concentration of Ni^2+ at equilibrium is of the order 10^{-17} M. Option (a) is the closest to the calculated result, hence correct.

Step 1: Understand the Redox Reaction

Zinc granules react with nickel nitrate in a redox reaction where zinc is oxidized and nickel ions are reduced. The corresponding half-reactions are: Zn -> Zn^2+ + 2e^- for zinc oxidation and Ni^2+ + 2e^- -> Ni for nickel reduction.

Step 2: Use the Nernst Equation

To find the equilibrium concentration of Ni^2+, we can use the Nernst equation which relates standard reduction potential, temperature, the number of electrons involved in the reaction to the concentration of reactants and products.

Step 3: Calculate the Change in Reduction Potential

Subtract the standard reduction potential of Zn^2+/Zn from that of Ni^2+/Ni to get the potential difference: E_cell = E_Ni/Ni^2+ - E_Zn/Zn^2+. This difference drives the reaction towards equilibrium.

Step 4: Apply the Given Formula

The given formula 2.303RT/F = 0.06 relates to the Nernst equation at 25 degrees C, which can be written as E = E^0 - (0.06/n) log(Q) where Q is the reaction quotient.

Step 5: Set Up the Nernst Equation for Equilibrium

At equilibrium, the cell potential E is zero. So, we can set E^0 - (0.06/2) log(Q) = 0 to solve for Q, where E^0 is the potential difference calculated in step 3 and Q = [Ni^2+]/[Zn^2+]. Since Zn is added in excess, we can assume [Zn^2+] to be constant and [Ni^2+] to be unknown.

Step 6: Calculate the Concentration of Ni^2+

Solve the equilibrium equation for [Ni^2+] using logs and exponentiation. Remember to address the equilibrium constant as log(Q) = nE^0/(0.06), where n is the number of moles of electrons transferred in the reaction, which is 2.

Step 7: Find the Correct Answer

Calculate the equilibrium concentration of Ni^2+ and choose the option that matches.

Key Concepts

Nernst Equation

Understanding the Nernst equation is essential when studying electrochemistry or dealing with chemical reactions that include electron transfers, as seen in redox reactions. The Nernst equation provides a quantitative relationship that we can use to calculate the electromotive force (EMF) of a cell under non-standard conditions. It incorporates the effect of ion concentration on the cell's potential.

The general form of the Nernst equation is: \[ E = E^{\text{0}} - \frac{RT}{nF} \ln{Q} \]
where:

• \(E\) is the cell potential under non-standard conditions,
• \(E^{\text{0}}\) is the standard cell potential,
• \(R\) is the universal gas constant,
• \(T\) is the temperature in kelvin,
• \(n\) is the number of moles of electrons transferred in the reaction,
• \(F\) is the Faraday's constant, and
• \(Q\) is the reaction quotient, which represents the ratio of product activities to reactant activities.

In practical applications, when the temperature is 25 degrees Celsius, the equation simplifies as the term \((RT/F)\) becomes a constant, commonly represented as 0.0592 V at 298 K, or sometimes approximated as 0.06 V for easier calculations as in the given exercise.

By using this equation, one can predict the direction in which a redox reaction will proceed as well as calculate equilibrium concentrations of the involved species.

Redox Reaction

A redox reaction, short for reduction-oxidation reaction, is a type of chemical reaction involving a transfer of electrons between two substances. It's composed of two interconnected parts: oxidation, where an element loses electrons, and reduction, where an element gains electrons.

For example, in the reaction between zinc granules and nickel nitrate, the zinc is oxidized:\[ \text{Zn} \rightarrow \text{Zn}^{2+} + 2e^- \]
Meanwhile, the nickel ions are reduced:\[ \text{Ni}^{2+} + 2e^- \rightarrow \text{Ni} \]
Redox reactions are fundamental to various processes, including biochemical functions like metabolism and industrial processes like electroplating. When these reactions occur, they often create a flow of electric current that can be harnessed for work, thus connecting redox reactions to electrochemical cells.

In a redox reaction, there's a change in the oxidation state of the involved elements. By analyzing oxidation states, scientists can determine what is oxidized and what is reduced—a critical step in predicting the course of the reaction and balancing redox equations.

Standard Reduction Potential

Standard reduction potential, often denoted as \(E^{\text{0}}\), is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. Each half-reaction in a redox process has its own standard reduction potential, which can be found in tables of standard electrode potentials.

These values are determined relative to a standard hydrogen electrode (SHE), which is assigned a potential of 0.00 V. The more positive the standard reduction potential, the greater the substance's affinity for electrons, and hence its ability to be reduced. Conversely, a negative standard reduction potential indicates a greater tendency to lose electrons and be oxidized.

In the exercise, the standard reduction potentials for zinc and nickel are given as -0.75 V and -0.24 V, respectively. These values imply that zinc more readily loses electrons compared to nickel, hence zinc serves as the anode (oxidation site) and nickel as the cathode (reduction site). By comparing the standard reduction potentials, we can predict the direction of electron flow in the redox reaction and determine the spontaneity of the process. When calculating the cell potential using these standard potentials, it's crucial to use them correctly; indeed, a common error is to misunderstand their sign or how they relate to the overall cell potential.