Problem 47 - A Complete Resource Book in Chemistry for JEE Main

Question

Which one of the following condition will increase the voltage of the cell represented by the equation \(\mathrm{Cu}(\mathrm{s})+2 \mathrm{Ag}^{+}(\mathrm{aq}) \rightleftharpoons \mathrm{Cu}^{2+}(\mathrm{aq})+2 \mathrm{Ag}(\mathrm{s}) ?\) (a) increase in the dimension of Cu electrode (b) increase in the dimension of Ag electrode (c) increase in the concentration of \(\mathrm{Cu}^{2+}\) ions (d) increase in the concentration of \(\mathrm{Ag}^{+}\)ions

Step-by-Step Solution

Answer

Increasing the concentration of \([\mathrm{Ag^+}]\) ions (option d) will increase the voltage of the cell.

Step 1: Understand the Nernst Equation

To determine which condition will increase the voltage of the cell, we need to use the Nernst Equation:\[E = E^0 - \frac{RT}{nF} \ln Q\]where:- \(E\) is the cell potential under non-standard conditions.- \(E^0\) 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 Faraday's constant.- \(Q\) is the reaction quotient.

Step 2: Identify Reaction Participants

The cell reaction is:\[\mathrm{Cu(s)} + 2 \mathrm{Ag^+(aq)} \rightleftharpoons \mathrm{Cu^{2+}(aq)} + 2 \mathrm{Ag(s)}\]Thus, participants are \([\mathrm{Cu^{2+}}]\) and \([\mathrm{Ag^+}]\) ions which affect the reaction quotient \(Q\). The solid metals do not contribute to \(Q\).

Step 3: Analyze Reaction Quotient

The reaction quotient \(Q\) is:\[Q = \frac{[\mathrm{Cu^{2+}}]}{[\mathrm{Ag^+}]^2}\]The effect of \([\mathrm{Cu^{2+}}]\) and \([\mathrm{Ag^+}]\) concentrations on \(E\) can be observed by understanding that increasing \(Q\) will decrease \(E\), and decreasing \(Q\) will increase \(E\) based on the Nernst equation.

Step 4: Evaluate Each Condition

(a) Increase in the dimension of Cu electrode does not affect \(Q\) as it is a solid.(b) Increase in the dimension of Ag electrode does not affect \(Q\) as it is a solid.(c) Increase in the concentration of \(\mathrm{Cu^{2+}}\) ions will increase \(Q\), thus decreasing \(E\).(d) Increase in the concentration of \(\mathrm{Ag^+}\) ions will decrease \(Q\), thus increasing \(E\).

Key Concepts

Cell Potential

The cell potential, often referred to as electromotive force (EMF), is a measure of the voltage difference between two half-cells in an electrochemical cell. This potential indicates how much work the cell can perform. It is dependent on the types of electrodes and the concentrations of the reactants and products involved. With the Nernst Equation, we can predict how the cell potential will change under non-standard conditions. The potential changes because the activity or concentration of the species in the cell alters the balance of the cell's chemical reactions. In standard conditions, the cell potential is denoted by the symbol \(E^0\). However, real-world applications require us to calculate the cell potential using variables that account for non-standard conditions, symbolized as \(E\).

Reaction Quotient

The reaction quotient, represented by \(Q\), is crucial in determining the direction and extent of a reaction at any given point. It is a ratio of the concentrations of products to reactants, each raised to the power of their respective stoichiometric coefficients. Specifically for the reaction \(\mathrm{Cu} + 2 \mathrm{Ag^+} \rightleftharpoons \mathrm{Cu^{2+}} + 2 \mathrm{Ag}\), the reaction quotient is defined as:

• \(Q = \frac{[\mathrm{Cu^{2+}}]}{[\mathrm{Ag^+}]^2} \)

Changes in \(Q\) directly impact the cell potential when substituted into the Nernst Equation. A higher \(Q\) suggests more products, aligning with a decrease in the cell's potential, while a lower \(Q\) indicates more reactants, resulting in an increased cell potential.

Electrochemistry

Electrochemistry focuses on the reactions that involve the transfer of electrons between species, fundamentally linking the realms of chemistry and electricity. In an electrochemical cell, spontaneous redox reactions produce electrical energy. In galvanic or voltaic cells, such as the one in our problem, chemical energy is converted into electrical energy through redox reactions. Understanding the principles of electrochemistry is essential for predicting how changes in concentration and other conditions affect the voltage of the cell using the Nernst Equation. It's also critical for solving practical problems related to batteries, corrosion, and electroplating.

Concentration Effect

The concentration effect refers to how changes in concentration of reactants or products influence the voltage of an electrochemical cell. Specifically, according to the Nernst Equation, increasing the concentration of the ions that are products in a reaction generally increases the reaction quotient \(Q\), which decreases the cell potential \(E\). Conversely, increasing the concentration of reactants decreases \(Q\), thereby increasing the cell potential. In our reaction:

• Increasing \([\mathrm{Cu^{2+}]}\) will lead to higher \(Q\), thus lowering \(E\).
• Increasing \([\mathrm{Ag^+}]\) will reduce \(Q\), leading to higher \(E\).

This illustrates why adjustments in concentration are critical in managing and optimizing cell performance for practical applications.