Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

The emf of a Daniell cell at \(298 \mathrm{~K}\) is \(E_{i}\) \(\mathrm{Zn}\left|\mathrm{ZnSO}_{4} \| \mathrm{CuSO}_{4}\right| \mathrm{Cu}\) \((0.01 \mathrm{M}) \quad(1.0 \mathrm{M})\) when the concentration of \(\mathrm{ZnSO}_{4}\) is \(1.0 \mathrm{M}\) and that of \(\mathrm{CuSO}_{4}\) is \(0.01 \mathrm{M}\), the emf changed to \(E_{2}\) What is the relationship between \(E_{1}\) and \(E_{2} ?\) (a) \(E_{1}=E_{2}\) (b) \(E_{2} \neq E_{i}\) (c) \(E_{1}>E_{2}\) (d) \(E_{1}

Step-by-Step Solution

Answer

(c) \( E_1 > E_2 \)

Step 1: Define the Daniell Cell Reaction

The cell notation for the Daniell cell is \( \mathrm{Zn} | \mathrm{Zn}^{2+} \| \mathrm{Cu}^{2+} | \mathrm{Cu} \), which corresponds to the reaction \( \mathrm{Zn(s)} + \mathrm{Cu}^{2+}(aq) \rightarrow \mathrm{Zn}^{2+}(aq) + \mathrm{Cu(s)} \). In this cell, zinc is oxidized and copper is reduced.

Step 2: Use Nernst Equation to Understand EMF Dependence

The Nernst equation relates the cell EMF to concentrations: \[ E = E^\circ - \frac{RT}{nF} \ln\left( \frac{[Zn^{2+}]}{[Cu^{2+}]} \right) \] where \( E^\circ \) is the standard EMF, \( R \) is the gas constant, \( T \) is temperature, \( n \) is the number of electrons transferred, and \( F \) is Faraday's constant.

Step 3: Original Concentrations and EMF, \(E_i\)

For \(E_i\), [ZnSO4] = 0.01 M and [CuSO4] = 1.0 M. Hence, \[ E_i = E^\circ - \frac{0.059}{2} \log\left( \frac{0.01}{1} \right) \] This simplifies to: \[ E_i = E^\circ + \frac{0.059}{2} \cdot 2 \] \[ E_i = E^\circ + 0.059 \]

Step 4: Changed Concentrations and EMF, \(E_2\)

For \(E_2\), [ZnSO4] = 1.0 M and [CuSO4] = 0.01 M. Thus, \[ E_2 = E^\circ - \frac{0.059}{2} \log\left( \frac{1}{0.01} \right) \] This simplifies to: \[ E_2 = E^\circ - \frac{0.059}{2} \cdot 2 \] \[ E_2 = E^\circ - 0.059 \]

Step 5: Compare \(E_1\) and \(E_2\)

Based on the calculations: - \( E_1 = E^\circ + 0.059 \) - \( E_2 = E^\circ - 0.059 \) Since \( E_1 > E_2 \), meaning \(E_1\) is greater than \(E_2\).

Key Concepts

Nernst Equation

The Nernst Equation is a fundamental tool in electrochemistry, illuminating the relationship between the cell potential (EMF) and the concentrations of the chemical species involved. This equation is particularly helpful in predicting how changes in concentration can affect the EMF of an electrochemical cell. It is given by:\[ E = E^\circ - \frac{RT}{nF} \ln\left( \frac{[Products]}{[Reactants]} \right) \]Here,

• \( E \) is the electrode potential under non-standard conditions.
• \( E^\circ \) represents the standard cell potential.
• \( R \) is the universal gas constant \( (8.314 \, \text{J}\, \text{mol}^{-1}\,\text{K}^{-1}) \).
• \( T \) is the temperature in Kelvin.
• \( n \) is the number of moles of electrons transferred in the reaction.
• \( F \) is Faraday’s constant (approximately \( 96485 \, \text{C}\,\text{mol}^{-1} \)).

This equation accounts for the deviation of the cell potential from the standard state due to concentration changes. It shows that as the concentration of reactants or products changes, the EMF alters accordingly, impacting the cell's efficiency and behavior.

Cell EMF

Electromotive force, commonly abbreviated as EMF, is the driving force behind the flow of electric current in a cell. It represents the maximum potential difference between the cathode and anode of the cell under open circuit conditions. In simpler terms, it is the voltage developed by a voltaic cell when no current is drawn from it, a measure of its power to drive charge.

• For standard conditions (where reactant and product concentrations are 1 M at \( 298 \, \text{K} \)), the EMF is referred to as the standard cell potential \( E^\circ \).
• The standard EMF is determined by the inherent properties of the substances involved in the reaction.
• In a Daniell cell, the EMF can be calculated using the specific half-reaction potentials of the zinc and copper electrodes.

The magnitude of EMF provides insights into the cell's ability to perform electrical work. A high EMF means more work can be performed per unit charge transported, indicating the cell's efficiency.

Electrochemistry

Electrochemistry is a branch of chemistry that explores the relationship between electricity and chemical change. It plays a crucial role in many technologies such as batteries, fuel cells, and corrosion prevention.

Key Concepts in Electrochemistry:

• Oxidation and Reduction: Fundamental to electrochemical processes where oxidation involves the loss of electrons, and reduction involves the gain of electrons.
• Redox Reactions: Combined oxidation-reduction reactions. In a Daniell cell, zinc undergoes oxidation, while copper undergoes reduction.
• Electrodes: Solid conductors that facilitate the transfer of electrons. The anode is where oxidation occurs, and the cathode is where reduction takes place.
• Electrolytes: Conductive solutions or substances that allow ion flow, necessary for completing the electrochemical circuit.

Electrochemistry's practical applications range from energy storage systems to industrial electroplating methods, emphasizing its pivotal role in modern science and engineering.

Concentration Effect on EMF

The concentration of ions in an electrochemical cell can significantly impact its EMF. According to the Nernst equation, even slight variations in concentration can shift the cell's potential. The cell potential is directly influenced by how electrochemical reactions respond to changes in concentration.

How Concentration Affects EMF:

• When ion concentrations are high in the electrolyte, the reaction is closer to its standard conditions, typically yielding a potential close to \( E^\circ \).
• Conversely, a decreased concentration usually lowers EMF since the reaction tends to shift backwards, generating less potential difference.
• In the Daniell cell example, changing the concentrations of \( \text{CuSO}_4 \) and \( \text{ZnSO}_4 \) alters the EMF.
• This is marked by a higher EMF when copper ion concentration is high and lower when it is reduced, as seen in the step-by-step solution.

Understanding this concentration effect is crucial for designing more efficient electrochemical cells and predicting their behavior under various conditions.