Calculate The Electrode Potential (E.P) Of A Zn Electrode Dipped In A 0.1 M Solution Of Its Salt At 25°C, Assuming The Electrolyte Is 20% Dissociated. Given E^0_{Zn^{2+}/Zn} = -0.76 V.

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Calculating the electrode potential of a metal in a solution of its ions is a fundamental concept in electrochemistry. This article delves into the calculation of the electrode potential (E.P) for a zinc (Zn) electrode immersed in a 0.1 M solution of its salt at 25°C, considering the electrolyte's dissociation. We will utilize the Nernst equation and discuss the importance of factors like the degree of dissociation in determining the electrode potential. Understanding these concepts is crucial for various applications, including battery design, corrosion studies, and electrochemical sensors. This comprehensive guide aims to provide a clear and detailed explanation of the process, making it accessible to students and professionals alike.

Introduction to Electrode Potential

In electrochemistry, electrode potential is a crucial concept that describes the tendency of a metal electrode to lose or gain electrons when it is in contact with a solution of its own ions. This tendency is what drives electrochemical reactions, which are the basis for batteries, fuel cells, and corrosion processes. The electrode potential is influenced by several factors, including the nature of the metal, the concentration of its ions in the solution, and the temperature. When a metal electrode is immersed in a solution containing its ions, an equilibrium is established between the metal atoms on the electrode surface and the metal ions in the solution. This equilibrium involves the transfer of electrons between the metal and the solution, resulting in the formation of an electrical double layer at the electrode-solution interface. The potential difference across this double layer is the electrode potential. The standard electrode potential (E⁰) is a reference value measured under standard conditions (298 K, 1 atm pressure, and 1 M concentration). However, in real-world scenarios, the concentrations of ions often deviate from 1 M, and the temperature may not be 298 K. To calculate the electrode potential under non-standard conditions, we use the Nernst equation, which takes into account the effects of concentration and temperature on the electrode potential. This equation is essential for understanding and predicting the behavior of electrochemical systems in various applications.

Problem Statement: Calculating the Electrode Potential of a Zinc Electrode

Let's consider a specific scenario: a zinc (Zn) electrode is dipped in a 0.1 M solution of its salt at 25°C. The electrolyte is 20% dissociated, and we are given the standard electrode potential (E⁰) for the Zn²⁺/Zn couple as -0.76 V. Our objective is to calculate the electrode potential (E.P) for this setup. This problem highlights the practical application of electrochemical principles and the importance of considering factors such as dissociation in determining the electrode potential. The dissociation of the electrolyte plays a crucial role because it affects the actual concentration of Zn²⁺ ions in the solution. A 20% dissociation means that only 20% of the zinc salt molecules have broken down into Zn²⁺ and counter ions. This reduced concentration of Zn²⁺ ions will directly impact the electrode potential, as described by the Nernst equation. To accurately calculate the electrode potential, we must first determine the actual concentration of Zn²⁺ ions in the solution, taking into account the degree of dissociation. Then, we can apply the Nernst equation using this adjusted concentration to find the electrode potential under the given conditions. This step-by-step approach ensures a clear understanding of the problem and the application of the relevant electrochemical concepts.

The Nernst Equation: The Key to Non-Standard Conditions

The Nernst equation is the cornerstone for calculating electrode potentials under non-standard conditions. This equation elegantly relates the electrode potential (E) to the standard electrode potential (E⁰), temperature (T), the number of electrons transferred (n), and the reaction quotient (Q). The Nernst equation is expressed as:

E = E⁰ - (RT / nF) * ln(Q)

Where:

  • E is the electrode potential under non-standard conditions
  • E⁰ is the standard electrode potential
  • R is the ideal gas constant (8.314 J/mol·K)
  • T is the temperature in Kelvin
  • n is the number of moles of electrons transferred in the cell reaction
  • F is the Faraday constant (96485 C/mol)
  • Q is the reaction quotient

The reaction quotient (Q) represents the ratio of products to reactants at a given time, raised to their stoichiometric coefficients. For the zinc electrode in a solution of its ions, the relevant half-cell reaction is:

Zn²⁺(aq) + 2e⁻ ⇌ Zn(s)

Therefore, the reaction quotient (Q) for this half-cell reaction is:

Q = 1 / [Zn²⁺]

Where [Zn²⁺] is the concentration of zinc ions in the solution. The Nernst equation is crucial because it allows us to quantify how changes in concentration and temperature affect the electrode potential. In the case of the zinc electrode, a lower concentration of Zn²⁺ ions will result in a more negative electrode potential, as predicted by the Nernst equation. This is because the system will try to compensate for the lower concentration of Zn²⁺ ions by shifting the equilibrium towards the reduction of Zn²⁺, making the electrode more negative. Similarly, an increase in temperature will also affect the electrode potential, as the term (RT / nF) in the Nernst equation becomes larger. Understanding the Nernst equation is therefore essential for predicting the behavior of electrochemical systems in various scenarios.

Calculating the Concentration of Zn²⁺ Ions

Before we can apply the Nernst equation, we need to determine the actual concentration of Zn²⁺ ions in the solution. The problem states that the electrolyte is 20% dissociated. This means that only 20% of the zinc salt molecules in the 0.1 M solution have dissociated into Zn²⁺ ions and their counter ions. To calculate the concentration of Zn²⁺ ions, we multiply the initial concentration of the salt by the degree of dissociation:

[Zn²⁺] = Initial concentration × Degree of dissociation

In this case, the initial concentration is 0.1 M, and the degree of dissociation is 20%, or 0.20. Therefore:

[Zn²⁺] = 0.1 M × 0.20 = 0.02 M

So, the actual concentration of Zn²⁺ ions in the solution is 0.02 M. This value is significantly lower than the initial concentration of the salt, highlighting the importance of considering the degree of dissociation when calculating electrode potentials. The dissociation of the electrolyte is a crucial factor because it directly affects the number of Zn²⁺ ions available to participate in the electrode reaction. If we were to use the initial concentration of 0.1 M in the Nernst equation, we would obtain an inaccurate value for the electrode potential. The lower concentration of 0.02 M means that the equilibrium of the half-cell reaction (Zn²⁺(aq) + 2e⁻ ⇌ Zn(s)) will be shifted towards the left, making it more difficult for Zn²⁺ ions to be reduced to Zn. This shift will result in a more negative electrode potential compared to what would be calculated using the initial concentration. Therefore, accurately calculating the concentration of Zn²⁺ ions is a critical step in determining the electrode potential under non-standard conditions.

Applying the Nernst Equation to the Zinc Electrode

Now that we have the concentration of Zn²⁺ ions ([Zn²⁺] = 0.02 M), we can apply the Nernst equation to calculate the electrode potential (E.P) for the zinc electrode. Recall the Nernst equation:

E = E⁰ - (RT / nF) * ln(Q)

For the zinc electrode, the standard electrode potential (E⁰) is given as -0.76 V, the number of electrons transferred (n) is 2 (as Zn²⁺ gains two electrons to become Zn), and the reaction quotient (Q) is 1 / [Zn²⁺]. Let's plug in the values:

E = -0.76 V - (8.314 J/mol·K × 298 K / (2 × 96485 C/mol)) * ln(1 / 0.02)

First, we calculate the term (RT / nF):

(RT / nF) = (8.314 J/mol·K × 298 K) / (2 × 96485 C/mol) ≈ 0.01285 V

Next, we calculate ln(1 / 0.02):

ln(1 / 0.02) = ln(50) ≈ 3.912

Now, we can substitute these values back into the Nernst equation:

E = -0.76 V - 0.01285 V * 3.912

E = -0.76 V - 0.05027 V

E ≈ -0.810 V

Therefore, the electrode potential (E.P) for the zinc electrode under these conditions is approximately -0.810 V. This result shows that the electrode potential is more negative than the standard electrode potential (-0.76 V), which is expected due to the lower concentration of Zn²⁺ ions in the solution. The Nernst equation allows us to accurately predict this change in electrode potential based on the actual conditions of the system. This calculation demonstrates the power of the Nernst equation in understanding and quantifying the behavior of electrochemical systems.

Result: The Calculated Electrode Potential

Based on our calculations, the electrode potential (E.P) for the zinc electrode in a 0.1 M solution of its salt at 25°C, with a 20% dissociation, is approximately -0.810 V. This result is significant because it demonstrates how the concentration of ions in a solution affects the electrode potential. The standard electrode potential (E⁰) for Zn²⁺/Zn is -0.76 V, which is the potential under standard conditions (1 M concentration). However, in this scenario, the concentration of Zn²⁺ ions is only 0.02 M due to the 20% dissociation of the electrolyte. This lower concentration shifts the equilibrium of the half-cell reaction (Zn²⁺(aq) + 2e⁻ ⇌ Zn(s)) to the left, making it more difficult for Zn²⁺ ions to be reduced to Zn. As a result, the electrode potential becomes more negative. The difference between the calculated electrode potential (-0.810 V) and the standard electrode potential (-0.76 V) highlights the importance of using the Nernst equation to account for non-standard conditions. The Nernst equation allows us to accurately predict the electrode potential under various conditions, which is crucial for many applications, including the design of batteries, fuel cells, and electrochemical sensors. By considering factors such as the degree of dissociation and the concentration of ions, we can gain a deeper understanding of the behavior of electrochemical systems.

Importance of Dissociation in Electrode Potential Calculations

The degree of dissociation of the electrolyte plays a vital role in accurately calculating the electrode potential. As we have seen in this example, the electrolyte being only 20% dissociated significantly impacts the concentration of Zn²⁺ ions in the solution. If we were to neglect the dissociation and assume a 100% dissociation, we would use the initial concentration of 0.1 M in the Nernst equation, which would lead to an incorrect electrode potential. The actual concentration of Zn²⁺ ions, which is 0.02 M, is much lower, resulting in a more negative electrode potential. This difference highlights the importance of considering the dissociation of the electrolyte when performing electrochemical calculations. In many real-world scenarios, electrolytes do not completely dissociate in solution. The degree of dissociation depends on several factors, including the nature of the electrolyte, the solvent, and the temperature. Weak electrolytes, for example, dissociate only to a small extent in solution. Therefore, it is crucial to account for the degree of dissociation to obtain accurate results. The Nernst equation provides a powerful tool for this purpose, as it allows us to relate the electrode potential to the actual concentration of ions in the solution. By carefully considering the dissociation of the electrolyte, we can gain a more accurate understanding of the electrochemical behavior of the system. This understanding is essential for various applications, such as predicting the performance of batteries, designing corrosion protection strategies, and developing electrochemical sensors.

Conclusion: Mastering Electrode Potential Calculations

In conclusion, calculating the electrode potential of a metal in a solution of its ions requires a careful consideration of several factors, including the standard electrode potential, the temperature, the number of electrons transferred, the concentration of ions, and the degree of dissociation of the electrolyte. The Nernst equation is the key tool for performing these calculations under non-standard conditions. In the case of the zinc electrode in a 0.1 M solution of its salt at 25°C with a 20% dissociation, we found that the electrode potential is approximately -0.810 V. This result differs from the standard electrode potential (-0.76 V) due to the lower concentration of Zn²⁺ ions resulting from the partial dissociation of the electrolyte. This example highlights the importance of accurately determining the concentration of ions in the solution and using the Nernst equation to account for non-standard conditions. Mastering electrode potential calculations is essential for a wide range of applications in electrochemistry, including the design and optimization of electrochemical devices, the study of corrosion processes, and the development of new electrochemical technologies. By understanding the principles and techniques discussed in this article, students and professionals can confidently tackle electrode potential calculations and apply them to solve real-world problems in electrochemistry. The ability to accurately calculate electrode potentials is a fundamental skill for anyone working in this field, and it opens the door to a deeper understanding of electrochemical phenomena and their applications.