Problem 142 - A Complete Resource Book in Chemistry for JEE Main

Question

The reversible reduction potential of pure water is \(-0.413 \mathrm{~V}\) under \(1.00 \mathrm{~atm} \mathrm{H}_{2}\) pressure. If the reduction is considered to be \(2 \mathrm{H}^{+}+2 \mathrm{e}^{2} \longrightarrow \mathrm{H}_{2}\), calculate \(\mathrm{pH}\) of pure water. (a) 6 (b) 7 (c) 3 (d) 5

Step-by-Step Solution

Answer

The pH of pure water is approximately 7.

Step 1: Understanding the Nernst Equation

The Nernst equation relates the reduction potential to the concentrations of the ions involved in a half-cell reaction. It is given by \[ E = E^0 - \frac{RT}{nF} \ln Q \]where \( E \) is the electrode potential, \( E^0 \) is the standard electrode potential, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons exchanged, and \( F \) is the Faraday constant. \( Q \) is the reaction quotient.

Step 2: Simplifying the Nernst Equation for pH

Since we're dealing with the hydrogen evolution reaction, the Nernst equation simplifies to:\[ E = E^0 - \frac{0.0592}{n} \log \left(\frac{[H_2]}{[H^+]^2}\right) \]At standard conditions, \( E^0 \) for hydrogen is 0, \( n = 2 \), and pressure of \( H_2 \) is 1 atm, hence the equation reduces to:\[ -0.413 = 0 - \frac{0.0592}{2} \log \left(\frac{1}{[H^+]^2}\right) \]

Step 3: Solving for [H⁺] concentration

Rearrange the equation to solve for \([H^+]^2\):\[ -0.413 = -0.0296 \log \left(\frac{1}{[H^+]^2}\right) \]\[ 0.413 = 0.0296 \log \left([H^+]^2\right) \]\[ \log \left([H^+]^2\right) = \frac{0.413}{0.0296} \approx 13.949 \]Taking the antilog:\[ [H^+]^2 = 10^{13.949} \approx 10^{-13.949} \]

Step 4: Calculating pH from [H⁺] concentration

Recall that \( pH = -\log[H^+] \). Therefore:\[ [H^+] = 10^{-6.975} \]\[ pH = 6.975 \]The pH would be approximately 7 for pure water.

Key Concepts

Nernst Equation

The Nernst equation is a fundamental principle in electrochemistry that allows us to calculate the cell potential of an electrochemical cell under non-standard conditions. It is expressed as: \[ E = E^0 - \frac{RT}{nF} \ln Q \] Where:

• \(E\) is the cell potential.
• \(E^0\) is the standard electrode potential.
• \(R\) is the universal gas constant (8.314 J/mol K).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of electrons transferred in the reaction.
• \(F\) is the Faraday constant (96485 C/mol).
• \(Q\) is the reaction quotient, a ratio of concentrations of products and reactants.

This equation helps us understand how changes in ion concentrations or pressure can influence the electrode potential. In the context of pH calculation, the Nernst equation assists in linking the cell potential to hydrogen ion concentration, enabling the calculation of pH.

Hydrogen Evolution Reaction

The hydrogen evolution reaction (HER) is an important process in electrochemistry where hydrogen ions (\(H^+\)) in solution are reduced to form hydrogen gas (\(H_2\)). The reaction is typically represented as:\[ 2H^+ + 2e^- \rightarrow H_2 \] This reaction is crucial for various applications, including hydrogen gas production and the functioning of hydrogen fuel cells. During the HER, protons gain electrons to form molecular hydrogen. The efficiency and speed of this reaction are affected by several factors:

• The electrode material - Platinum is a common catalyst as it facilitates efficient electron transfer.
• pH level - The concentration of hydrogen ions directly influences the reaction rate.
• Electrode potential - A more negative electrode potential speeds up the reaction.

In a pH calculation context, understanding the HER is key, as the generation of hydrogen gas under certain conditions allows us to link it to the concentration of hydrogen ions, from which pH can be determined.

Standard Electrode Potential

The standard electrode potential, denoted as \(E^0\), is a fundamental measure in electrochemistry that indicates the intrinsic potential of a half-cell under standard conditions. These conditions typically refer to:

• Solutions having concentrations of 1M.
• A pressure of 1 atm for gases.
• A temperature of 298 K (25°C).

For the hydrogen evolution reaction, the standard electrode potential is \(E^0 = 0 \, V\) because it serves as a reference point by convention. This value implies that under standard conditions, there is no driving energy difference across the hydrogen electrode relative to it self. Having the standard electrode potential as a starting value enables us to use the Nernst equation for practical calculations. It is crucial to understanding how the potential changes when we deviate from these standard conditions, which is key in processes like determining the pH of a solution.

Reaction Quotient

The reaction quotient, denoted as \(Q\), is a crucial concept that helps us understand how far a reaction has proceeded from its equilibrium state. It's calculated using the concentrations of the reactants and products involved in the reaction: \[ Q = \frac{\text{{[products]}}^{\text{{coefficients}}}}{\text{{[reactants]}}^{\text{{coefficients}}}} \] For the hydrogen evolution reaction that involves hydrogen ions and hydrogen gas, this simplifies to:\[ Q = \frac{[H_2]}{[H^+]^2} \] The reaction quotient provides a snapshot of the reaction's momentary state; if it differs from the equilibrium constant, the system may shift towards the equilibrium.Understanding \(Q\) is essential for using the Nernst equation effectively, as it helps relate the concentrations of substances in a reaction to the potential difference across an electrochemical cell. This relationship between \(Q\) and potential difference is vital when assessing the \(pH\) of a solution because it showcases how close or far the solution is from achieving balance.