Problem 128 - A Complete Resource Book in Chemistry for JEE Main

Question

If the pressure of hydrogen gas is increased from 1 arm to \(100 \mathrm{~atm}\), keeping the hydrogen ion concentration constant at \(1 \mathrm{M}\), the voltage of the hydrogen half cell at \(25^{\circ} \mathrm{C}\) will be (a) \(-0.059 \mathrm{~V}\) (b) \(+0.059 \mathrm{~V}\) (c) \(5.09 \mathrm{~V}\) (d) \(0.259 \mathrm{~V}\)

Step-by-Step Solution

Answer

The voltage is +0.059 V; choose option (b).

Step 1: Understand the Problem

We need to determine the voltage change of a hydrogen half-cell when the pressure of hydrogen gas changes from 1 atm to 100 atm while keeping the hydrogen ion concentration constant at 1 M at 25°C.

Step 2: Identify the Nernst Equation

The Nernst equation relates the voltage of an electrochemical cell to the concentration of ions and pressure of gases. The equation is: \[ E = E^0 - \frac{0.059}{n} \log \frac{P_{ ext{products}}}{P_{ ext{reactants}}} \] where \( E^0 \) is the standard cell potential, \( n \) is the number of moles of electrons, and \( P \) represents pressures or concentrations.

Step 3: Apply Nernst Equation to Hydrogen Half-Cell

For the hydrogen half-cell reaction, \(2H^+(aq) + 2e^- \rightarrow H_2(g)\), the standard hydrogen electrode potential \( E^0 = 0 \text{ V} \). Plug in the given pressure conditions (100 atm for hydrogen gas now) into the equation:\[ E = 0 - \frac{0.059}{2} \log \frac{1}{100} \] where \(n = 2\) because 2 electrons are involved.

Step 4: Calculate the Nernst Equation

Now, compute the logarithmic term:\( \log \frac{1}{100} = \log 0.01 = -2\).Substitute back into the Nernst equation:\[ E = 0 - \frac{0.059}{2} \times (-2) = 0.059 \text{ V} \]

Step 5: Analyze Result and Choose Answer

We calculated that the new hydrogen half-cell voltage is \(+0.059 \text{ V}\). This corresponds to option (b) in the list of given answers.

Key Concepts

Electrochemical cell

An electrochemical cell is a fundamental concept in chemistry that combines chemical reactions with the flow of electricity. It works by converting chemical energy into electrical energy using redox reactions.
In a typical electrochemical cell, two different metals or metal compounds serve as electrodes, each in a separate compartment called the half-cell. Each half-cell contains an electrode where a specific reaction occurs. The two half-cells are connected by a salt bridge or a porous barrier, allowing ions to flow but preventing the mixing of different solutions.

• One of the electrodes acts as the anode, where oxidation occurs and electrons are released.
• The other electrode serves as the cathode, where reduction happens and electrons are accepted.

Electrical current flows through an external circuit connecting the two electrodes. The movement of electrons provides usable energy. Electrochemical cells are the basis for batteries and other devices that harness power from chemical reactions.

Standard cell potential

Standard cell potential, denoted as \(E^0\), is a crucial aspect of electrochemical cells. It refers to the voltage or electrical charge difference between two electrodes when an electrochemical cell operates under standard conditions. These conditions typically involve a 1 M concentration for solutions, 1 atm pressure for gases, and a temperature of 25°C.
The standard cell potential measures the tendency of redox reactions to occur spontaneously. Each electrode in the electrochemical cell has a specific reduction potential, which is a measure of the electrode's ability to gain electrons and be reduced.
To find the overall standard cell potential for the cell, you take the difference between the reduction potentials of the cathode and anode:
\[E^0_{cell} = E^0_{cathode} - E^0_{anode}\] This calculation helps predict the cell's overall voltage and efficiency in generating electrical power under ideal conditions.

Hydrogen half-cell

A hydrogen half-cell, often labeled as the standard hydrogen electrode (SHE), holds a central place in electrochemistry. It serves as a reference electrode with an assigned potential of 0 volts.
The hydrogen half-cell consists of a platinum electrode immersed in a 1 M solution of hydrogen ions (\[H^+\]) with hydrogen gas bubbled over it at 1 atm pressure. The platinum electrode acts as a catalyst, facilitating the exchange of electrons.
Employing SHE as a standard makes it possible to measure the potentials of other electrodes in comparison. The reaction occurring at the hydrogen half-cell is:
\[2H^+(aq) + 2e^- \rightarrow H_2(g)\]
This reaction demonstrates the conversion of hydrogen ions into hydrogen gas. By adjusting the pressure of hydrogen gas or concentration of \(H^+\), the voltage of the half-cell can change according to the Nernst equation, providing insights into electrochemical processes.

Pressure effect on cell voltage

The pressure of gases within an electrochemical cell can alter the cell voltage. This factor is accounted for using the Nernst Equation, which adjusts the standard cell potential based on changes in pressure or concentration.
For gaseous reactants or products like hydrogen in a hydrogen half-cell, increasing the pressure affects the equilibrium position of the reaction. According to Le Chatelier's principle, an increase in the pressure of a gas on one side of a reversible reaction shifts the equilibrium to reduce this effect.
As per the Nernst Equation:
\[ E = E^0 - \frac{0.059}{n} \log \frac{P_{\text{products}}}{P_{\text{reactants}}}\]
We see that changing the pressure of hydrogen gas from 1 atm to 100 atm modifies the cell voltage, increasing it in this case by 0.059 volts. This relationship highlights the sensitivity of cell potential to external conditions, a vital consideration in practical applications of galvanic cells and fuel cells.