Problem 70 - Chemistry Principles and Practice

Question

What is the voltage of a concentration cell of \(\mathrm{Cl}^{-}\) ions where the concentrations are 1.045 and \(0.085 \mathrm{M}\) ? What is the spontaneous reaction?

Step-by-Step Solution

Answer

Voltage is approximately 0.118 V; Cl- ions flow from 1.045 M to 0.085 M spontaneously.

Step 1: Identify the Electrochemical Cell Type

The problem involves Cl- ions at two different concentrations in a concentration cell. Concentration cells generate voltage using a single type of ion diffusing from high to low concentration.

Step 2: Write the Nernst Equation

The Nernst equation for a concentration cell is given by: \[ E = \frac{RT}{nF} \ln \left( \frac{\text{[Low Concentration]}}{\text{[High Concentration]}} \right) \] where \( R \) is the gas constant (8.314 J/mol K), \( T \) is the temperature in Kelvin (assumed to be 298 K if not specified), \( n \) is the number of electrons transferred in the half-reaction (\( n = 1 \) for Cl-), and \( F \) is Faraday's constant (96485 C/mol).

Step 3: Calculate Q (Reaction Quotient)

Identify the concentrations of Cl- ions: high concentration is 1.045 M and low concentration is 0.085 M. The reaction quotient, \( Q \), is the concentration ratio: \[ Q = \frac{0.085}{1.045} \]

Step 4: Calculate the Cell Voltage

Substitute values into the Nernst Equation: \[ E = \frac{(8.314 \times 298)}{(1 \times 96485)} \ln \left( \frac{0.085}{1.045} \right) \] This evaluates to \( E \approx 0.118 \, \text{V} \).

Step 5: Determine the Spontaneous Reaction Direction

The spontaneous reaction occurs as the Cl- ions move from higher concentration (1.045 M) to lower concentration (0.085 M). The cell voltage being positive confirms this direction of spontaneity.

Key Concepts

Nernst Equation

The Nernst equation is a vital tool in electrochemistry. It allows us to calculate the electrical potential (voltage) of a cell. This equation is especially handy for concentration cells where the potential is determined by concentration differences, not redox reactions. The equation is formulated as:\[ E = \frac{RT}{nF} \ln \left( \frac{[\text{Low Concentration}]}{[\text{High Concentration}]} \right) \]In this equation:

• \( R \) represents the gas constant, which is 8.314 J/mol K.
• \( T \) stands for temperature, typically considered to be 298 K if not specified.
• \( n \) is the number of electrons transferred per ion in the electrochemical cell. For Cl⁻ ions, \( n = 1 \).
• \( F \) is Faraday's constant, equaling 96485 C/mol, accounting for charge transfer.

The main takeaway is that the equation shows how the voltage depends on the ratio of concentrations. With higher differences, the voltage changes.

Reaction Quotient

The concept of the reaction quotient, denoted as \( Q \), is pivotal in understanding concentration cells. It describes the ratio of concentrations in a system before any significant change in concentration occurs. This is different from equilibrium, where concentrations are stable.For a concentration cell involving chloride ions, \( Q \) is calculated by the formula:\[ Q = \frac{[\text{Low Concentration}]}{[\text{High Concentration}]} \]In the example provided, the concentrations of chloride ions are 0.085 M and 1.045 M. Thus:\[ Q = \frac{0.085}{1.045} \]This provides the foundation for the Nernst equation. As \( Q \) deviates from 1, it signifies the potential for generating voltage due to concentration differences. A smaller \( Q \) means a significant potential exists, as the system is far from equilibrium.

Electrochemical Cell

An electrochemical cell is a system where chemical energy is converted into electrical energy or vice versa. It consists of two electrodes placed in different ionic environments and can be divided into two main types: galvanic (or voltaic) and electrolytic cells. In this context, we're dealing with a concentration cell. This is a special kind of galvanic cell using the same electrolyte at different concentrations. It relies on the principle that ions move from areas of high concentration to low concentration to generate a voltage. Key characteristics of electrochemical cells:

• They have electrodes generally made of metals, like platinum in inert setups.
• Each half-cell contains a solution with a specific ion concentration.
• A salt bridge is often used to maintain charge balance.

The operation of these cells is governed by thermodynamics and kinetic principles. As ions migrate, they generate a current, making the process useful in various batteries and chemical applications.

Spontaneous Reaction

A spontaneous reaction in an electrochemical context means that the reaction proceeds naturally without an external input. In concentration cells, this is driven by entropy and the natural tendency for systems to level out ion concentrations.In our chloride ion cell example, the ion movement from higher (1.045 M) to lower concentration (0.085 M) signifies a spontaneous process. This is validated by the positive cell voltage calculated from the Nernst equation.Important aspects of spontaneous reactions include:

• They occur without external energy, driven by a decrease in free energy.
• In electrochemical cells, a positive voltage is a good indicator of spontaneity.
• Systems naturally proceed towards equilibrium where the reaction quotient \( Q \) heads towards 1.

Understanding spontaneity helps in predicting the direction and feasibility of reactions in both natural and industrial processes.