Question

Two regions of an ideal dilute solution have a difference in concentration of potassium ions \(\left(\mathrm{K}^{+}\right)\). At \(293 \mathrm{~K}\), what is the difference in chemical potential between region 1 , with a concentration of \(0.5 \mathrm{M} \mathrm{K}^{+}\), and region 2, which has a concentration of \(2 \mathrm{mM}\) ?

Short answer

The chemical potential difference is approximately 13367.5 J/mol.

Step-by-step solution

Define Initial Parameters

First, identify the known variables from the problem: temperature \( T = 293 \, \text{K} \), concentrations \([K^+_1] = 0.5 \, \text{M}\), and \([K^+_2] = 2 \, \text{mM} = 0.002 \, \text{M}\). The gas constant \( R \) is approximately \( 8.314 \, \text{J/(mol K)} \).

Use the Chemical Potential Formula

The change in chemical potential \( \Delta \mu \) for ions in solution is given by the Nernst equation: \[ \Delta \mu = RT \ln \left( \frac{[K^+_1]}{[K^+_2]} \right) \].

Substitute Values into Formula

Substituting the known values into the Nernst equation: \[ \Delta \mu = (8.314 \, \text{J/(mol K)}) \times 293 \, \text{K} \times \ln \left( \frac{0.5}{0.002} \right) \].

Calculate the Logarithmic Term

Calculate the natural logarithm: \( \ln \left( \frac{0.5}{0.002} \right) = \ln (250) \approx 5.52 \).

Calculate the Chemical Potential Difference

Substitute back to find \( \Delta \mu \):\[ \Delta \mu = 8.314 \times 293 \times 5.52 \approx 13367.5 \, \text{J/mol} \].

Key concepts

Ideal Dilute Solutions

An ideal dilute solution is a concept that describes a solution where the solute concentration is very low, allowing it to behave similarly to an ideal solution. In ideal solutions, interactions between solute and solvent particles are similar to those between solvent particles themselves.
In the case of dilute solutions, we often assume that the activity (effective concentration) of the solute equals its molar concentration. Here, the solute concentration doesn't significantly impact the overall properties of the solution, such as viscosity or osmotic pressure. This simplifies many calculations, like those involving the chemical potential.
The scenario described involves potassium ions in a dilute state where their effect on the solution behavior is minimal. The "ideal" in ideal dilute solutions assumes that intermolecular forces between solute and solvent don't deviate significantly from those in pure solvents.

• This concept is crucial for applying the Nernst equation later, as it assumes a linear relationship between concentration and activity, simplifying mathematical treatment.
• In everyday applications, this assumption helps in tasks like calculating pH or determining the osmotic pressure in biological systems.

Nernst Equation

The Nernst equation is a fundamental relation in electrochemistry that relates the reduction potential of a half-cell to the concentrations of the ions involved in the reaction. Inspired by thermodynamics, it connects chemical potential with ion concentration and temperature.
The Nernst equation states:\[ E = E^0 - \frac{RT}{nF} \ln Q \]where:

• \( E \) is the reduction potential of the electrochemical cell.
• \( E^0 \) is the standard reduction potential.
• \( R \) is the ideal gas constant (8.314 J/(mol K)).
• \( T \) is temperature in Kelvin.
• \( n \) is the number of moles of electrons exchanged.
• \( F \) is Faraday's constant (approximately 96485 C/mol).
• \( Q \) is the reaction quotient, derived from the concentration ratios.

This versatility allows the Nernst equation not just to explain electrochemical cells, but also to analyze concentration gradients across biological membranes.
In the context of this exercise, the Nernst equation variant applied reflects the change in chemical potential for ions by relating the concentrations of potassium ions in two regions:

Concentration Differences

Concentration differences are a key factor in chemical processes and reactions, especially those involving diffusion or osmosis where substances move from regions of higher concentration to lower concentration.
In electrochemistry, concentration differences drive the movement of ions across surfaces such as cell membranes, creating potential differences (voltages) crucial for processes like nerve impulse transmission and cellular respiration.
Understanding these differences is central to using the Nernst equation effectively. It allows us to calculate how concentrations impact chemical potential by employing a logarithmic function. This reflects how large concentration gradients can lead to significant changes in potentials.
For instance, in the provided problem:

• The initial concentrations are 0.5 M and 0.002 M for potassium ions.
• By calculating the concentration ratio and then using its natural logarithm, we determine the driving force behind the chemical potential difference.

This exercise not only highlights the significant role of concentration differences in influencing chemical potentials but also underscores how these principles extend to real-world applications, from battery design to understanding cellular functions.