Question

The potential near a plane. You have a uniformly charged flat plate in contact with a \(0.02 \mathrm{M} \mathrm{NaCl}\) solution at \(25^{\circ} \mathrm{C}\). At a distance of \(3 \mathrm{~nm}\) from the plate, the potential is \(30 \mathrm{mV}\). (a) What is the Debye length \(1 / \kappa\) in the solution? (b) What is the surface potential of the charged plane in \(\mathrm{mV}\), and in units of \(k T / e\) ? Use the linear PoissonBoltzmann equation. (c) If you had used the nonlinear Poisson-Boltzmann equation, would you find the surface potential to be larger or smaller than you found under (b)? Why? (d) Use the surface potential from (b) to find the surface charge density \(\sigma\) of the plane.

Short answer

The Debye length is 0.714 nm. The surface potential is 1700 mV (66.15 \frac{kT}{e}). The non-linear Poisson-Boltzmann potential is larger due to less screening. Surface charge density \sigma \is \1.7 \times 10^{-9} ext{C/m}^2.

Step-by-step solution

- Find the Debye Length

The Debye length \(1/\kappa\) for an electrolyte solution is given by the formula: \[\frac{1}{\kappa} = \sqrt{\frac{\varepsilon \varepsilon_0 k_B T}{2 N_A e^2 I}}\], where \(I\) is the ionic strength of the solution, and \(N_A\), \(e\), \(\varepsilon_0\), and \(k_B\) are constants. For a 1:1 electrolyte such as \text{NaCl}\, \(I = 0.5 \times c\). With \(c = 0.02 \text{M}\):\ I = 0.5 \times 0.02 = 0.01 \text{M}\.\ Substituting known values at \text{25}^{\circ} \text{C}\: \(T = 298\: K\; \varepsilon \varepsilon_0 = 7.08 \times 10^{-10} \text{C/(V m)},\; k_B = 1.38 \times 10^{-23} \text{J/K},\; e = 1.6 \times 10^{-19} \text{C},\; N_A = 6.022 \times 10^{23} \text{mol}^{-1}\):\ \kappa = \sqrt{\frac{2 \times (0.01) \times (6.022 \times 10^{23}) \times (1.6 \times 10^{-19})^2}{(7.08\times 10^{-10}) \times (1.38 \times 10^{-23}) \times 298}}\.\ \kappa = 1.4 \text{nm}^{-1}\.\ Therefore: \frac{1}{\kappa} = \frac{1}{1.4} \text{nm} \approx 0.714 \text{nm}\.

- Calculate Surface Potential Using Linear Poisson-Boltzmann Equation

For the linear Poisson-Boltzmann equation, the potential \psi at a distance \text{d} from a charged surface is given by: \[\psi(d) = \psi_0 \cdot \exp(- \kappa d )\.\] \Here, we know: \psi(d=3 \text{nm}) = 30 \text{mV},\ \text{and} \kappa = 1.4 \text{nm}^{-1}. \Therefore: \psi(3 \text{nm}) = \psi_0 \exp(-1.4 \times 3) = 30 \text{mV}\.\ \Solving for \psi_0,\ \30 = \psi_0 \cdot \exp(-4.2),\ \psi_0 = 30 \text{mV} \cdot \exp(4.2) \approx 1700 \text{mV}\. You also need to convert \psi_0 to units of \frac{kT}{e},\ \frac{kT}{e} \approx 25.7 \text {mV} at \text{298 K}.\ Therefore: \psi_0 \text{ in units of } \frac{kT}{e} = \frac{1700 \text{mV} }{25.7 } \approx 66.15\.

- Compare Surface Potentials from Linear and Non-linear Poisson-Boltzmann Equations

In the nonlinear Poisson-Boltzmann equation, the potential decays more slowly because the field created by high surface charge causes ion accumulation close to the surface. Hence, the surface potential found using nonlinear Poisson-Boltzmann equation is generally larger than that obtained using linear approximation due to less screening at high potentials.

- Calculate Surface Charge Density \sigma

Surface charge density \( \sigma \) is given by: \[\sigma = \varepsilon \varepsilon_0 \kappa \psi_0,\]\where \varepsilon\ is the relative permittivity (80 for water at 25\degree C). \varepsilon_0 = 8.85 \times 10^{-12} \text{F/\text{m}}, \kappa = 1.4.\ \ Therefore: \sigma = 80 \times8.85 \times 10^{-12 } \times1.4 \times 1.7 \times 10^{-3} \approx 1.7 \times 10^{-9} \text{C/\text{m}^2}.\

Key concepts

Poisson-Boltzmann equation

The Poisson-Boltzmann equation is a fundamental equation used to describe the electric potential in a fluid containing charged particles around a charged surface. This equation takes into account both the distribution of the electric field and the distribution of ions in the solution. There are two forms of this equation: linear and nonlinear.

The linear form is used for low potentials and simplifies the computations. It assumes that the potential decays exponentially away from a charged surface. The potential, \(\psi\), at a distance, \(d\), from the surface is given by:
\[ \psi(d) = \psi_0 \cdot \exp(- \kappa d) \]
Here, \(\psi_0\) represents the surface potential, and \(\kappa\) is the inverse of the Debye length. The Debye length provides a measure for the thickness of the charge cloud around the particle.

On the other hand, the nonlinear Poisson-Boltzmann equation is more accurate for high potentials. It accounts for the fact that ion distribution is not purely exponential due to ion accumulation close to the surface. This results in a more gradual potential decay. Hence, the nonlinear version is typically used for strongly charged surfaces.

Surface potential

Surface potential, often denoted as \(\psi_0\), is the electric potential at the charged surface within the electrolyte solution. It is a vital parameter in determining the interaction forces between charged particles and the surrounding ions.

For instance, in our exercise, the surface potential \(\psi_0\) was calculated using the linear Poisson-Boltzmann equation. Given the potential at a distance of 3 nm from the plate is 30 mV, and the Debye length \( \frac{1}{\kappa}\) is approximately 0.714 nm, we use the formula:
\[ \psi(d) = \psi_0 \cdot \exp(- \kappa d) \]
Solving for \(\psi_0\), we found \(\psi_0\) to be 1700 mV.

Converting \(\psi_0\) to units of \( \frac{kT}{e}\), we know that at 298 K, \( \frac{kT}{e} \approx 25.7 \) mV. Therefore, the surface potential in these units is approximately 66.15.

Understanding the surface potential helps predict the behavior of ions in the vicinity of the charged surface, which is crucial for various applications, such as in colloidal dispersions and electrochemistry.

Ionic strength

Ionic strength \(I\) is a measure of the total concentration of ions in an electrolyte solution. It influences the electric interactions in the solution and is crucial in the calculation of several properties, such as the Debye length.

Ionic strength is calculated using the formula:
\[ I = 0.5 \sum_j c_j z_j^2 \]
Here, \(c_j\) and \(z_j\) are the molar concentration and charge number of the \(j^{th}\) ion, respectively. For a 1:1 electrolyte like NaCl, where the concentration of the solution, \(c \), is given as 0.02 M:
\[ I = 0.5 \times c = 0.5 \times 0.02 = 0.01 \text{ M} \]

Ionic strength directly affects the Debye length, which is inversely proportional to the square root of ionic strength. In our example, we used this ionic strength value to calculate the Debye length, providing us with the screening length around the charged surface. This is vital to understand how quickly the electric potential decays in the solution.

Surface charge density

Surface charge density, represented as \( \sigma \), is the amount of electric charge per unit area on the surface of a material. It is an essential parameter when dealing with electrostatic phenomena in electrolyte solutions.

Surface charge density can be calculated using the relation:
\[ \sigma = \varepsilon \varepsilon_0 \kappa \psi_0 \]
Here, \( \varepsilon \) is the relative permittivity of the medium, \( \varepsilon_0 \) is the permittivity of free space, \( \kappa \) is the inverse of the Debye length, and \( \psi_0 \) is the surface potential.

For our exercise, using values: \( \varepsilon \) = 80 (for water), \( \varepsilon_0 \) = 8.85 \( \times 10^{-12} \) \text{ F/m}, \( \kappa = 1.4 \) \text{ nm}^{-1}, and \( \psi_0 \) = 1700 mV (or 1.7 \( \times 10^{-3} \) V considering units):
\[ \sigma = 80 \times 8.85 \times 10^{-12} \times 1.4 \times 1.7 \times 10^{-3} \approx 1.7 \times 10^{-9} \text{ C/m}^2 \]

Understanding surface charge density is important in fields like colloid chemistry and material science, as it dictates how particles within a solution interact with each other and their surroundings.