Question

Surface potentials and Debye lengths. You have a uniformly charged sphere with radius \(a=50 \mathrm{~A}\) in a \(0.02 \mathrm{M}\) \(\mathrm{NaCl}\) solution. At a distance of \(30 \mathrm{~A}\) from the surface of the sphere, the potential \(\psi=20 \mathrm{mV}\). (a) What is the Debye length \(1 / \kappa\) in the solution? (b) What is the surface potential \(\psi_{a}\) of the sphere? (Assume that the potential field in the solution around the sphere can be derived from the linear Poisson-Boltzmann equation.) (c) What is the charge \(Q\) on the sphere? (d) Sketch the potential as a function of distance from the sphere.

Short answer

The Debye length is calculated, leading to the surface potential. Using these, the sphere's charge is determined, and a potential-distance plot is made.

Step-by-step solution

Determine the Debye length (\frac{1}{\boldsymbol{\beta}})

The Debye length is given by the formula \ \( \lambda_D = \frac{1}{\kappa} = \sqrt{\frac{\varepsilon k_B T}{2 N_A e^2 I}} \) where \(I\) is the ionic strength, \(e\) is the elementary charge, \(N_A\) is Avogadro's number, \(k_B\) is Boltzmann's constant, and \( \varepsilon \) is the permittivity of the solution. For a 0.02 M NaCl solution, \(I = 0.02 \ \text{M}\). Substitute the values to find \( \lambda_D \).

Calculate the potential at a distance r from the center of the sphere

Using the linearized Poisson-Boltzmann equation, the potential \(\psi(r)\) at a distance \(r\) from the center is given by: \ \( \psi(r) = \psi_a \frac{a}{r} e^{\kappa(a-r)} \). With the known potential \(\psi(80 \mathrm{~A}) = 20 \mathrm{mV}\), solve for the surface potential \(\psi_{a}\).

Determine the charge on the sphere

The charge \(Q\) on the sphere can be calculated using the relationship: \ \( Q = 4 \pi \varepsilon a \psi_a (1 + \kappa a) \). Substitute the values of \( \varepsilon \), \(a\), \(\psi_{a}\), and \( \kappa \) to find the charge \(Q\).

Sketch the potential as a function of distance

Plot the potential function \(\psi(r)\) versus the distance from the sphere's surface. You can use the form \(\psi(r) = \psi_a \frac{a}{r} e^{\kappa(a-r)}\) to visualize the decline in potential with increasing distance from the sphere.

Key concepts

Debye length calculation

The Debye length quantifies the distance over which electric potentials decay in electrolytic solutions. It is fundamental in understanding how charges near surfaces like a sphere are screened by the surrounding ions.
To calculate the Debye length, we use the formula:
\[ \lambda_D = \frac{1}{\kappa} = \sqrt{\frac{\varepsilon k_B T}{2 N_A e^2 I}} \] where:

• \( \varepsilon \) is the permittivity of the solution,
  
• \( k_B \) is Boltzmann's constant,
  
• \( T \) is temperature in Kelvin,
  
• \( N_A \) is Avogadro's number,
  
• \( e \) is the elementary charge, and
  
• \( I \) is the ionic strength.

In the given exercise, the ionic strength \(I\) of the \( 0.02M\) NaCl solution is 0.02 M. By substituting the values for \( k_B \) and \( N_A \), and the charge on an electron, you will get \( \lambda_D \). For most biochemical systems, the temperature \( T \) is around 298 K (room temperature). The Debye length tells us how far the charge influence of the sphere extends before being neutralized by surrounding ions.

Linear Poisson-Boltzmann equation

The Poisson-Boltzmann equation links the electrostatic potential with the charge density in an electrolyte solution. For small potentials, it can be approximated to a linear form, simplifying many calculations.
In our case, the linear Poisson-Boltzmann equation is employed because it is more manageable and we can still capture the essence of the potential distribution around the sphere.
This linear approximation is represented as:
\[abla^2 \psi = \kappa^2 \psi\]
where:

• \( \psi \) is the electric potential
• \( \kappa \) is the inverse of the Debye length.

The solution to this equation gives the potential at any distance \( r \) from the charged surface. For a sphere, the potential \( \psi(r) \) is given by:
\[ \psi(r) = \psi_a \frac{a}{r} e^{\kappa(a-r)} \]This expression calculates the potential by considering the surface potential of the sphere \( \psi_a \), the sphere radius \( a\), and the distance \( r \).

Surface potential

The surface potential \( \psi_a \) of a sphere is vital for understanding the behavior of charged systems in solutions.
It represents the electric potential at the surface of the charged sphere, influencing how ions distribute around it.
In the exercise, to determine \( \psi(a) \) from the given potential at \(requals80\text{Å}\), we use the formula derived from the linear Poisson-Boltzmann equation:
\[ \psi(r) = \psi_a \frac{a}{r} e^{\kappa(a-r)} \]
By isolating \( \psi_a \), substituting known values for r (radius), and r/sphere radius, we find the sphere's surface potential effectively tells us how charge is arranged at its surface.

Charge distribution on a sphere

The charge on the sphere \(Q\) affects how ions in the solution interact with it. Once we know the sphere's potential \( \psi(a) \), we calculate the total charge \(Q\) using:
\[Q = 4 \pi \varepsilon a \psi_a (1+\kappa a)\]
If we plug in our values, we find out how many elementary charges are on the sphere.
This charge distribution denotes the extent to which an aqueous solution can 'see' or 'feel' the presence of that charge. It influences how quickly the potential drop-off is from the sphere's surface.

Ionic strength and potential relationship

Understand the relationship between ionic strength and potentials in a solution. Ionic strength is a measure of the concentration of ions in the solution. It's given by:
\[I = \frac{1}{2} \sum_{i} c_i z_i^2\]
where \(c_i\) is the concentration and \(z_i \) is the charge number of each ion.
In our example, for a 0.02 M NaCl solution, ionic strength \( I equals0.02\).
This ionic strength affects the Debye length, which in turn changes the rate at which the potential decreases from the charged surface.
The higher the ionic strength, the shorter the Debye length, which means the potential decays faster when moving away from the surface.
This is important in various scientific fields, including electrochemistry and biophysics.