8.3.P11

A positively charged gold nanoparticle with charge Q is placed in an ionic fluid at room

 temperature. Use dimensional analysis and the form of the equations to answer all questions below. Be sure to show your work. It should be neat and orderly for full credit.

Given: The electrostatic potential (measured in volts) outside the nanoparticle is given by

where r is the distance from the center of the nanoparticle, the parameter κ is the fluid dielectric, k_C is the Coulomb constant, and  

is defined to be "the Debye length." Here, k_B is Boltzmann's constant and the charge on each ion in the fluid is q, and c₀ is a quantity to be determined below. Let's see how much can be figured out about this system from these equations and some basic considerations.

(a) Use dimensional analysis and the first expression to determine the dimensions of the fluid dielectric. 

(b) Use dimensional analysis and the first expression to determine the dimensions of λ_D.

(c) Use dimensional analysis to determine the dimensions of c₀.

(d) In light of these findings, give two plausible interpretations for what the quantity c₀ might represent. 

(e) The radial component of the electric field associated with a charge Q (i.e., the component of the electric field vector that points away from the charge) is

Use this expression together with what you know about the electrostatic potential due to a single charge (not immersed in an ionic fluid) to show that a formula for the radial component of the electric field of an ion with charge Q  is given by

in the absence of an ionic fluid. Let us call this quantity E₀.

(f) Use the formula for the electrostatic potential of a gold particle in an ionic fluid (provided at the very beginning of this problem) and the information from part (e) to show that the radial component of the electric field associated with a charge Q that is immersed in an ionic fluid is given by

This ratio is shown in the figure below, for the case of κ = 1. 

(g) Which of the following statements are true? State your reasoning in each case.

     (1) The electric field magnitude outside the nanoparticle is everywhere reduced by the presence of the ionic fluid when κ > 1.

     (2) The electric field magnitude outside the nanoparticle is everywhere reduced by the presence of the ionic fluid when κ < 1. 

     (3) The ionic fluid reduces the strength of the electric field more at larger distances from the gold.

     (4) The ionic fluid reduces the strength of the electric field less at larger distances from the gold.

Bill Dorland 3/3/19