To calculate the electrical double layer thickness also known as the Debye length of a silica particle, we need to consider the Debye-Hückel theory. The Debye length can be calculated using the following formula: = rRT / 2FI ^1/2 where: = vacuum permittivity 8.854 x 10 F/m r = relative permittivity of water approximately 78.5 at room temperature R = gas constant 8.314 J/molK T = temperature in Kelvin assume room temperature, 298 K F = Faraday's constant 96485 C/mol I = ionic strength of the solution in mol/m To find the ionic strength I , we need to know the concentration of ions in the solution. Since we are given the surface charge density of the silica particle, we can estimate the ionic strength using the Gouy-Chapman theory. The Gouy-Chapman equation is: = 2 * z * e * N * C * sinh e * / 2kT where: = surface charge density -0.05 C/m z = valence of counterions assume monovalent counterions, z = 1 e = elementary charge 1.602 x 10 C N = Avogadro's number 6.022 x 10 mol C = concentration of counterions in mol/m = surface potential in V k = Boltzmann's constant 1.381 x 10 J/K We can rearrange the Gouy-Chapman equation to solve for the concentration of counterions C :C = / 2 * z * e * N * sinh e * / 2kT Assuming that the surface potential is large enough that sinh e * / 2kT e * / 2kT, we can simplify the equation to:C = / 2 * z * e * N * e * / 2kT Now, we can plug in the given values and constants:C = -0.05 C/m / 2 * 1 * 1.602 x 10 C * 6.022 x 10 mol * 1.602 x 10 C * / 2 * 1.381 x 10 J/K * 298 K Since we don't have the surface potential , we cannot find the exact concentration of counterions C . However, we can still express the Debye length in terms of : = rRT / 2FI ^1/2 = rRT / 2F * 2C ^1/2 Plugging in the values and constants: = 8.854 x 10 F/m * 78.5 * 8.314 J/molK * 298 K / 2 * 96485 C/mol * 2 * -0.05 C/m / 2 * 1 * 1.602 x 10 C * 6.022 x 10 mol * 1.602 x 10 C * / 2 * 1.381 x 10 J/K * 298 K ^1/2 This equation gives the electrical double layer thickness Debye length of the silica particle in terms of the surface potential . To find the exact value of the Debye length, we would need the surface potential or the concentration of counterions in the solution.