problem set 2 solutions

# problem set 2 solutions - 1 BME C101/C201 INTRODUCTION TO...

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Unformatted text preview: 1. BME C101/C201. INTRODUCTION TO BIOMEDICAL ENGINEERING FALL 2010 PROBLEM SET 2 Due October 14, 2010 (Thursday) Recall the charged planar wall example from lecture (see ﬁgure below). In this analysis, we used a boundary condition at the surface (x=0) and at xéinﬁnity. In reality, you are analyzing a charged surface in an aqueous solution in a beaker, and the beaker has ﬁnite dimensions. Accordingly, you are never at an inﬁnite distance away from the charged surface. Can you provide an estimate of what “inﬁnity” is in a real situation? ' + + + + + + + + + + + + + + + + + + Consider a spherical micelle comprised of the surfactant dioctanoyl phosphatidylcholine (Cg—lecithin). This is a zwitterionic surfactant with both positive and negative charges as shown below. In one spherical micelle, there are N surfactant monomers. As shown below, the micelle may be modeled as two spheres with the entire negative charge of the micelle smeared over the surface area of the inner sphere of radius Rm and the entire positive charge of the micelle smeared over the surface area of the outer sphere of radius Rm. Assuming the salt ions in solution corresponding to a particular value of the Debye- Hﬁckel screening length (16’) cannot penetrate the region between 0 and Row, derive expressions for the electrostatic potentials in Regions a, ,8, and 5 shown below. Do not solve for the constants of integration. However, you need to derive the boundary conditions that are required to evaluate the constants of integration. Although the expressions for the electrostatic potentials and the boundary conditions may not be a function of all of the following parameters and variables, these equations can be a function of k3, T, e (the electronic charge), N, Row, Rm, r, K, constants of integration, and the dielectric constants in the 0c, ,8, and 6 regions (i.e. 8a, 8/}, and 85). In this problem, use the Linearized Poisson-Boltzmann equation. n (1)6 (13H3 EB CH3—(CH2)7-C~O—('3H—CHZ—O—ﬁ—O—CHz-CHz—N—CH3 CH3-(CH2)7-ﬁ—O-CH2 0 CH3 model as 3. Do the electrostatic interactions between two proteins increase or decrease as more potassium chloride is added to the solution? Explain why. 4. Derive an expression for the electrostatic potential in the aqueous solution outside a sphere having a radius R and a surface potential We at r=R. The ionic strength of the solution outside the sphere is known, and it’s value is on in units of M. 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