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problem set 3 solutions

# problem set 3 solutions - 1 BME C101/C201.‘ INTRODUCTION...

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Unformatted text preview: 1. BME C101/C201.‘ INTRODUCTION TO BIOMEDICAL ENGINEERING FALL 2010 PROBLEM SET 3 Due October 21, 2010 (Thursday) Derive an expression for the electrostatic potential in the aqueous solution outside a sphere having a radius R and a surface charge density of o. The ionic strength of the solution outside the sphere is known, and its value is on in units of M. Use the Linearized Poisson-Boltzmann equation with the dielectric constant not a function of position. Note that you will not need to solve for the potential inside the sphere if you use Gauss’s Law just outside the sphere for one of the boundary conditions. 2. Derive an expression for the electrostatic potential in the aqueous solution outside a cylinder of radius R and length L, where R<<L. The charges are uniformly distributed on the outside of the cylinder, and the surface potential qJ(r=R+) is known as 010- The ionic strength of the solution outside the sphere is known, and its value is on in units of M. Use the Linearized Poisson—Boltzmann equation with the dielectric constant not a function of . position. Submit a separate MATLAB script m—ﬁle on CourseWeb for each of the following problems. Please make sure that your ﬁles run ﬁne before submitting them. a. Write an m-ﬁle that asks the user to input the concentrations of MgClz and NaCl in units of molarity. MATLAB should then compute and display the ionic strength of the solution with the ionic strength variable included in the display. b. The same as Part a except that only the value of the ionic strength should be displayed, i.e. do not display the name of the ionic strength variable. ' 0. Write an m-ﬁle that computes the value of the electrostatic potential at x = 0, 1x10~10 meters (m), 2x10~10 m, 3x10"lo m,... 10x10'Io m, and displays all ofthose values. You are also given w = we exp(- KX) where W0 is 0.0257 J/C or 25.7 mV, and K is given by K = (3.29x10?z csan‘”) m" for a symmetric electrolyte as shown in Problem Set 1, where z is equal to z+ and 2+ is equal to —z., while Csalt is the molar concentration of the salt. Note that NaCl C. is your symmetric electrolyte at a concentration of 2 M. For this m-ﬁle, please generate your 1—by—11 array for x by listing all values in brackets. Note that: [a1 a2 a3 a4]*b=[a1b azb aab a4b] The same as Part c except that the colon notation should be used to specify the 1~ by-ll array for x. Also, display only the 2nd, 3rd, 4th, and 5th elements of the resulting electrostatic potential array. The same as Part c except that the linspace notation should be used to specify the l-by-ll array for x. Also, display only the 3rd element of the resulting electrostatic potential array. Write an m-file that computes the value of the electrostatic potential at x = O, 0.1x10'IO meters (m), 0.2xlO'10 m, 0.3x10'm m,... 10x10'10 m. You are also given w .= we exp(— KX) where we is 0.0257 J/C or 25.7 mV, and K is given by K = (3.29x1092 Cm”) m-1 for a symmetric electrolyte as shown in Problem Set 1, where z is equal to 2+ and 2+ is equal to ~z-, while Csalt is the molar concentration of the salt. Note that NaCl is your symmetric electrolyte at a concentration of 2 M. Display only the 50‘h element of the resulting electrostatic potential array. 4. Solve the following math problems. Write coth(u) in terms of exponentials, where coth is the hyperbolic cotangent. 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C 14‘”) £KF[X) + 6; £2; u. :0 ’ k W 4/5,; ‘5'”(1— ﬁx}; mama/e ”70?“th 0 saw, ﬂaw/Mar Via {avid/a}? fax-Rf 74m” +5.; Amm’ﬂﬁfaf 3024’!” Kl," inmaus m 7"“ Warn/11nd,)” Mm/Jre, C, =0 5pm : (2 mix) 91/00.) ccakokr) ”7/"; 56'”- HKR) = 62 KO (KR); V; V6 C2 Mme) I W, \$V[Kr )“ - 00m) (Kr) 530) It was Show/r in (tofu/e % script M—file problemset3_3a.m MgClZ_conc = input (‘Enter the concentration of MgClZ salt added in molarity units > '); NaClmconc = input ('Enter the concentration of NaCl salt added in molarity units > '); Mg_conc = MgC12_conc; Na_conc = NaClaconc; Cl_conc = 2*MgCl2_conc + NaCl_conc; Mg_valence = 2; Na_valence = l; Cl_valence = «l; ionic_strength = (l/2)*(Mg_conc*Mg_valence“2 + Cl_conc*Cl_valence“2 + Na_conc*Na_valence“2 ) >> problemset3_3a Enter the concentration of MgClZ salt added in molarity units > 2 Enter the concentration of NaCl salt added in molarity units > 3 ionic_strength = 9 % script M~file problemset3 3b.m MgClZHconc = input ('Enter the concentration of MgClZ salt added in molarity units > '); NaCl_conc = input ('Enter the concentration of NaCl~salt added in molarity units > '); Mg_conc = MgC12_conc; Na_conc = NaCl_conc; Clmconc = 2*MgCl2~conc + NaClwconc; Mg_valence = 2; Na_valence = l; Clwvalence = ~l; ionic_strength = (l/2)*(Mg_conc*Mg_valenceA2 + Cl_conc*Cl_valence“2 + Na_conc*Na~valence“2 ); disp(ionic_strength) >> problemset3_3b Enter the concentration of MgClZ salt added in molarity units > 2 Enter the concentration of NaCl salt added in molarity units > 3 9 % script M—file problemsetBﬂBc.m psi_0 = 0.0257; 2 = l; 2; kappa = (3.29*lO“9*z*C_saltAO.5); O U) m [.1 (1. ll X = [0 l 2 3 4 5 6 7 8 9 10]*(lOA(~lO)); psi = psi_0 * exp(—kappa*x) >> problemset3_3c psi = Columns 1 through 6 0.0257 0.0161 0.0101 Columns 7 through 11 0.0016 0.0010 0.0006 % script M~file problemset3_3d.m psi 0 = 0.0257; C salt = 2; kappa = (3.29*10“9*Z*C_salt“0.5); x = (O:l:lO)*10“(—10); psi = psi_0 * exp(—kappa*x); psi = psi(2:5) >> problemset3_3d psi = 0.0161 0.0101 0.0064 0.0064 0.0004 0.0040 0.0040 0.0002 0.0025 % script M-file problemset3_39.m 2; kappa = (3.29*10“9*2*C_salt“0.5); O U} {D }._l (‘f II x = linspace(0,10,ll)*10“(~10); psi = psi_0 * exp(—kappa*x); >> problemset3ﬁ3e ans = 0.0101 % script M~file problemsetB»3f.m psi_0 = 0.0257; 2 = l; C_salt = 2; kappa = (3.29*10“9*Z*C‘salt“0.5); x = linspace(0,lO,lOl)*10“(~10); psi = psi_0 * exp(—kappa*x); psi(50) >> problemset3_3f ans = 0.0026 lo. 9 xplu J 1" exp (4,0 Mg}. 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problem set 3 solutions - 1 BME C101/C201.‘ INTRODUCTION...

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