BME201_2010_HWK2b_soln

# BME201_2010_HWK2b_soln - NAME 125:201 INTRODUCTION TO...

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NAME:_______________________________ 1 125:201 INTRODUCTION TO BIOMEDICAL ENGINEERING, Fall 2010 Bioelectrical Phenomena Instructor: Nada Boustany HOMEWORK #2- Due THURSDAY Dec. 2 SOLUTIONS Problem 1: Relationships, variables and constants: σ : conductivity in ( .m) -1 = |z|.F.u.c Einstein relationship: D/u= (RT)/(|z| F) E= -d φ /dx J= Γ .z.F z = valence of the ion. c = concentration in moles/cm 3 u = mobility in cm 2 /volt/s D = diffusivity in cm 2 /s F is Faraday’s constant= 96,500 C/mole R is the gas constant = 8.31 J/mole/K T is the temperature in Kelvin (K). J is the current density in C/cm 2 /s φ is the electrical potential in Volts Γ is the molecular flux in moles/cm 2 /s ρ is the charge density in Coul/cm 3 A cell has a membrane, which is permeable only to K + . The concentrations of K+ are 5mM outside and 145mM inside. x 0 δ ------------ + + + + + + + + Outside inside [K + ] out = 5 mM [K + ] in = 145 mM ∆Φ = Φ (in)- Φ (out) J drift J diffusion

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NAME:_______________________________ 2 a. Using Ohm’s law, J drift = σ E, rewrite the electrical drift current, J drift , for K+ as a function of mobility, concentration, the electric potential and x. J K,drift = - |z K |.F.u K .c K .d Φ /dx b. Using Fick’s law, Γ diff = -D.dc/dx, derive the diffusion current, J diff , for K+ as a function of diffusivity, concentration and x. J K,diffusion = -D K .z K .F.dc K /dx c. Assuming that K+ ions are at equilibrium, write an expression relating J drift to J diff . At equilibrium: J drift + J diffusion =0 Æ -D K+ .z K+ .F.dc K+ /dx - |z K+ |.F.u K+ .c K+ .d Φ /dx= 0 d. Integrate the equation found in Part c and derive an expression for the Nernst potential for K+. The Nerst potential is the potential resulting from J drift + J diffusion =0 for a given ion at equilibrium. Starting from the differential equation found in Part c, separating the variables, c and Φ , and integrating both sides from the outside of the cell to the inside: in out dc K+ /c K+ = in out -[u K+ / D K+ ] [|z K+ | / z K+ ] d Φ ln[c K+ (in)] - ln[c K+ (out)] = -[u K+ / D K+ ] [|z K+ | / z K+ ] Φ (in) - Φ (out) ∆Φ K+, Nernst = Φ (in) - Φ (out)= - [D K+ / u K+ ] [|z K+ | / z K+ ] ln [c K+ (in) / c K+ (out)] Note that |z K+ | / z K+ = z K+ / |z K+ |. With the Einstein relationship: D i /u i = (RT)/(|z i |F) ∆Φ K+, Nernst = Φ (in) - Φ (out)= (RT/F) (1/ z K+ ) ln [c K+ (out) / c K+ (in)] e. Calculate the Nernst potential for K+. Assume the membrane is at 25 o C. ∆Φ K+, Nernst = Φ (in) - Φ (out) = [(8.31)(273+37)/(96,500)(1/ 1) ln[5 /145]= (0.0257)(-3.37)= -0.0865V= -86.5 mV
NAME:_______________________________ 3 f. Assuming that the electric field to be constant across the membrane, we can rewrite d Φ /dx as ∆Φ / δ . Thus, for 0 x ≤δ we can re-write: J diffusion = -D.z.F.dc/dx J drift = - |z|.F.u.c.( ∆Φ Nernst / δ  ) Based on these equations, show that the concentration is an exponential function of x for 0 x . At equilibrium: J drift + J diffusion =0 Æ -D K+ .z K+ .F.dc K+ /dx - |z K+ |.F.u K+ .c K+ .( ∆Φ / δ) = 0 x 0 dc K+ /c K+ = x 0 -[u K+ / D K+ ] [|z K+ | / z K+ ] ( ∆Φ / δ) dx ln[c K+ (x)] - ln[c K+ (0)] = -[u K+ / D K+ ] [|z K+ | / z K+ ]( ∆Φ / δ) x ln[c K+ (x)] = -[u K+ / D K+ ] [|z K+ | / z K+ ]( ∆Φ / δ) x + ln[c K+ (0)] x RT F K K c x z z D u K e c x c e x c K K K K K ) / ]( / [ )} 0 ( ln ) / ]( / | ][| / [ { ) 0 ( ) ( ) ( δ ∆Φ + ∆Φ + + + + + + + + = = g. Are the values of [K + ] out = 5mM and [K + ] in = 145mM consistent with the exponential function of x found in Part f above?

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## This note was uploaded on 10/11/2011 for the course BIOMEDICAL 201 taught by Professor Berth during the Fall '10 term at Rutgers.

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BME201_2010_HWK2b_soln - NAME 125:201 INTRODUCTION TO...

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