What is the time constant of a 9.0-nm-thick membrane surrounding a 0.040-mm-diameter spherical cell? Assume the resistivity of the cell memb

Question

What is the time constant of a 9.0-nm-thick membrane surrounding a 0.040-mm-diameter spherical cell? Assume the resistivity of the cell membrane as 3.6×107 Ω⋅m and the dielectric constant is approximately 9.0. Express your answer with the appropriate units.

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Kim Chi 5 years 2021-08-04T12:38:41+00:00 1 Answers 216 views 0

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  1. Tryphena
    0
    2021-08-04T12:39:52+00:00
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    Given Information:

    Diameter of spherical cell = 0.040 mm

    thickness = L = 9 nm

    Resistivity =  ρ = 3.6×10⁷ Ω⋅m

    Dielectric constant = k = 9.0

    Required Information:

    time constant = τ = ?

    Answer:

    time constant = 2.87×10⁻³ seconds

    Explanation:

    The time constant is given by

    τ = RC

    Where R is the resistance and C is the capacitance.

    We know that resistivity of of any material is given by

    ρ = RA/L

    R =  ρL/A

    Where area of spherical cell is given by

    A = 4πr²

    A = 4π(d/2)²

    A = 4π(0.040×10⁻³/2)²

    A = 5.026×10⁻⁹ m²

    The resistance becomes

    R =  (3.6×10⁷*9×10⁻⁹)/5.026×10⁻⁹

    R = 6.45×10⁷ Ω

    The capacitance of the cell membrane is given by

    C = kεoA/L

    Where k = 9 is the dielectric constant and εo = 8.854×10⁻¹² F/m

    C = (9*8.854×10⁻¹²*5.026×10⁻⁹)/9×10⁻⁹

    C = 44.5 pF

    C = 44.5×10⁻¹² F

    Therefore, the time constant is

    τ = RC

    τ = 6.45×10⁷*44.5×10⁻¹²

    τ = 2.87×10⁻³ seconds

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