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.

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