The membrane that surrounds a certain type of living cell has a surface area of 2.13 x 10-9 m2 and a thickness of 1.94 x 10-8 m. Assume that the membrane behaves like a parallel plate capacitor and has a dielectric constant of 6.4.
(a) If the potential on the outer surface of the membrane is +57.2 mV greater than that on the inside surface, how much charge resides on the outer surface?
(b) If the charge in part (a) is due to K+ ions (charge +e), how many such ions are present on the outer surface?
To find the charge residing on the outer surface of the membrane, we can use the capacitance formula for a parallel plate capacitor:
C = (ε₀ * εᵣ * A) / d
Where:
C is the capacitance,
ε₀ is the permittivity of free space (8.85 x 10⁻¹² F/m),
εᵣ is the dielectric constant,
A is the surface area of the membrane,
d is the thickness of the membrane.
(a) To find the charge on the outer surface:
1. Calculate the capacitance using the given values:
C = (8.85 x 10⁻¹² F/m * 6.4 * 2.13 x 10⁻⁹ m²) / (1.94 x 10⁻⁸ m)
2. Find the charge using the formula for capacitance:
Q = C * ΔV, where ΔV is the potential difference across the capacitor.
Since we know that the potential on the outer surface is +57.2 mV greater than that on the inside surface, ΔV = +57.2 mV.
Convert the potential difference to volts: ΔV = 57.2 x 10⁻³ V
Q = C * ΔV
(b) To find the number of K+ ions on the outer surface:
1. Determine the charge of a single K+ ion:
Since K+ ions have a charge of +e, where e is the elementary charge (1.60 x 10⁻¹⁹ C), the charge of a K+ ion is +e.
2. Divide the total charge calculated in part (a) by the charge of a single K+ ion to find the number of ions present on the outer surface.
Now let's calculate the values:
(a) Calculating the charge on the outer surface:
1. Calculate the capacitance:
C = (8.85 x 10⁻¹² F/m * 6.4 * 2.13 x 10⁻⁹ m²) / (1.94 x 10⁻⁸ m)
C ≈ 5.44 x 10⁻¹ F
2. Calculate the charge:
Q = C * ΔV
Q = (5.44 x 10⁻¹³ F) * (57.2 x 10⁻³ V)
Q ≈ 3.11 x 10⁻¹⁴ C
(b) Calculating the number of K+ ions:
1. Determine the charge of a single K+ ion:
Charge of a single K+ ion = +e = 1.60 x 10⁻¹⁹ C
2. Calculate the number of ions:
Number of ions = Q / charge of a single K+ ion
Number of ions = (3.11 x 10⁻¹⁴ C) / (1.60 x 10⁻¹⁹ C)
Now you can calculate part (b) to find the number of K+ ions on the outer surface of the membrane.