A simplified model for an erythrocyte is a spherical capacitor with a positive inner surface of area A and a membrane of thickness b. Use potential difference of 98 mV across the membrane and a dielectric constant K = 5.0. If the capacitance of the membrane is C = 0.27 pF, what is the radius of the erythrocyte? The thickness of the membrane is b = 90 nm.
To solve this problem, we can use the formula for the capacitance of a spherical capacitor:
C = 4πεRK / (1 - (R/b))
Where:
C is the capacitance
εR is the permittivity of the dielectric medium
K is the dielectric constant
R is the radius of the inner surface of the capacitor
b is the thickness of the membrane
First, convert the capacitance C from picofarads to farads:
C = 0.27 pF = 0.27 × 10^(-12) F
Next, substitute the given values into the formula:
0.27 × 10^(-12) = 4πεR(5.0) / (1 - (R / 90 × 10^(-9)))
Now, we can solve the equation to find the radius R. Rearranging the equation, we get:
1 - (R / 90 × 10^(-9)) = 4πεR(5.0) / 0.27 × 10^(-12)
Let's simplify this equation step by step.
Multiply both sides of the equation by 0.27 × 10^(-12):
(0.27 × 10^(-12))(1 - (R / 90 × 10^(-9))) = 4πεR(5.0)
Expand both sides of the equation:
0.27 × 10^(-12) - (0.27 × 10^(-12))(R / 90 × 10^(-9)) = 20πεR
Rearrange the equation to isolate the term including the radius:
(0.27 × 10^(-12))(R / 90 × 10^(-9)) + 20πεR = 0.27 × 10^(-12)
Now, factor out R from both terms:
[R ((0.27 × 10^(-12)) / (90 × 10^(-9))) + 20πε] R = 0.27 × 10^(-12)
Combine the constants:
[R (3 × 10^(-12)) + 20πε] R = 0.27 × 10^(-12)
Multiply out R:
(3 × 10^(-12))R^2 + (20πε)R - 0.27 × 10^(-12) = 0
Now, we have a quadratic equation:
(3 × 10^(-12))R^2 + (20πε)R - 0.27 × 10^(-12) = 0
To solve this quadratic equation, you can use the quadratic formula:
R = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, the values are:
a = (3 × 10^(-12))
b = (20πε)
c = (-0.27 × 10^(-12))
Substitute these values into the quadratic formula and solve for R. The positive value of R will give us the radius of the erythrocyte.