A model of a red blood cell portrays the cell as a spherical capacitor, a positi

Question

A model of a red blood cell portrays the cell as a spherical capacitor, a positively charged liquid sphere of surface area A separated from the surrounding negatively charged fluid by a membrane of thickness t. Tiny electrodes introduced into the interior of the cell show a potential difference of 100 mV across the membrane. The membrane’s thickness is estimated to be 100 nm and has a dielectric constant of 5.00. A) if an average red blood cell has a mass of 1.00 x 10-12 kg, estimate the volume of the cell and thus find its surface area. The density of blood is 1100 kg/m3. B) Estimate the capacitance of the cell by assuming the membrane surfaces act as parallel plates. C) Calculate the charge on the surface of the membrane. How many electronic charges does the surface charge represent?

** The answers at the back of the book are: A) volume 9.09 x 10-16 m3, area 4.54 x 10-10 m2, b) 2.01 x 10-13 F, c) 2.01 x 10-14 C, 1.26x 105 electronic charges.

Please provide full solution.

Explanation / Answer

Part A)

Volume is found via the density formula = m/V

S0 V = m/

V = (1 X 10-12)/1100

V = 9.09 X 10-16 m3

Using that volume, we can find the radius of the blood cell sphere

The formula for the Volume of a sphere is V = 4/3(r3)

Solve for r

r = (3V/4)1/3

r = 6.01 X 10-6 m

Surface area of a sphere is found by the formula A = 4r2

So A = 4()(6.01 X 10-6)2

A = 4.54 X 10-10 m2

Part B)

For capacitance apply the formula C = kA/d

k = 5 and = 8.85 X 10 X 10-12 C2/Nm2

d = 100 nm

C = (5)(8.85 X 10-12)(4.54 X 10-10)/(100 X 10-9)

C = 2.01 X 10-13 F

Part C)

To find charge, we use Q = CV

Q = (2.01 X 10-13)(100 X 10-3)

Q = 2.01 X 10-14 C

To find the number of charges that involves, we know that one elememtal charge has a value of

1.6 X 10-19 C

Therefore the number of charges will be total charge divided by elemental charge

2.01 X 10-14/1.6 X 10-19

Total number of charges is 1.26 X 105

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