Question

In: Physics

A cell membrane has a resistance and a capacitance and thus a characteristic time constant.

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 ??m and the dielectric constant is approximately 9.0.

 

Solutions

Expert Solution

Concepts and reasons The concepts required to solve the problem are resistance, capacitance, and the time constant of an RC circuit. First, using the area and thickness, find the resistance and capacitance of the cell membrane. Then, use the relation between resistance and capacitance, and find the time constant of the cell membrane.

Fundamentals

“The capacitance of a capacitor is directly proportional to the dielectric constant and the area, and is inversely proportional to the thickness". The expression for the capacitance of parallel plate capacitor is, \(C=\frac{k \varepsilon_{0} A}{d}\)

Here, \(k\) is the dielectric constant, \(\varepsilon_{0}\) is the permittivity of free space, \(A\) is the area, and \(d\) is the separation distance of the plate. The resistance of a material is directly proportional to the length and inversely proportional to the area. The expression for the resistance of the material is, \(R=\frac{\rho L}{A}\)

Here, \(\rho\) is the resistivity, \(L\) is the length, \(\mathrm{R}\) is the resistance, and \(A\) is the cross-section. The time constant for an RC circuit is, \(\tau=R C\)

Here, \(R\) is the resistance, and \(C\) is the capacitance.

The area of the cell membrane is, \(A=4 \pi r^{2}\)

Here, \(r\) is the radius of the cell.

Replace \(d / 2\) for \(r\) in the above equation. \(A=4 \pi\left(\frac{d}{2}\right)^{2}\)

Here, \(\mathrm{d}\) is the diameter of the cell. Substitute \(0.040 \mathrm{~mm}\) for \(d\). \(A=4 \pi\left[\left(\frac{0.040 \mathrm{~mm}}{2}\right)\left(\frac{10^{-3} \mathrm{~m}}{1 \mathrm{~mm}}\right)\right]^{2}\)

\(=5.03 \times 10^{-9} \mathrm{~m}^{2}\)

The equation for capacitance is, \(C=\frac{k \varepsilon_{0} A}{d}\)

Substitute \(9.0\) for \(k, 8.854 \times 10^{-12} \mathrm{~F} / \mathrm{m}\) for \(\varepsilon_{0}, 5.03 \times 10^{-9} \mathrm{~m}^{2}\) for \(A\), and \(9.0 \mathrm{~nm}\) for \(d\). \(C=\frac{(9.0)\left(8.854 \times 10^{-12} \mathrm{~F} / \mathrm{m}\right)\left(5.03 \times 10^{-9} \mathrm{~m}^{2}\right)}{(9.0 \mathrm{~nm})\left(\frac{10^{-9} \mathrm{~m}}{1 \mathrm{~nm}}\right)}\)

\(=4.45 \times 10^{-11} \mathrm{~F}\)

The resistance of the cell membrane is, \(R=\frac{\rho L}{A}\)

Substitute \(3.6 \times 10^{7} \Omega \mathrm{m}\) for \(\rho, 5.03 \times 10^{-9} \mathrm{~m}^{2}\) for \(A\) and \(9.0 \mathrm{~nm}\) for \(L\). \(R=\frac{\left(3.6 \times 10^{7} \Omega \mathrm{m}\right)(9.0 \mathrm{~nm})\left(\frac{10^{-9} \mathrm{~m}}{1 \mathrm{~nm}}\right)}{5.03 \times 10^{-9} \mathrm{~m}^{2}}\)

\(=6.44 \times 10^{7} \Omega\)

The RC circuit has both a capacitor and resistor. The capacitance of the cell membrane depends on the dielectric constant, the absolute permittivity, and the area of the cross-section of the material. The resistance of the cell membrane is found from the length and the area of the cross-section.

The equation for the time constant is, \(\tau=R C\)

Substitute \(6.44 \times 10^{7} \Omega\) for \(R\) and \(4.45 \times 10^{-11} \mathrm{~F}\) for \(C\). \(\tau=\left(6.44 \times 10^{7} \Omega\right)\left(4.45 \times 10^{-11} \mathrm{~F}\right)\)

\(=2.87 \times 10^{-3} \mathrm{~s}\)

\(=2.87 \mathrm{~ms}\)

For RC circuits, the time constant is found from the resistance and the capacitance. The value of the time constant is obtained in milliseconds. The time constant is "the time required to charge the capacitor through the resistor from zero to \(63.2 \%\) of the applied DC voltage".


Thus, the time constant of the membrane is \(2.87 \mathrm{~ms}\).

Related Solutions

A cell membrane has a resistance and a capacitance and thus a characteristic time constant. What...
A cell membrane has a resistance and a capacitance and thus a characteristic time constant. What is the time constant of a 8.3 nm -thick membrane surrounding a 3.6×10−2 mm -diameter spherical cell? Assume the resistivity of the cell membrane as 3.6×106 Ω⋅m and the dielectric constant is approximately 9.0.
A cell membrane has a resistance and a capacitance and thus a characteristic time constant Part...
A cell membrane has a resistance and a capacitance and thus a characteristic time constant Part A What is the time constant of a 7.7 nm -thick membrane surrounding a 4.0×10−2 mm -diameter spherical cell? Assume the resistivity of the cell membrane as 3.6×106 Ω⋅m and the dielectric constant is approximately 9.0. Express your answer to two significant figures and include the appropriate units.
What supports the notion that the product of resistance and capacitance is the time constant for...
What supports the notion that the product of resistance and capacitance is the time constant for an RC circuit?
The membrane of a living cell is an insulator that separates two conducting fluids. Thus, it...
The membrane of a living cell is an insulator that separates two conducting fluids. Thus, it functions as a capacitor. The membrane is not a perfect insulator, however. It has a small conductance, making it a leaky capacitor. In this problem, you will estimate the RC time constant of the cell membrane. (a) A cell membrane typically has a capacitance per unit area on the order of 1 μF/cm2 — i.e., 1 cm2 of the membrane material would have a...
A coil (resistance Rc), conductor (capacitance C) and a resistor (resistance Rr) are connected in a...
A coil (resistance Rc), conductor (capacitance C) and a resistor (resistance Rr) are connected in a series a circuit with sinusoidal alternating current (amplitude A) power supply. Calculate the theoretical current in the circuit by using Kirchhoff's 2nd law. Use a sinusoidal trial function for the current to solve the differential equation. Prove, that the sinusoidal current's amplitude depends heavily on the frequency.
For a resistance and capacitance in series with a voltage source, show that it is possible...
For a resistance and capacitance in series with a voltage source, show that it is possible to draw a phasor diagram for the current and all voltages from magnitude measurement of these quantities only.Ilustrate your answer graphieally
36. Antimicrobial resistance mechanisms include A. removing efflux pumps from the cell membrane. B. modifying or...
36. Antimicrobial resistance mechanisms include A. removing efflux pumps from the cell membrane. B. modifying or reducing the amount of the target of the drug. C. inhibition of protein synthesis in the microbe. D. preventing attachment of the microbe. E. All of the above are antimicrobial resistance mechanisms. 37. Which of the following is TRUE about antibiotic interaction? A. Taking two antibiotics together may neutralize the effects of both. B. Synergism occurs when two antibiotics work better together than alone....
Describe the structure of the cell membrane. Explain how the characteristics of the cell membrane are...
Describe the structure of the cell membrane. Explain how the characteristics of the cell membrane are ideal for its function.
A RLC series circuit has a resistance of 104 ohms, inductance 0.7 H,and Capacitance 6.0 µF....
A RLC series circuit has a resistance of 104 ohms, inductance 0.7 H,and Capacitance 6.0 µF. If the amplitude of the voltage supplied by the ac source is 170 V, and the frequency is 60 Hertz, calculate the rms current.
Consider an RC circuit with resistance R and capacitance C. The circuit is stimulated with a...
Consider an RC circuit with resistance R and capacitance C. The circuit is stimulated with a pulse of amplitude A and width T. The purpose of this study is to understand what happens to the impulse response, capacitor voltage, and resistor voltage for various resistor values: 600 ?, 1000 ?, and 1200 ?. The name of the MATLAB script will be called project2a. Excite the circuit with a rectangular pulse voltage of amplitude=5 V and pulse width of 10 ms....
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT