Question

A cell membrane can be modeled as a capacitor. The thickness of

membrane is 7.0 nm.
How much energy is stored in a 16μm diameter spherical cell that
has a membrane potential of -90 mV?
For comparison

Answer 1

The energy stored in the spherical cell's membrane, modeled as a capacitor with a thickness of 7.0 nm and a membrane potential of -90 mV, is approximately
`1.52 × 10^(-16)` joules.

Step-by-step explanation:

The capacitance for a spherical capacitor is given by
`\(C = (4\pi\varepsilon R)/(d)\)` Substituting the known values, we get
`\(C = \frac{4\pi(8 * 10^(-6) \, \text{m})}{7 * 10^(-9) \, \text{m}}\).`

Next, using the energy formula
`\(U = (1)/(2) C V^2\)`, we substitute
`\(V = -90 * 10^(-3) \, \text{V}\)`to calculate the energy stored:
`\(U = (1)/(2) \left(\frac{4\pi(8 * 10^(-6) \, \text{m})}{7 * 10^(-9) \, \text{m}}\right) * (-90 * 10^(-3) \, \text{V})^2\)`. This yields an energy of approximately 1.52 ×
`10^(-16)` joules.

This calculated value represents the electrostatic energy stored in the cell membrane, reflecting the potential energy associated with the separation of charges across the membrane. It serves as a measure of the electrical storage capacity of the membrane, contributing to the understanding of biological systems and their electrical properties.

Answer 2

The energy stored in the cell is approximately 2.556 * 10^-16 joules.

Step-by-step explanation:

The energy stored in a cell can be calculated by using the formula: E = 0.5 * C * V^2, where E is the energy stored, C is the capacitance, and V is the voltage.

Since the cell membrane is being modeled as a capacitor, we can use the equation for the capacitance of a spherical capacitor: C = (4 * pi * epsilon * r) / d, where C is the capacitance, pi is a mathematical constant, epsilon is the permittivity of the medium between the capacitor plates, r is the radius of the capacitor plates, and d is the distance between the plates.

Plugging in the given values, we have: C = (4 * pi * epsilon * r) / d = (4 * 3.14 * 8.854 * 10^-12 * 8 * 10^-6) / 7 * 10^-9.

Now, we can substitute the value of C and the voltage (-90 mV) into the energy equation to calculate the energy stored in the cell: E = 0.5 * C * V^2 = 0.5 * (4 * 3.14 * 8.854 * 10^-12 * 8 * 10^-6) / 7 * 10^-9 * (-90 * 10^-3)^2 = 2.556 * 10^-16 J.

Therefore, the energy stored in the cell is approximately 2.556 * 10^-16 joules.