Question

A cell membrane can be modeled as a capacitor. The thickness of membrane is 7.0 nm. Part A How much energy is stored in a 16-μm-diameter spherical cell that has a membrane potential of -90 mV? Express your answer with the appropriate units. ANSWER: U= Part B For comparison, how many ATP molecules need to undergo hydrolysis to generate this much energy? In a cell, the energy released from the hydrolysis of ATP is approximately 60 kJ/mol. ANSWER: NATP =

Answer 1

The energy stored in the spherical cell's membrane can be calculated using the formula for the energy stored in a capacitor. By substituting the given values into the formula, the energy stored is approximately 1.03 * 10^{-11} J.

Step-by-step explanation:

A cell membrane can be modeled as a capacitor. The energy stored in a capacitor is given by the formula U = 1/2 * C * V^2, where C is the capacitance and V is the potential difference. The capacitance of a spherical capacitor is given by C = 4πε0 * (r1 * r2) / (r2 - r1), where r1 and r2 are the radii of the inner and outer surfaces of the capacitor, and ε0 is the permittivity of free space.

In this case, the radius of the spherical cell is 16 μm / 2 = 8 μm = 8 * 10^-6 m. The capacitance of the cell membrane can be calculated using the given thickness and the radius:

C = 4π * ε0 * (r1 + d/2) * (r2 + d/2) / d

Substituting the values, we get C ≈ 2.27 * 10^-11 F.

Now, we can calculate the energy stored in the capacitor:

U = 1/2 * C * V^2 = 1/2 * 2.27 * 10^{-11} * (-90 * 10^-3)^2 ≈ 1.03 * 10^{-11} J.

Answer 2

Main Answer:

A. The energy stored in a 16-μm-diameter spherical cell with a membrane potential of -90 mV is approximately
`\(1.14 * 10^(-9)\)` joules.

B. For comparison, the number of ATP molecules needed to generate this energy is approximately
`\(1.90 * 10^(11)\)` ATP molecules.

Step-by-step explanation:

The energy stored in the cell membrane, modeled as a capacitor, can be calculated using the formula
`\(U = (1)/(2) C V^2\), where \(U\)` is the energy,
`\(C\)` is the capacitance, and
`\(V\)` is the voltage. The capacitance of a spherical capacitor (cell) is given by
`\(C = (4 \pi \varepsilon r)/(d)\),`where \(r\) is the radius and \(d\) is the separation between the plates (membrane thickness).

For Part A:

`\[ C = \frac{4 \pi \varepsilon (8 * 10^(-6) \, \text{m})}{7 * 10^(-9) \, \text{m}} \]\[ U = (1)/(2) \left( \frac{4 \pi \varepsilon (8 * 10^(-6) \, \text{m})}{7 * 10^(-9) \, \text{m}} \right) \left( -90 * 10^(-3) \, \text{V} \right)^2 \]\[ U \approx 1.14 * 10^(-9) \, \text{joules} \]`

For Part B:

The energy released from the hydrolysis of ATP is approximately
`\(60 \, \text{kJ/mol}\).` To find the number of moles needed, divide the energy stored in the cell membrane by the energy per mole of ATP.

`\[ \text{NATP} = \frac{1.14 * 10^(-9) \, \text{joules}}{60 * 10^3 \, \text{joules/mol}} \]\[ \text{NATP} \approx 1.90 * 10^(11) \, \text{ATP molecules} \]`

In summary, a spherical cell with a membrane potential of -90 mV stores approximately
`\(1.14 * 10^(-9)\)` joules of energy, equivalent to the hydrolysis of about
`\(1.90 * 10^(11)\)` ATP molecules. This highlights the significance of ATP as the cellular energy currency.