(a) To find the resistivity of the fluid in the pore, we can use the formula for resistance:
Resistance = Resistivity × Length / Area
We are given the resistance of the ion channel (1× 10^11 Ω) and the length of the channel (12 nm). The area of the channel can be calculated using the formula for the area of a cylinder:
Area = π × radius²
The radius of the channel is given as 0.3 nm. Plugging these values into the equation, we can solve for the resistivity:
Resistivity = (Resistance × Area) / Length
Resistivity = (1× 10^11 Ω × π × (0.3 nm)²) / (12 nm)
Solving this equation will give you the value of the resistivity of the fluid in the pore.
(b) To find the number density of open ion channels in the membrane, we can use the formula for current density:
Current Density = Current / Area
We are given the current density (5 mA/cm^2) and the voltage across the membrane (50 mV). The current can be calculated using Ohm's law:
Current = Voltage / Resistance
We are also given the resistance of the ion channel (1× 10^11 Ω). Plugging these values into the equation, we can solve for the current.
Once we have the current, we can rearrange the formula for current density to find the number density of open ion channels:
Number Density = Current / (Current Density × Area)
The area in this case is given as 1/μm². Plugging in the values will give you the number density of open ion channels in the membrane.
(c) To find the time constant required when the ion channels are open, we can use the formula for the time constant:
Time Constant = Capacitance × Resistance
We are given the capacitance per unit area (1 μF/cm²) and the resistance of the ion channel (1×10^11 Ω).
The area in this case is given as 100/μm². Plugging in the values will give you the time constant required when the ion channels are open.