Determine the number of uncompensated ions (both total number and as a
percentage of the bulk ions) required to establish a charge of 100mV across a
membrane of a spherical cell that is 25 ƒÊm in diameter. Assume that the total number
of ions inside and outside the cell is .5M (or 500 mM)
Recall the capacitance of a patch of membrane is 1 ƒÊF/cm2
Recall that a Farad is the charge (in coulombs) required to establish 1 V of potential, so
the capacitance of a patch of membrane is 10-6 C/ V cm2 .
Also recall that the elemental charge (the coloumbs on 1 ion) is 1.6x10^-19 C
Well, let's take a humorous approach to this scientific question!
If we're talking about uncompensated ions, we're looking at the ions that are responsible for creating the charge across the membrane. They're like the rebellious teenagers of the ion world, refusing to follow the rules and creating all this electrical mischief.
Now, we know the diameter of the spherical cell is 25 ƒÊm. That's like the size of a tiny sprinkle on a cupcake. So, let's imagine this cell as a tiny cupcake just floating around in a sea of ions.
Given that the total number of ions inside and outside the cell is 500 mM (or 0.5M), we can start solving this puzzle. But first, we need to know how many ions are needed to establish a charge of 100mV across the membrane.
The capacitance of a patch of membrane is 1 ƒÊF/cm2, which basically tells us how much the membrane can hold in terms of charge. It's like the maximum number of ions the membrane can juggle without dropping any.
Now, since we're dealing with a spherical cell, the surface area of the membrane would be 4πr^2, where r is the radius. In this case, the radius would be half of the diameter, so 25/2 = 12.5 ƒÊm.
Let's convert that to cm because we're working with capacitance in cm2. So, 12.5 ƒÊm is equal to 0.00125 cm.
Now, let's plug the values into our formula: Capacitance = 10^-6 C/V cm2 * (4π * 0.00125 cm2) = 1.57 x 10^-8 C/V
We can rearrange the formula, considering that 1F is equal to 1C/V, to find the charge needed to establish a potential of 100mV.
Charge = Capacitance * Potential = (1.57 x 10^-8 C/V) * 0.1 V = 1.57 x 10^-9 C
Now we know the charge required, which is 1.57 x 10^-9 C. But wait, we also know that the elemental charge (the coloumbs on 1 ion) is 1.6x10^-19 C.
So, to find the number of ions needed, we divide the required charge by the elemental charge: Number of ions = (1.57 x 10^-9 C) / (1.6 x 10^-19 C) = 9.81 x 10^9 ions
Voila! We have the number of uncompensated ions needed to establish a charge of 100mV across the membrane. It's around 9.81 billion ions. That's a whole lot of rebellious teenagers causing electrical commotion inside and outside the cell!
As for the percentage of uncompensated ions compared to the bulk ions, we'll have to know the total number of bulk ions. Unfortunately, that information is missing from your question. But hey, let's throw in some clown magic and say the total number of bulk ions is 10 trillion. In that case, the percentage of uncompensated ions would be around 0.0981%.
Hope that answered your question with a touch of humor!
To determine the number of uncompensated ions required to establish a charge of 100mV across the membrane of a spherical cell, we can follow these steps:
1. Calculate the surface area of the membrane:
The formula for the surface area of a sphere is A = 4πr^2. Since the diameter of the cell is given as 25 μm, the radius (r) would be half of that, which is 12.5 μm or 1.25 x 10^-3 cm. Plugging this value into the formula, we get:
A = 4π(1.25 x 10^-3 cm)^2
A = 4π(1.5625 x 10^-6 cm^2)
A ≈ 1.962 x 10^-5 cm^2
2. Calculate the total charge required to establish an electric potential of 100mV (0.1V):
Recall that the capacitance of the membrane is given as 1 μF/cm^2. So, the charge (Q) required can be calculated using the formula Q = CV, where C is the capacitance and V is the electric potential.
Q = (1 x 10^-6 C/V cm^2) * (0.1 V) * (1.962 x 10^-5 cm^2)
Q ≈ 1.962 x 10^-12 C
3. Calculate the number of ions required:
Since the elemental charge on 1 ion is given as 1.6 x 10^-19 C, we can divide the total charge required by this elemental charge to get the number of ions:
Number of ions = (1.962 x 10^-12 C) / (1.6 x 10^-19 C)
Number of ions ≈ 1.22625 x 10^7 ions
Therefore, the total number of uncompensated ions required to establish a charge of 100mV across the membrane is approximately 1.22625 x 10^7 ions.
To determine the percentage of uncompensated ions relative to the bulk ions, we need to calculate the total number of bulk ions. Since the concentration both inside and outside the cell is 500 mM (millimolar), we can use Avogadro's number (6.022 x 10^23 ions/mol) to calculate the total number of bulk ions.
4. Calculate the total number of bulk ions:
Bulk ions = (500 mM) * (1 x 10^-3 L/mol) * (6.022 x 10^23 ions/mol)
Bulk ions ≈ 3.011 x 10^20 ions
5. Calculate the percentage of uncompensated ions:
Percentage of uncompensated ions = (Number of uncompensated ions / Total number of bulk ions) * 100
Percentage of uncompensated ions = (1.22625 x 10^7 ions / 3.011 x 10^20 ions) * 100
Percentage of uncompensated ions ≈ 4.074 x 10^-3 %
Therefore, the number of uncompensated ions required is approximately 1.22625 x 10^7 ions, and this represents around 0.004074% of the total number of bulk ions.