The concentration of potassium ions inside a nerve cell is around 20 to 30 times that outside the cell. Calculate the potential difference between inside and outside a nerve cell, if it depends only on this ionic imbalance.
This problem is worked the same way as the concentration cell problem I worked last night on iron rusting.
To calculate the potential difference between the inside and outside of a nerve cell, we can use the Nernst equation. The Nernst equation relates the potential difference (E) with the concentration gradient of a specific ion.
The Nernst equation is given as follows:
E = (RT/zF) * ln([C_out]/[C_in])
Where:
E is the potential difference (in volts),
R is the ideal gas constant (8.314 J/(mol·K)),
T is the temperature in Kelvin,
z is the valence of the ion (in this case, +1 for potassium),
F is the Faraday constant (96,485 C/mol),
[C_out] is the concentration of potassium ions outside the cell (in moles per liter),
[C_in] is the concentration of potassium ions inside the cell (in moles per liter), and
ln represents the natural logarithm.
Given that the concentration of potassium ions outside the cell is 20 to 30 times less than inside the cell, we can assume an average ratio of 25:1 for simplicity.
Let's assume the concentration of potassium ions outside the cell is 1 M. Therefore, the concentration inside the cell would be 25 M.
Plugging these values into the Nernst equation and assuming a physiological temperature of 37°C (or 310 K), we can calculate the potential difference:
E = (8.314 J/(mol·K) * 310 K / (1 * 96,485 C/mol)) * ln(1/25)
E = 0.025 V * ln(0.04)
Using a scientific calculator, we find ln(0.04) ≈ -3.2188.
E ≈ 0.025 V * -3.2188
E ≈ -0.080 V
Therefore, the potential difference between inside and outside a nerve cell, due to the concentration imbalance of potassium ions, is approximately -0.080 V. Note that the negative sign indicates the inside of the cell is relatively negative compared to the outside.