Nerve impulses are electrical "signals" that pass through neurons in the body. The electrical potential is created by the differences in the concentration of \(\mathrm{Na}^{+}\) and \(\mathrm{K}^{+}\) ions across the nerve cell membrane. We can think about this potential as being caused by a concentration gradient, similar to what we see in a concentration cell (keep in mind that this is a very simple explanation of how nerves work; there is much more involved in the true biologic process). A typical nerve cell has a resting potential of about \(-70 \mathrm{mV}\). Let's assume that this resting potential is due only to the \(\mathrm{K}^{+}\) ion concentration difference. In nerve cells, the \(\mathrm{K}^{+}\) concentration inside the cell is larger than the \(\mathrm{K}^{+}\) concentration outside the cell. Calculate the \(\mathrm{K}^{+}\) ion concentration ratio necessary to produce a resting potential of \(-70 . \mathrm{mV}\). $$\frac{\left[\mathrm{K}^{+}\right]_{\text {inside }}}{\left[\mathrm{K}^{+}\right]_{\text {outside }}}=?$$

Step by step solution

01

Convert mV to V

First, we need to convert the resting potential from mV to V. To do this, divide the given value by 1000: \(-70 mV = -70 \times 10^{-3} V = -0.07 V\)

02

Substitute the known values into the Nernst equation

Now, we substitute the known values into the Nernst equation: E=-0.07 V, R=8.314 J/mol*K, T=298 K, z=1, F=96,485 C/mol \(-0.07 = \frac{8.314 \times 298}{1\times 96,485} \ln \frac{[\mathrm{K}^{+}]_\text{outside}}{[\mathrm{K}^{+}]_\text{inside}}\)

03

Solve for the ion concentration ratio

Now, we solve for the ratio of potassium ion concentrations: \(\ln \frac{[\mathrm{K}^{+}]_\text{outside}}{[\mathrm{K}^{+}]_\text{inside}} = \frac{-0.07 \times 1 \times 96,485}{8.314 \times 298}\) Calculate the value on the right side of the equation: \(\frac{-0.07 \times 1 \times 96,485}{8.314 \times 298} \approx -0.299\) Now, take the exponent of both sides to remove the natural logarithm: \(\frac{[\mathrm{K}^{+}]_\text{outside}}{[\mathrm{K}^{+}]_\text{inside}} \approx e^{-0.299}\) Calculate the exponent: \(e^{-0.299} \approx 0.74\)

04

Write the final answer

The required potassium ion concentration ratio to produce a resting potential of -70 mV is approximately: \(\frac{[\mathrm{K}^{+}]_\text{inside}}{[\mathrm{K}^{+}]_\text{outside}} \approx \frac{1}{0.74} = 1.35\) Thus, the potassium ion concentration inside the nerve cell must be about 1.35 times the concentration outside the cell.

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