The Nernst Equation and Membrane Potential

The equilibrium potential (often denoted as $V$ or $E_{ion}$) is the specific membrane voltage at which these two opposing forces perfectly cancel each other out. At this voltage, there is no net movement of that specific ion across the membrane.

The Nernst Equation

The Nernst equation is the mathematical tool used to calculate this equilibrium potential for a single ion species. It accounts for the concentration gradient, the temperature, and the physical properties of the ion itself.

The general formula is expressed as:

$$V = \frac{RT}{zF} \ln \frac{C_{ext}}{C_{int}}$$

To use this equation, you must understand its constituent variables:

• $V$: The equilibrium potential (measured in Volts).
• $R$: The gas constant (approximately $2 \text{ cal mol}^{-1} \text{ K}^{-1}$).
• $T$: The absolute temperature (measured in Kelvin).
• $z$: The ion valence, which represents the electrical charge of the ion (e.g., $+1$ for $\text{K}^+$, $-1$ for $\text{Cl}^-$).
• $F$: The Faraday constant (approximately $2.3 \times 10^4 \text{ cal V}^{-1} \text{ mol}^{-1}$).
• $C_{ext}$ and $C_{int}$: The concentration of the ion outside and inside the cell, respectively.

Exam Tip: Pay close attention to the valence ($z$). If you are calculating the potential for a cation like $\text{K}^+$, $z$ is $+1$. If you are calculating for an anion like $\text{Cl}^-$, $z$ is $-1$. This sign change is crucial for determining whether the equilibrium potential will be positive or negative.

Biological Application: The Potassium Ion ($\text{K}^+$)

In most human cells, the resting membrane potential is primarily determined by the movement of the $\text{K}^+$ ion (potassium). This is because the cell membrane at rest is highly permeable to potassium, allowing it to be the "principal ion" that dictates the electrical state of the cell.

Because $\text{K}^+$ concentration is much higher inside the cell than outside, $\text{K}^+$ tends to leak out of the cell. As these positive ions leave, they leave behind negative charges, creating a negative resting potential. For a typical human cell, this resulting potential usually falls between $-60 \text{ mV}$ and $-90 \text{ mV}$.

The Simplified Equation

In biological contexts, where temperature is relatively constant (body temperature), scientists often use a simplified version of the Nernst equation specifically for animal membranes where $\text{K}^+$ is the main ion:

$$V_{(en \text{ mV})} = 58 \log \frac{[Ext]}{[Int]}$$

This version uses the base-10 logarithm and combines the constants ($R, T, F$) into a single coefficient (58) to make rapid calculations easier for students and researchers.

Summary Table of Variables

By mastering the Nernst equation, you gain the ability to predict how changes in ion concentrations (such as those occurring during disease or metabolic shifts) will directly impact the electrical excitability of a cell.