Nernst Equation Calculator: Cell Membrane Potential

What Is the Nernst Equation Calculator?

The Nernst equation calculator finds the equilibrium potential of a single ion across a cell membrane, or the potential of an electrochemical cell. It uses the ion's charge, the temperature, and the ion's concentration on each side of the membrane. The result is a voltage, usually shown in millivolts (mV).

The equation is named after German chemist Walther Nernst, who described it in 1889. In biology it explains why cells hold an electrical charge across their membrane, which is needed for nerve signals, muscle contraction, and other processes that depend on ion movement.

The Nernst Equation Formula

$E = E^{\circ} - \frac{RT}{zF} \ln\left(\frac{[C]_{\text{inside}}}{[C]_{\text{outside}}}\right)$

Where:

• $E$ = the cell or membrane potential, in volts
• $E^{\circ}$ = the standard potential, usually taken as 0 for a biological membrane
• $R$ = the gas constant, 8.314 J·mol⁻¹·K⁻¹
• $T$ = temperature in kelvin
• $z$ = the ion's charge (its valence), such as +1 for Na⁺ or -1 for Cl⁻
• $F$ = the Faraday constant, 96,485 C·mol⁻¹
• $[C]_{\text{inside}}$, $[C]_{\text{outside}}$ = the ion's concentration inside and outside the cell

For biology, $E^{\circ}$ is usually set to 0 and the result is converted to millivolts. Swapping the ratio to outside-over-inside removes the leading minus sign:

$E = \frac{RT}{zF} \ln\left(\frac{[C]_{\text{outside}}}{[C]_{\text{inside}}}\right) \times 1000 \text{ mV}$

The variables, explained

Temperature (T). Measured in kelvin, where K = °C + 273.15. Body temperature is 310.15 K (37 °C). Room temperature is 298.15 K (25 °C).

Ion charge (z). Sodium (Na⁺) and potassium (K⁺) are +1. Calcium (Ca²⁺) and magnesium (Mg²⁺) are +2. Chloride (Cl⁻) is -1.

Concentrations. The amount of the ion on each side of the membrane, usually in millimolar (mM). Both values must use the same unit and both must be greater than zero.

Gas constant and Faraday constant. Fixed physical constants: R = 8.314 J/(mol·K) and F = 96,485 C/mol.

How to Calculate Membrane Potential

1. Enter the temperature in kelvin. The default is body temperature, 310.15 K.
2. Enter the ion's charge, such as 1 for potassium or -1 for chloride.
3. Enter the concentration outside the cell, in mM.
4. Enter the concentration inside the cell, in mM.
5. Read the result in millivolts, shown automatically as the values change.

Worked Example: Potassium

Potassium (K⁺) sits at about 5 mM outside a typical cell and 140 mM inside. At body temperature, with charge z = +1:

$E = \frac{8.314 \times 310.15}{1 \times 96{,}485} \times \ln\left(\frac{5}{140}\right) \times 1000$

$E \approx 0.02673 \text{ V} \times (-3.332) \times 1000 \approx -89.1 \text{ mV}$

The calculator itself uses more precise physical constants and returns -89.06 mV for these inputs. The result is negative because the concentration is much higher inside the cell than outside, so the natural log term is negative.

Typical Ion Potentials at Body Temperature

These are commonly cited textbook concentrations for a mammalian cell, all calculated at 310.15 K:

How Temperature Affects the Result

Because temperature appears directly in the equation, a warmer system produces a larger potential (in either direction). Using potassium's values (5 mM outside, 140 mM inside, z = +1):

• At 25 °C (298.15 K): -85.61 mV
• At 37 °C (310.15 K): -89.06 mV
• At 42 °C (315.15 K): -90.49 mV

Each 10 °C rise increases the size of the potential by roughly 3%, since the equation scales directly with absolute temperature.

What a Positive or Negative Result Means

The result is the voltage of the inside of the cell, relative to the outside, at which the ion has no net flow across the membrane.

• For a positive ion, a negative result means the ion is more concentrated inside (like K⁺), and a positive result means it is more concentrated outside (like Na⁺ and Ca²⁺).
• For a negative ion such as Cl⁻, the signs are reversed.
• A result of zero means the concentrations on both sides are equal.

Which way the ion actually flows depends on the cell's real membrane voltage. The further that voltage is from the ion's Nernst potential, the stronger the pull on the ion.

Where the Nernst Equation Is Used

In biology, it is used to work out the resting and action potentials of neurons, the electrical behavior of heart cells, and ion movement in muscle and kidney tissue. Because real cells respond to several ions at once rather than one in isolation, researchers studying overall resting membrane potential often extend the Nernst equation using the Goldman-Hodgkin-Katz equation, which accounts for multiple ions and their different permeabilities.

In chemistry and engineering, the same equation describes the voltage produced by batteries and fuel cells, predicts metal corrosion, and underlies ion-selective sensors used to measure water and soil quality.

A Short History

Walther Nernst formulated the equation in 1889 while working on electrochemical cells at the University of Leipzig. He won the 1920 Nobel Prize in Chemistry for related work in thermochemistry. In the 1940s and 1950s, Alan Hodgkin and Andrew Huxley applied Nernst's principles to nerve cells, work that led to their own Nobel Prize and to the Goldman-Hodgkin-Katz equation used in neuroscience today.

Frequently Asked Questions

What does the Nernst equation calculate? It calculates the equilibrium potential of a single ion across a membrane, or the potential of an electrochemical cell, based on concentration, charge, and temperature.

What units does the calculator use? Temperature is in kelvin, concentrations are typically in millimolar, and the result is in millivolts. Any concentration unit works as long as both sides use the same one.

What happens when the two concentrations are equal? The ratio of the concentrations becomes 1, the natural log of 1 is 0, and the calculated potential is 0 mV. This is electrochemical equilibrium.

Why is the potassium potential negative but the sodium potential positive? Because potassium is far more concentrated inside the cell than outside, while sodium is more concentrated outside than inside. Whether the ratio inside the log term is above or below 1 sets the sign of the result.

How is the Nernst equation different from the Goldman-Hodgkin-Katz equation? The Nernst equation covers one ion at a time. The Goldman-Hodgkin-Katz equation combines several ions, weighted by how easily each crosses the membrane, to estimate a cell's actual resting potential.

Does the Nernst equation account for active transport? No. It describes passive movement down an electrochemical gradient only. Pumps that use energy (such as ATP) to move ions against their gradient are not part of this equation.

References

1. National Institute of Standards and Technology. CODATA value: molar gas constant. NIST CODATA R — R = 8.314 462 618 J mol⁻¹ K⁻¹.
2. National Institute of Standards and Technology. CODATA value: Faraday constant. NIST CODATA F — F = 96 485.332 12 C mol⁻¹.
3. National Institute of Standards and Technology. SI Units – Temperature. NIST temperature units — 0 °C equals 273.15 K.
4. The Nobel Prize. Walther Nernst – Biographical. Nobel Prize biography — in 1889 Nernst set out the theory of galvanic cells while working with Wilhelm Ostwald at Leipzig.
5. The Nobel Prize. Walther Nernst – Facts. Nobel Prize facts — 1920 Nobel Prize in Chemistry for his work in thermochemistry.
6. The Nobel Prize. The Nobel Prize in Physiology or Medicine 1963. Nobel Prize 1963 summary — Eccles, Hodgkin and Huxley, for discoveries on the ionic mechanisms of nerve cell membranes.