Free Nernst Equation Calculator - Calculate Membrane Potential

Calculate cell membrane potential instantly with our free Nernst equation calculator. Input temperature, ion charge & concentrations for precise electrochemical results.

Nernst Equation Calculator

Calculate the electric potential in a cell using the Nernst equation.

Input Parameters

Temperature*

temperatureHelper: 0°C = 273.15K, 25°C = 298.15K, 37°C = 310.15K

Ion Charge (z)*

Concentration Outside Cell*

Concentration Inside Cell*

Result

Cell Potential:

0.00 mV

Copy

What is the Nernst Equation?

The Nernst equation relates the reduction potential of a cell to the standard cell potential, temperature, and the reaction quotient.

Equation Visualization

Nernst Equation

E = E° - (RT/zF) × ln([ion]_out/[ion]_in)

Variables

E: Cell Potential (mV)
E°: Standard Potential (0 mV)
R: Gas Constant (8.314 J/(mol·K))
T: Temperature (310.15 K)
z: Ion Charge (1)
F: Faraday Constant (96485 C/mol)
[ion]_out: Outside Concentration (145 mM)
[ion]_in: Inside Concentration (12 mM)

Calculation

RT/zF = (8.314 × 310.15) / (1 × 96485) = 0.026725

ln([ion]_out/[ion]_in) = ln(145/12) = 2.491827

(RT/zF) × ln([ion]_out/[ion]_in) = 0.026725 × 2.491827 × 1000 = 66.59 mV

E = 0 - 66.59 = 0.00 mV

cellDiagram

insideCell

[12 mM]

outsideCell

[145 mM]

ionFlowDirection

Interpretation

A zero potential indicates that the system is at equilibrium.

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Documentation

Nernst Equation Calculator: Calculate Cell Membrane Potential Online

Calculate cell membrane potential instantly with our free Nernst equation calculator. Simply input temperature, ion charge, and concentrations to determine electrochemical potentials for neurons, muscle cells, and electrochemical systems.

What is the Nernst Equation Calculator?

The Nernst equation calculator is an essential tool for calculating the electric potential across cell membranes based on ion concentration gradients. This fundamental electrochemistry calculator helps students, researchers, and professionals determine membrane potential values by inputting temperature, ion charge, and concentration differences.

Whether you're studying action potentials in neurons, designing electrochemical cells, or analyzing ion transport in biological systems, this cell potential calculator provides precise results using principles established by Nobel Prize-winning chemist Walther Nernst.

The Nernst equation relates electrochemical reaction potential to standard electrode potential, temperature, and ion activities. In biological contexts, it's essential for understanding how cells maintain electrical gradients—critical for nerve impulse transmission, muscle contraction, and cellular transport processes.

The Nernst Equation Formula

The Nernst equation is expressed mathematically as:

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

Where:

$E$ = Cell potential (volts)
$E^{\circ}$ = Standard cell potential (volts)
$R$ = Universal gas constant (8.314 J·mol⁻¹·K⁻¹)
$T$ = Absolute temperature (Kelvin)
$z$ = Valence (charge) of the ion
$F$ = Faraday constant (96,485 C·mol⁻¹)
$[C]_{\text{inside}}$ = Concentration of the ion inside the cell (molar)
$[C]_{\text{outside}}$ = Concentration of the ion outside the cell (molar)

For biological applications, the equation is often simplified by assuming a standard cell potential ( $E^{\circ}$ ) of zero and expressing the result in millivolts (mV). The equation then becomes:

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

The negative sign and inverted concentration ratio reflect the convention in cellular physiology, where the potential is typically measured from inside to outside the cell.

Inside Cell [K⁺] = 140 mM

Outside Cell [K⁺] = 5 mM

K⁺

E = -61 log([K⁺]outside/[K⁺]inside) mV

Variables Explained

Temperature (T): Measured in Kelvin (K), where K = °C + 273.15. Body temperature is typically 310.15K (37°C).
Ion Charge (z): The valence of the ion, which can be:
- +1 for sodium (Na⁺) and potassium (K⁺)
- +2 for calcium (Ca²⁺) and magnesium (Mg²⁺)
- -1 for chloride (Cl⁻)
- -2 for sulfate (SO₄²⁻)
Ion Concentrations: Measured in millimolar (mM) for biological systems. Typical values:
- K⁺: 5 mM outside, 140 mM inside
- Na⁺: 145 mM outside, 12 mM inside
- Cl⁻: 116 mM outside, 4 mM inside
- Ca²⁺: 1.5 mM outside, 0.0001 mM inside
Constants:
- Gas constant (R): 8.314 J/(mol·K)
- Faraday constant (F): 96,485 C/mol

How to Calculate Membrane Potential: Step-by-Step Guide

Our Nernst equation calculator simplifies complex electrochemical calculations into an intuitive interface. Follow these steps to calculate cell membrane potential:

Enter the Temperature: Input the temperature in Kelvin (K). The default is set to body temperature (310.15K or 37°C).
Specify the Ion Charge: Enter the valence (charge) of the ion you're analyzing. For example, enter "1" for potassium (K⁺) or "-1" for chloride (Cl⁻).
Input Ion Concentrations: Enter the concentration of the ion:
- Outside the cell (extracellular concentration) in mM
- Inside the cell (intracellular concentration) in mM
View the Result: The calculator automatically computes the membrane potential in millivolts (mV).
Copy or Analyze: Use the "Copy" button to copy the result for your records or further analysis.

Example Calculation

Let's calculate the Nernst potential for potassium (K⁺) at body temperature:

Temperature: 310.15K (37°C)
Ion charge: +1
Extracellular concentration: 5 mM
Intracellular concentration: 140 mM

Using the Nernst equation: $E = -\frac{8.314 \times 310.15}{1 \times 96485} \ln\left(\frac{5}{140}\right) \times 1000$

$E = -\frac{2580.59}{96485} \times \ln(0.0357) \times 1000$

$E = -0.02675 \times (-3.33) \times 1000$

$E = 89.08 \text{ mV}$

This positive potential indicates that potassium ions tend to flow out of the cell, which aligns with the typical electrochemical gradient for potassium.

Understanding Your Nernst Potential Results

The calculated membrane potential provides crucial insights into ion movement across cell membranes:

Positive Potential: Ion tends to flow out of the cell (efflux)
Negative Potential: Ion tends to flow into the cell (influx)
Zero Potential: System at equilibrium with no net ion flow

The potential magnitude reflects electrochemical driving force strength. Larger absolute values indicate stronger forces driving ion movement across the membrane.

Nernst Equation Applications in Science and Medicine

The Nernst equation has extensive applications across biology, chemistry, and biomedical engineering:

Cellular Physiology and Medicine

Neuroscience Research: Calculate resting membrane potential and action potential thresholds in neurons for understanding brain function
Cardiac Physiology: Determine electrical properties of heart cells essential for normal cardiac rhythm and arrhythmia research
Muscle Physiology: Analyze ion gradients controlling muscle contraction and relaxation in skeletal and smooth muscle
Kidney Function Studies: Investigate ion transport in renal tubules for electrolyte balance and kidney disease research

Electrochemistry

Battery Design: Optimizing electrochemical cells for energy storage applications.
Corrosion Analysis: Predicting and preventing metal corrosion in various environments.
Electroplating: Controlling metal deposition processes in industrial applications.
Fuel Cells: Designing efficient energy conversion devices.

Biotechnology

Biosensors: Developing ion-selective electrodes for analytical applications.
Drug Delivery: Engineering systems for controlled release of charged drug molecules.
Electrophysiology: Recording and analyzing electrical signals in cells and tissues.

Environmental Science

Water Quality Monitoring: Measuring ion concentrations in natural waters.
Soil Analysis: Assessing ion exchange properties of soils for agricultural applications.

Alternative Approaches

While the Nernst equation is powerful for single-ion systems at equilibrium, more complex scenarios may require alternative approaches:

Goldman-Hodgkin-Katz Equation: Accounts for multiple ion species with different permeabilities across the membrane. Useful for calculating the resting membrane potential of cells.
Donnan Equilibrium: Describes ion distribution when large, charged molecules (like proteins) cannot cross the membrane.
Computational Models: For non-equilibrium conditions, dynamic simulations using software like NEURON or COMSOL may be more appropriate.
Direct Measurement: Using techniques like patch-clamp electrophysiology to directly measure membrane potentials in living cells.

History of the Nernst Equation

The Nernst equation was developed by German chemist Walther Hermann Nernst (1864-1941) in 1889 while studying electrochemical cells. This groundbreaking work was part of his broader contributions to physical chemistry, particularly in thermodynamics and electrochemistry.

Key Historical Developments:

1889: Nernst first formulated his equation while working at the University of Leipzig, Germany.
1890s: The equation gained recognition as a fundamental principle in electrochemistry, explaining the behavior of galvanic cells.
Early 1900s: Physiologists began applying the Nernst equation to biological systems, particularly to understand nerve cell function.
1920: Nernst was awarded the Nobel Prize in Chemistry for his work in thermochemistry, including the development of the Nernst equation.
1940s-1950s: Alan Hodgkin and Andrew Huxley extended Nernst's principles in their groundbreaking work on action potentials in nerve cells, for which they later received the Nobel Prize.
1960s: The Goldman-Hodgkin-Katz equation was developed as an extension of the Nernst equation to account for multiple ion species.
Modern Era: The Nernst equation remains fundamental in fields ranging from electrochemistry to neuroscience, with computational tools making its application more accessible.

Programming Examples

Here are examples of how to implement the Nernst equation in various programming languages:

1def calculate_nernst_potential(temperature, ion_charge, conc_outside, conc_inside):
2    """
3    Calculate the Nernst potential in millivolts.
4    
5    Args:
6        temperature: Temperature in Kelvin
7        ion_charge: Charge of the ion (valence)
8        conc_outside: Concentration outside the cell in mM
9        conc_inside: Concentration inside the cell in mM
10        
11    Returns:
12        Nernst potential in millivolts
13    """
14    import math
15    
16    # Constants
17    R = 8.314  # Gas constant in J/(mol·K)
18    F = 96485  # Faraday constant in C/mol
19    
20    # Avoid division by zero
21    if ion_charge == 0:
22        ion_charge = 1
23    
24    # Check for valid concentrations
25    if conc_inside <= 0 or conc_outside <= 0:
26        return float('nan')
27    
28    # Calculate Nernst potential in millivolts
29    nernst_potential = -(R * temperature / (ion_charge * F)) * math.log(conc_outside / conc_inside) * 1000
30    
31    return nernst_potential
32
33# Example usage
34temp = 310.15  # Body temperature in Kelvin
35z = 1  # Potassium ion charge
36c_out = 5  # mM
37c_in = 140  # mM
38
39potential = calculate_nernst_potential(temp, z, c_out, c_in)
40print(f"Nernst potential: {potential:.2f} mV")
41

1function calculateNernstPotential(temperature, ionCharge, concOutside, concInside) {
2  // Constants
3  const R = 8.314;  // Gas constant in J/(mol·K)
4  const F = 96485;  // Faraday constant in C/mol
5  
6  // Avoid division by zero
7  if (ionCharge === 0) {
8    ionCharge = 1;
9  }
10  
11  // Check for valid concentrations
12  if (concInside <= 0 || concOutside <= 0) {
13    return NaN;
14  }
15  
16  // Calculate Nernst potential in millivolts
17  const nernstPotential = -(R * temperature / (ionCharge * F)) * 
18                          Math.log(concOutside / concInside) * 1000;
19  
20  return nernstPotential;
21}
22
23// Example usage
24const temp = 310.15;  // Body temperature in Kelvin
25const z = 1;  // Potassium ion charge
26const cOut = 5;  // mM
27const cIn = 140;  // mM
28
29const potential = calculateNernstPotential(temp, z, cOut, cIn);
30console.log(`Nernst potential: ${potential.toFixed(2)} mV`);
31

1public class NernstCalculator {
2    // Constants
3    private static final double R = 8.314;  // Gas constant in J/(mol·K)
4    private static final double F = 96485;  // Faraday constant in C/mol
5    
6    public static double calculateNernstPotential(
7            double temperature, 
8            double ionCharge, 
9            double concOutside, 
10            double concInside) {
11        
12        // Avoid division by zero
13        if (ionCharge == 0) {
14            ionCharge = 1;
15        }
16        
17        // Check for valid concentrations
18        if (concInside <= 0 || concOutside <= 0) {
19            return Double.NaN;
20        }
21        
22        // Calculate Nernst potential in millivolts
23        double nernstPotential = -(R * temperature / (ionCharge * F)) * 
24                                 Math.log(concOutside / concInside) * 1000;
25        
26        return nernstPotential;
27    }
28    
29    public static void main(String[] args) {
30        double temp = 310.15;  // Body temperature in Kelvin
31        double z = 1;  // Potassium ion charge
32        double cOut = 5;  // mM
33        double cIn = 140;  // mM
34        
35        double potential = calculateNernstPotential(temp, z, cOut, cIn);
36        System.out.printf("Nernst potential: %.2f mV%n", potential);
37    }
38}
39

1Function NernstPotential(temperature As Double, ionCharge As Double, concOutside As Double, concInside As Double) As Variant
2    ' Constants
3    Const R As Double = 8.314  ' Gas constant in J/(mol·K)
4    Const F As Double = 96485  ' Faraday constant in C/mol
5    
6    ' Avoid division by zero
7    If ionCharge = 0 Then
8        ionCharge = 1
9    End If
10    
11    ' Check for valid concentrations
12    If concInside <= 0 Or concOutside <= 0 Then
13        NernstPotential = CVErr(xlErrValue)
14        Exit Function
15    End If
16    
17    ' Calculate Nernst potential in millivolts
18    NernstPotential = -(R * temperature / (ionCharge * F)) * _
19                      Application.Ln(concOutside / concInside) * 1000
20End Function
21
22' Example usage in a cell:
23' =NernstPotential(310.15, 1, 5, 140)
24

1function potential = nernstPotential(temperature, ionCharge, concOutside, concInside)
2    % Calculate the Nernst potential in millivolts
3    %
4    % Inputs:
5    %   temperature - Temperature in Kelvin
6    %   ionCharge - Charge of the ion (valence)
7    %   concOutside - Concentration outside the cell in mM
8    %   concInside - Concentration inside the cell in mM
9    %
10    % Output:
11    %   potential - Nernst potential in millivolts
12    
13    % Constants
14    R = 8.314;  % Gas constant in J/(mol·K)
15    F = 96485;  % Faraday constant in C/mol
16    
17    % Avoid division by zero
18    if ionCharge == 0
19        ionCharge = 1;
20    end
21    
22    % Check for valid concentrations
23    if concInside <= 0 || concOutside <= 0
24        potential = NaN;
25        return;
26    end
27    
28    % Calculate Nernst potential in millivolts
29    potential = -(R * temperature / (ionCharge * F)) * ...
30                log(concOutside / concInside) * 1000;
31end
32
33% Example usage
34temp = 310.15;  % Body temperature in Kelvin
35z = 1;  % Potassium ion charge
36c_out = 5;  % mM
37c_in = 140;  % mM
38
39potential = nernstPotential(temp, z, c_out, c_in);
40fprintf('Nernst potential: %.2f mV\n', potential);
41

1#include <iostream>
2#include <cmath>
3
4double calculateNernstPotential(double temperature, double ionCharge, 
5                               double concOutside, double concInside) {
6    // Constants
7    const double R = 8.314;  // Gas constant in J/(mol·K)
8    const double F = 96485;  // Faraday constant in C/mol
9    
10    // Avoid division by zero
11    if (ionCharge == 0) {
12        ionCharge = 1;
13    }
14    
15    // Check for valid concentrations
16    if (concInside <= 0 || concOutside <= 0) {
17        return std::nan("");
18    }
19    
20    // Calculate Nernst potential in millivolts
21    double nernstPotential = -(R * temperature / (ionCharge * F)) * 
22                             std::log(concOutside / concInside) * 1000;
23    
24    return nernstPotential;
25}
26
27int main() {
28    double temp = 310.15;  // Body temperature in Kelvin
29    double z = 1;          // Potassium ion charge
30    double cOut = 5;       // mM
31    double cIn = 140;      // mM
32    
33    double potential = calculateNernstPotential(temp, z, cOut, cIn);
34    std::cout << "Nernst potential: " << potential << " mV" << std::endl;
35    
36    return 0;
37}
38

1using System;
2
3class NernstCalculator
4{
5    // Constants
6    private const double R = 8.314;  // Gas constant in J/(mol·K)
7    private const double F = 96485;  // Faraday constant in C/mol
8    
9    public static double CalculateNernstPotential(
10        double temperature, 
11        double ionCharge, 
12        double concOutside, 
13        double concInside)
14    {
15        // Avoid division by zero
16        if (ionCharge == 0)
17        {
18            ionCharge = 1;
19        }
20        
21        // Check for valid concentrations
22        if (concInside <= 0 || concOutside <= 0)
23        {
24            return double.NaN;
25        }
26        
27        // Calculate Nernst potential in millivolts
28        double nernstPotential = -(R * temperature / (ionCharge * F)) * 
29                                 Math.Log(concOutside / concInside) * 1000;
30        
31        return nernstPotential;
32    }
33    
34    static void Main()
35    {
36        double temp = 310.15;  // Body temperature in Kelvin
37        double z = 1;          // Potassium ion charge
38        double cOut = 5;       // mM
39        double cIn = 140;      // mM
40        
41        double potential = CalculateNernstPotential(temp, z, cOut, cIn);
42        Console.WriteLine($"Nernst potential: {potential:F2} mV");
43    }
44}
45

Numerical Examples

Here are some practical examples of Nernst equation calculations for different ions at body temperature (37°C = 310.15K):

Example 1: Potassium (K⁺)

Ion charge (z): +1
Outside concentration: 5 mM
Inside concentration: 140 mM
Nernst potential: +89.08 mV

Example 2: Sodium (Na⁺)

Ion charge (z): +1
Outside concentration: 145 mM
Inside concentration: 12 mM
Nernst potential: -67.71 mV

Example 3: Calcium (Ca²⁺)

Ion charge (z): +2
Outside concentration: 1.5 mM
Inside concentration: 0.0001 mM
Nernst potential: -143.70 mV

Example 4: Chloride (Cl⁻)

Ion charge (z): -1
Outside concentration: 116 mM
Inside concentration: 4 mM
Nernst potential: -89.08 mV

Example 5: Temperature Effect

Calculating potassium potential at different temperatures:

At 25°C (298.15K): +85.66 mV
At 37°C (310.15K): +89.08 mV
At 42°C (315.15K): +90.51 mV

This demonstrates how temperature affects the Nernst potential, with higher temperatures resulting in larger absolute potential values.

Frequently Asked Questions About Nernst Equation Calculations

What is the Nernst equation calculator used for?

The Nernst equation calculator computes electric potential differences across cell membranes and electrochemical cells. It's essential for understanding how neurons maintain electrical gradients for signal transmission, muscle contraction, and cellular transport processes.

How do I calculate membrane potential using the Nernst equation?

To calculate membrane potential, input the temperature (Kelvin), ion charge (valence), and concentrations inside and outside the cell into our calculator. The tool automatically computes the equilibrium potential in millivolts.

How does temperature affect Nernst potential calculations?

Temperature directly influences membrane potential calculations. Higher temperatures increase absolute potential values by approximately 3-4% per 10°C increase, reflecting greater thermal energy driving ion movement.

What happens when ion concentrations are equal in Nernst calculations?

When inside and outside concentrations are equal, the Nernst potential equals zero. This represents electrochemical equilibrium with no net driving force for ion movement across the membrane.

Can I use the Nernst equation calculator for any ion?

Yes, the Nernst calculator works for any ion with known charge and concentrations. Common biological ions include potassium (K⁺), sodium (Na⁺), calcium (Ca²⁺), and chloride (Cl⁻).

What's the difference between Nernst and Goldman equation calculations?

The Nernst equation calculates equilibrium potential for single ions, while the Goldman-Hodgkin-Katz equation handles multiple ions with different permeabilities for overall membrane potential calculation.

How accurate is the Nernst equation for cell membrane calculations?

The Nernst equation provides precise results for equilibrium conditions and single-ion systems. For complex biological membranes with multiple ions and active transport, Goldman-Hodgkin-Katz equations offer better approximations.

What does a negative result mean in Nernst potential calculations?

A negative Nernst potential indicates ion influx tendency (positive ions flow into cell), while positive values suggest efflux (ions flow out). The sign follows electrophysiology conventions measuring inside-to-outside potential.

How does the Nernst equation relate to resting membrane potential?

Resting membrane potential results from multiple ion Nernst potentials weighted by their membrane permeabilities. The Goldman-Hodgkin-Katz equation extends Nernst principles for multi-ion membrane potential calculations.

Can the Nernst calculator predict active ion transport?

No, the Nernst equation calculator only models passive ion movement down electrochemical gradients. Active transport using ATP energy to move ions against gradients requires different calculations.

What concentration units work with the Nernst equation calculator?

The calculator accepts any concentration units (mM, μM, nM) provided both inside and outside values use identical units:

Millimolar (mM): 1 mM = 10⁻³ mol/L
Micromolar (μM): 1 μM = 10⁻⁶ mol/L
Nanomolar (nM): 1 nM = 10⁻⁹ mol/L

How do I calculate potassium equilibrium potential?

For potassium potential calculation, use typical values: 310.15K temperature, +1 charge, 5mM outside concentration, 140mM inside concentration. The result is approximately +89mV.

What are normal sodium membrane potential values?

Sodium equilibrium potential typically calculates to approximately -67mV using: 310.15K temperature, +1 charge, 145mM outside, 12mM inside concentrations.

How do I calculate calcium membrane potential?

For calcium potential calculation, input: 310.15K temperature, +2 charge, 1.5mM outside, 0.0001mM inside. The calculated potential is approximately -144mV.

References

Nernst, W. (1889). "Die elektromotorische Wirksamkeit der Ionen." Zeitschrift für Physikalische Chemie, 4, 129-181.
Hille, B. (2001). Ion Channels of Excitable Membranes (3rd ed.). Sinauer Associates.
Lodish, H., Berk, A., Kaiser, C. A., Krieger, M., Scott, M. P., Bretscher, A., Ploegh, H., & Matsudaira, P. (2008). Molecular Cell Biology (6th ed.). W. H. Freeman.
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., & Walter, P. (2014). Molecular Biology of the Cell (6th ed.). Garland Science.
Hodgkin, A. L., & Huxley, A. F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve." The Journal of Physiology, 117(4), 500-544.
Goldman, D. E. (1943). "Potential, impedance, and rectification in membranes." The Journal of General Physiology, 27(1), 37-60.
Kandel, E. R., Schwartz, J. H., Jessell, T. M., Siegelbaum, S. A., & Hudspeth, A. J. (2013). Principles of Neural Science (5th ed.). McGraw-Hill Education.
Karp, G. (2009). Cell and Molecular Biology: Concepts and Experiments (6th ed.). John Wiley & Sons.

Calculate Membrane Potential Now - Free Nernst Equation Calculator

Our free Nernst equation calculator makes membrane potential calculations simple and precise. Perfect for students studying cellular physiology, researchers analyzing ion transport, and professionals in electrochemistry and neuroscience.

Start calculating immediately: Input your temperature, ion charge, and concentrations to get instant Nernst potential results. Use these calculations to understand cellular processes, design experiments, or solve electrochemical problems in biology and chemistry.

Why Choose Our Nernst Calculator?

✅ Instant results - Calculate membrane potentials in seconds
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✅ User-friendly interface - No complex formulas to remember
✅ Free access - No registration or payment required
✅ Mobile responsive - Calculate on any device

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