Nernst and Goldman Equations: Calculating Equilibrium Potential

Diffusion is the movement of molecules from areas of higher concentration to lower concentration, driven by concentration gradients. Ion diffusion rate depends on hydration shell size—ions with smaller hydration shells diffuse more rapidly. When ions diffuse across a membrane, they create charge separation called diffusion potential. This potential is temporary because the electrical force generated counteracts the concentration gradient, preventing further net movement. The balance between chemical and electrical forces determines ion movement, with ions experiencing both types of forces simultaneously.

Ions possess two types of gradients: concentration gradients (difference in ion concentration between compartments) and electrical gradients (difference in charge between compartments). The concentration gradient drives molecules from areas of higher concentration to areas of lower concentration, similar to perfume diffusion. The electrical gradient exerts forces on charged particles based on their charge—positively charged ions are attracted to negatively charged regions and repelled by positively charged regions. These two forces act independently and can either reinforce or oppose each other. The combined electrochemical gradient determines the direction and rate of ion movement across membranes.

Ion transport in electrolytes occurs through migration (movement in electric field) and diffusion (movement due to concentration gradients). Migration can be eliminated by adding supporting electrolyte, which increases conductivity and reduces the electric field. During electrochemical reactions, the concentration of reacting ions near the electrode surface changes. The diffusion layer is the region near the electrode where concentration gradients exist, with thickness (δ) depending on time, stirring, and electrode geometry.

Diffusion is the passive movement of ions from areas of higher concentration to lower concentration until equilibrium is reached. This chemical gradient creates a driving force independent of electrical forces. When both electrical and concentration gradients act on the same ion, they may work together or oppose each other, determining the net direction of ion movement.

Ions diffuse across cell membranes according to concentration gradients: potassium diffuses out, sodium diffuses in, and chloride diffuses in. However, the movement of charged particles creates electrical forces that oppose or support diffusion. At electrochemical equilibrium, the chemical gradient equals the electrical gradient, and there is no net ion movement. The Nernst equation calculates equilibrium potential: E = -61 × log([ion]out/[ion]in).

Membrane potential is the electrical potential difference across a biological membrane, arising from uneven ion distribution and selective membrane permeability. At rest, cells maintain specific ion gradients: potassium is ~20 times more concentrated inside, sodium ~15 times more concentrated outside, and chloride outside. The inside becomes negatively charged while the outside becomes positive. Resting potential typically ranges from -60 to -90 mV. This potential is generated by concentration gradients, selective permeability (P_K=1, P_Na=0.04, P_Cl=0.45), and negatively charged intracellular proteins. The Nernst equation calculates equilibrium potential for specific ions.

Membrane potential is the electrical voltage difference across a cell membrane, measured in millivolts. It exists when there is an imbalance of charges between the inside and outside of the cell. The electrochemical gradient combines two forces driving ion movement: the chemical gradient (concentration difference) and the electrical gradient (membrane potential). Positive ions are attracted to negative charges and repelled by positive charges. When ions can move across a membrane, they follow their concentration gradient, but the membrane potential adds an additional force. The electrochemical gradient becomes stronger when membrane potential works with the concentration gradient, and weaker when it opposes it. Membrane potential itself is the electrical voltage based on the unequal distribution of ions between the two sides.

Membrane potential is the electrical potential difference between the interior and exterior of a biological cell. The plasma membrane, composed of a lipid bilayer, acts as both an electrical insulator and a diffusion barrier to ion movement. This barrier creates unequal distribution of ions: sodium and potassium are cations, while chloride is an anion. Under resting conditions, the cytosol is more negatively charged than the extracellular fluid, creating an electrical gradient across the membrane. This charge difference constitutes the membrane potential.

Membrane potential is fundamental to all cellular functions including heartbeat, movement, and cognition. Solids diffuse following chemical gradients (concentration differences), but when charges are added, displacement depends on both chemical and electrical gradients. Positive charges repel each other, and electrical gradients depend only on overall charge differences, not specific ion types. Chemical and electrical gradients balance simultaneously with no priority order. For each ion concentration difference, an electric potential can compensate for the chemical gradient. The cell membrane separates intracellular and extracellular compartments with different ion concentrations: potassium concentrated inside, sodium outside, and chlorine mainly outside.

Membrane potential (or electrical potential of the membrane) refers to the electrical voltage that exists across the cell membrane. This occurs because the intracellular fluid contains excess negative charges while the extracellular fluid contains excess positive charges. The attraction between these opposite charges creates an electrical force that tries to bring them together, but the lipid bilayer of the cell membrane prevents their movement, resulting in a separation of charges and the establishment of an electrical potential difference.

Common logarithm (log) has a base of 10 by default. Natural logarithm (ln) has a base of e (approximately 2.718). When converting between them, use the formula: ln(x) = 2.303 × log(x) and log(x) = ln(x) / 2.303.

Logarithm is defined by the BRP formula: log_B(R) = P, where B is the base, R is the result, and P is the power. Logarithm calculates the power to which the base must be raised to obtain the result. For example, log_10(100) = 2 because 10^2 = 100. The natural logarithm (ln) is logarithm to the base e (approximately 2.718), and ln and log to the base e are equivalent. The key property is that when you have the same base and same result on both sides of an equation, you can eliminate them and keep only the powers. This principle applies to both log and ln equations.

This comprehensive lesson covers logarithm fundamentals for secondary students. The common logarithm (log) has base 10 and does not require writing the base explicitly. Key properties include: log(10ⁿ) = n, log(10⁻ⁿ) = -n, and log(1) = 0. The natural logarithm (ln) has base e (approximately 2.71828) and follows the same properties. The exponential function y = eˣ and natural logarithm y = ln(x) are inverse functions. To solve equations involving natural logarithms: apply ln to both sides, use ln(eˣ) = x to eliminate the logarithm, and solve the resulting algebraic equation.

Logarithms (log) are base-10 inverses of exponents: log(x) = y means 10^y = x. Natural logs (ln) use base e (≈2.71828): ln(x) = y means e^y = x. To solve log equations, raise 10 to both sides; for ln equations, take natural log of both sides. For example, log(1000) = 3 because 10³ = 1000, and ln(x) = 3 means x = e³.

The common logarithm (log) has base 10. When no base is specified, it is assumed to be 10. To evaluate, determine what power of 10 gives the argument: log(100) = 2 since 10^2 = 100, and log(0.01) = -2 since 10^(-2) = 0.01. The natural logarithm (ln) has base e (approximately 2.718), Euler's number. For example, ln(e^3) = 3 since e^3 is the argument. To evaluate, determine what power of e gives the argument: ln(e^(-2)) = -2. When solving equations with common logarithms, convert to exponential form: log(x) = 9 gives x = 10^9. For ln(x) = 1/2, x = e^(1/2) = √e. For equations with natural logarithms, convert to exponential form: ln(x) = 5 gives x = e^5. For equations with common logarithms, convert to exponential form: log(a+b) = 2 gives a+b = 10^2 = 100, so a = 100 - b. For equations with natural logarithms, convert to exponential form: e^(x+2) = 3 gives x+2 = ln(3), so x = ln(3) - 2.