Enter the chemical equation in its unbalanced form. The calculator will attempt to balance it by finding the smallest integer coefficients.
Understanding Chemical Equation Balancing
Chemical reactions involve the rearrangement of atoms. The Law of Conservation of Mass, a fundamental principle in chemistry, states that matter cannot be created or destroyed in a closed system. This means that the number of atoms of each element must be the same on both the reactant side (left side of the arrow) and the product side (right side of the arrow) of a chemical equation. Balancing an equation ensures that this law is upheld.
Why Balance Equations?
Conservation of Mass: Ensures that the total mass of reactants equals the total mass of products.
Stoichiometry: Provides the correct molar ratios for quantitative predictions in chemical reactions (e.g., calculating the amount of product formed from a given amount of reactant).
Predicting Reaction Outcomes: Helps in understanding the precise quantities involved in a chemical process.
How to Balance Equations (The Method Used by this Calculator)
This calculator primarily uses an algebraic method to balance equations. Here's a simplified breakdown of the process:
Identify Reactants and Products: Separate the chemical species on the left side (reactants) from those on the right side (products) using an arrow.
Assign Variables: Assign a variable (coefficient, usually denoted by letters like a, b, c, d…) to each chemical species in the equation. For example: a H2 + b O2 = c H2O
Count Atoms: For each element present in the equation, create an equation that represents the total number of atoms of that element on both sides. The number of atoms of an element in a compound is the product of its coefficient and the subscript of that element within the compound.
For H2, the coefficient is a and the subscript for H is 2, so total H atoms are 2a.
For O2, the coefficient is b and the subscript for O is 2, so total O atoms are 2b.
For H2O, the coefficient is c. For H, the subscript is 2, so total H atoms are 2c. For O, the subscript is 1, so total O atoms are 1c.
Formulate Algebraic Equations:
For Hydrogen (H): 2a = 2c
For Oxygen (O): 2b = c
Solve the System of Equations:
Set one variable to a simple value, usually 1. Let's set c = 1.
Substitute c = 1 into the oxygen equation: 2b = 1, so b = 1/2.
Substitute c = 1 into the hydrogen equation: 2a = 2(1), so 2a = 2, and a = 1.
Convert to Smallest Integers: The coefficients are currently a=1, b=1/2, c=1. To get the smallest whole numbers, multiply all coefficients by the least common denominator of the fractions (in this case, 2).
a = 1 * 2 = 2
b = 1/2 * 2 = 1
c = 1 * 2 = 2
The balanced equation is 2 H2 + 1 O2 = 2 H2O, which is typically written as 2 H2 + O2 = 2 H2O.
Limitations
This calculator uses a direct algebraic approach. It works well for many common chemical equations but may struggle with highly complex reactions, redox reactions that require specific balancing methods (like half-reaction method), or equations with ambiguous structures. It assumes standard chemical formulas and simple reactions.
function balanceEquation() {
var equationInput = document.getElementById("equationInput").value.trim();
var resultDiv = document.getElementById("result");
resultDiv.textContent = ""; // Clear previous results
resultDiv.classList.remove("error");
if (!equationInput) {
resultDiv.textContent = "Please enter an equation.";
resultDiv.classList.add("error");
return;
}
try {
// Basic parsing and validation
var sides = equationInput.split('=');
if (sides.length !== 2) {
throw new Error("Equation must contain exactly one '=' sign.");
}
var reactantsStr = sides[0].trim();
var productsStr = sides[1].trim();
var reactants = parseChemicalSide(reactantsStr, "reactant");
var products = parseChemicalSide(productsStr, "product");
if (reactants.length === 0 || products.length === 0) {
throw new Error("Invalid chemical formulas provided.");
}
var allElements = new Set();
reactants.forEach(function(chem) {
Object.keys(chem.elements).forEach(function(el) { allElements.add(el); });
});
products.forEach(function(chem) {
Object.keys(chem.elements).forEach(function(el) { allElements.add(el); });
});
var elementsArray = Array.from(allElements);
var numChems = reactants.length + products.length;
var numElements = elementsArray.length;
// Construct the matrix for the system of linear equations
// Matrix dimensions: numElements x numChems
var matrix = [];
for (var i = 0; i < numElements; i++) {
matrix.push(new Array(numChems).fill(0));
}
var elementIndexMap = {};
elementsArray.forEach(function(el, index) {
elementIndexMap[el] = index;
});
var chemIndex = 0;
reactants.forEach(function(chem) {
Object.keys(chem.elements).forEach(function(el) {
var elementIdx = elementIndexMap[el];
matrix[elementIdx][chemIndex] -= chem.elements[el]; // Reactants are negative on one side
});
chemIndex++;
});
products.forEach(function(chem) {
Object.keys(chem.elements).forEach(function(el) {
var elementIdx = elementIndexMap[el];
matrix[elementIdx][chemIndex] += chem.elements[el]; // Products are positive on the other side
});
chemIndex++;
});
// Solve the system of linear equations using Gaussian elimination
// We are looking for a non-trivial solution.
// The last column of the augmented matrix is implicitly zero.
var solution = gaussianElimination(matrix, numElements, numChems);
if (!solution) {
throw new Error("Could not find a unique balanced solution. Check equation or complexity.");
}
// Normalize to smallest integers
var normalizedSolution = normalizeCoefficients(solution);
// Format the output
var balancedEquation = formatBalancedEquation(reactants, products, normalizedSolution);
resultDiv.textContent = balancedEquation;
} catch (error) {
resultDiv.textContent = "Error: " + error.message;
resultDiv.classList.add("error");
}
}
// Parses a string like "H2O + O2" into an array of chemical objects
function parseChemicalSide(sideStr, type) {
var chemicals = [];
var components = sideStr.split('+').map(function(c) { return c.trim(); });
components.forEach(function(comp) {
if (!comp) return;
var formula = comp;
var coefficient = 1; // Default coefficient if not specified
// Check for coefficient (e.g., "2H2O")
var coeffMatch = formula.match(/^(\d+)/);
if (coeffMatch) {
coefficient = parseInt(coeffMatch[1], 10);
formula = formula.substring(coeffMatch[0].length).trim();
}
var elements = parseFormula(formula);
if (Object.keys(elements).length === 0) {
throw new Error("Invalid chemical formula: " + formula);
}
chemicals.push({ formula: formula, coefficient: coefficient, elements: elements, type: type });
});
return chemicals;
}
// Parses a chemical formula like "H2SO4" into an element map {H: 2, S: 1, O: 4}
function parseFormula(formula) {
var elements = {};
var regex = /([A-Z][a-z]*)(\d*)/g;
var match;
while ((match = regex.exec(formula)) !== null) {
var elementName = match[1];
var count = match[2] === '' ? 1 : parseInt(match[2], 10);
elements[elementName] = (elements[elementName] || 0) + count;
}
return elements;
}
// Solves a system of linear equations Ax = 0 using Gaussian elimination
// 'matrix' is an m x n matrix, representing numElements x numChems
function gaussianElimination(matrix, numRows, numCols) {
var augmentedMatrix = matrix.map(function(row) { return row.slice(); }); // Copy matrix
var rank = 0;
var pivotCol = 0;
// Forward elimination
for (var r = 0; r < numRows && pivotCol < numCols; r++) {
var iMax = r;
for (var i = r + 1; i Math.abs(augmentedMatrix[iMax][pivotCol])) {
iMax = i;
}
}
if (Math.abs(augmentedMatrix[iMax][pivotCol]) < 1e-10) { // Pivot is close to zero
pivotCol++;
r–; // Re-evaluate current row with next column
continue;
}
// Swap rows
var temp = augmentedMatrix[r];
augmentedMatrix[r] = augmentedMatrix[iMax];
augmentedMatrix[iMax] = temp;
// Normalize pivot row
var pivotElement = augmentedMatrix[r][pivotCol];
for (var j = pivotCol; j < numCols; j++) {
augmentedMatrix[r][j] /= pivotElement;
}
// Eliminate other rows
for (var i = 0; i < numRows; i++) {
if (i !== r) {
var factor = augmentedMatrix[i][pivotCol];
for (var j = pivotCol; j < numCols; j++) {
augmentedMatrix[i][j] -= factor * augmentedMatrix[r][j];
}
}
}
pivotCol++;
rank++;
}
// Check for solvability (non-trivial solution means rank < numCols)
if (rank < numCols) {
// Extract solution from the Reduced Row Echelon Form (RREF)
// We assume the last variable is the free variable and set it to 1.
var solution = new Array(numCols).fill(0);
var freeVarIndex = -1;
// Find the first column without a leading 1 (free variable)
var leadingOnes = 0;
for(var c = 0; c < numCols; c++) {
var foundLeadingOne = false;
for(var r = 0; r < numRows; r++) {
if (Math.abs(augmentedMatrix[r][c] – 1) < 1e-10) {
// Check if it's the first non-zero in the row
var isLeading = true;
for(var k=0; k 1e-10) {
isLeading = false;
break;
}
}
if(isLeading) {
foundLeadingOne = true;
leadingOnes++;
break;
}
} else if (Math.abs(augmentedMatrix[r][c]) > 1e-10) {
// Non-zero element but not 1, might be part of a solution row later
break;
}
}
if (!foundLeadingOne && !solution[c]) { // If no leading one found in this col, it's potentially a free variable
freeVarIndex = c;
break; // Assume only one free variable for simplicity in this context
}
}
if (freeVarIndex === -1) { // If all are leading variables (rank == numCols), implies only trivial solution or error
// Try setting the last variable as free if no other found
freeVarIndex = numCols – 1;
}
solution[freeVarIndex] = 1; // Set the free variable to 1
for (var r = numRows – 1; r >= 0; r–) {
var pivotFound = false;
var pivotCol = -1;
for (var c = 0; c 1e-10) {
pivotCol = c;
pivotFound = true;
break;
}
}
if (pivotFound && pivotCol !== freeVarIndex) {
var sumOfKnowns = 0;
for (var c = pivotCol + 1; c < numCols; c++) {
sumOfKnowns += augmentedMatrix[r][c] * solution[c];
}
solution[pivotCol] = -sumOfKnowns;
}
}
return solution;
} else {
// Rank equals numCols, implies only the trivial solution (all zeros)
return null;
}
}
// Finds the least common multiple (LCM) of two numbers
function lcm(a, b) {
return Math.abs(a * b) / gcd(a, b);
}
// Finds the greatest common divisor (GCD) of two numbers
function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b) {
var t = b;
b = a % b;
a = t;
}
return a;
}
// Normalizes the solution vector to the smallest possible positive integers
function normalizeCoefficients(solution) {
if (!solution || solution.length === 0) return [];
// Find the smallest fractional coefficient and the LCM of denominators
var denominators = [];
var minPositiveValue = Infinity;
solution.forEach(function(val) {
if (Math.abs(val) 1e-6) {
var fraction = val.toString().split('.');
if (fraction.length > 1) {
var decimalPart = fraction[1];
var denominator = Math.pow(10, decimalPart.length);
denominators.push(denominator);
}
}
if (val > 0 && val 0) {
var commonDenominator = denominators.reduce(function(acc, val) {
return lcm(acc, val);
});
multiplier = commonDenominator;
}
// Also multiply by the inverse of the smallest positive coefficient to make it 1
// (or ensure all coefficients are positive integers)
if (minPositiveValue !== Infinity && minPositiveValue > 1e-10) {
// We want the smallest integers, not necessarily starting with 1.
// Example: 0.5, 1, 1.5 -> should become 1, 2, 3
// LCM of denominators (2, 2) is 2. Multiplier = 2. -> 1, 2, 3. Correct.
// Example: 2, 3, 4 -> LCM is irrelevant. minPositiveValue = 2.
// If we multiply by 1/2, we get 1, 1.5, 2. Not good.
// So we should only use LCM of denominators if fractions are involved.
// A simpler approach: Find GCD of all integer parts after multiplying by LCM of denominators
var scaledCoefficients = solution.map(function(val) { return val * multiplier; });
var absIntCoeffs = scaledCoefficients.map(function(val) { return Math.round(Math.abs(val)); });
var finalGcd = absIntCoeffs[0];
for(var i = 1; i 1) {
multiplier = multiplier / finalGcd;
}
}
var finalCoefficients = solution.map(function(val) {
var scaled = val * multiplier;
// Round to nearest integer, handle potential floating point inaccuracies
return Math.round(scaled * 10000) / 10000;
});
// Ensure all coefficients are positive and integers
var minVal = Infinity;
finalCoefficients.forEach(function(val) {
if (Math.abs(val) 1e-10) {
minVal = Math.abs(val);
}
});
if (minVal !== Infinity && minVal > 1e-10) {
var finalMultiplier = 1 / minVal;
finalCoefficients = finalCoefficients.map(function(val) {
var result = val * finalMultiplier;
return Math.round(result * 10000) / 10000; // Round again
});
}
// Make sure they are positive integers
var finalIntCoeffs = [];
var denominatorLCM = 1;
finalCoefficients.forEach(function(val) {
if (Math.abs(val) 1) {
finalCoefficients = finalCoefficients.map(function(val) { return val * denominatorLCM; });
}
// Find GCD of the integer coefficients to simplify further
var absCoeffs = finalCoefficients.map(function(val) { return Math.round(Math.abs(val)); });
var commonDivisor = absCoeffs[0];
for (var i = 1; i 1) {
finalCoefficients = finalCoefficients.map(function(val) { return val / commonDivisor; });
}
// Ensure positive
var allPositive = finalCoefficients.every(function(val) { return val >= 0; });
if (!allPositive) {
// If some are negative, it might indicate an issue or a need for sign flip.
// For balancing, we expect positive coefficients. Let's try to flip signs if needed.
var signFlipMultiplier = 1;
if (finalCoefficients.some(function(val) { return val 1e-6; });
if (firstPositiveCoeff !== undefined) {
signFlipMultiplier = Math.abs(firstPositiveCoeff) / firstPositiveCoeff; // Should be 1 or -1
} else {
// All coefficients are effectively zero or negative. Check if this is valid.
// If all are zero, the equation might be trivial or invalid.
// If all negative, flip them.
if (finalCoefficients.every(function(val) { return val < 1e-6; })) {
signFlipMultiplier = -1;
}
}
}
finalCoefficients = finalCoefficients.map(function(val) { return Math.round(val * signFlipMultiplier * 10000) / 10000; });
}
return finalCoefficients.map(function(val) { return Math.round(val); }); // Final rounding to integer
}
// Formats the balanced equation string
function formatBalancedEquation(reactants, products, coefficients) {
var numReactants = reactants.length;
var numProducts = products.length;
var balanced = [];
for (var i = 0; i < numReactants; i++) {
var coeff = coefficients[i];
balanced.push((coeff === 1 ? "" : coeff) + reactants[i].formula);
}
var equationString = balanced.join(" + ");
equationString += " = ";
balanced = [];
for (var i = 0; i < numProducts; i++) {
var coeff = coefficients[numReactants + i];
balanced.push((coeff === 1 ? "" : coeff) + products[i].formula);
}
equationString += balanced.join(" + ");
return equationString;
}