Chemical reactions involve the rearrangement of atoms. The Law of Conservation of Mass states that matter cannot be created or destroyed in a chemical reaction. This means that the number of atoms of each element must be the same on both sides of a chemical equation – the reactant side (what you start with) and the product side (what you end up with). Balancing a chemical equation is the process of adjusting the coefficients (the numbers in front of the chemical formulas) to ensure this conservation of mass.
Why Balance Chemical Equations?
Adherence to the Law of Conservation of Mass: This is the fundamental principle. Every atom present before the reaction must be accounted for after the reaction.
Stoichiometry: Balanced equations are crucial for quantitative calculations. They tell us the exact molar ratios in which substances react and are produced. This is vital for predicting yields, determining limiting reactants, and understanding reaction efficiency in laboratories and industrial processes.
Predicting Reaction Outcomes: A balanced equation provides a clear picture of the molecular transformations occurring.
How to Balance a Chemical Equation (The General Process)
Balancing chemical equations typically involves a systematic approach, often referred to as the "inspection method" or "algebraic method" for more complex cases. Here's a breakdown of the inspection method:
Write the Unbalanced Equation: Start with the correct chemical formulas for all reactants and products. For example, the combustion of methane:
CH4 + O2 → CO2 + H2O
Count Atoms of Each Element: Tally the number of atoms of each element on both the reactant and product sides.
Reactants: C=1, H=4, O=2
Products: C=1, H=2, O=3 (2 from CO2, 1 from H2O)
Balance Elements One by One: Start with elements that appear in only one reactant and one product. Often, it's best to leave elements like oxygen or hydrogen (especially if they appear in multiple compounds or as diatomic molecules like O2) until last.
Carbon (C) is already balanced (1 on each side).
Hydrogen (H): There are 4 H atoms on the left and 2 on the right. Place a coefficient of 2 in front of H2O on the product side:
CH4 + O2 → CO2 + 2H2O
Now recount atoms:
Reactants: C=1, H=4, O=2
Products: C=1, H=4 (2*2), O=4 (2 from CO2, 2*1 from 2H2O)
Oxygen (O): There are 2 O atoms on the left and 4 on the right. Place a coefficient of 2 in front of O2 on the reactant side:
CH4 + 2O2 → CO2 + 2H2O
Verify: Do a final count to ensure all elements are balanced.
Reactants: C=1, H=4, O=4 (2*2)
Products: C=1, H=4, O=4 (2 from CO2, 2*1 from 2H2O)
The equation is now balanced.
Simplify Coefficients: Ensure the coefficients are the smallest possible whole numbers. In this case, they already are.
How This Calculator Works
This calculator uses a simplified approach, primarily parsing the input formulas and attempting to balance them using a brute-force or iterative method combined with some heuristics. It identifies the unique elements present and then tries to find integer coefficients that satisfy the atom counts for each element on both sides of the reaction.
Input Format:
Separate reactants with a + sign.
Separate products with a + sign.
Use standard chemical notation for formulas (e.g., H2O, CO2, Fe2O3, Al(OH)3). Parentheses are handled by distributing the subscript outside the parentheses to all elements within.
The calculator will automatically detect elements and their counts.
Example:
To balance the reaction of hydrogen and oxygen to form water:
Reactants:H2+O2
Products:H2O
The calculator will output: 2H2 + O2 → 2H2O
This tool is designed to assist students and chemists in quickly verifying or determining the stoichiometric coefficients for chemical reactions, reinforcing the fundamental principles of chemical equation balancing.
function getElementCounts(formula) {
var counts = {};
var regex = /([A-Z][a-z]*)(\d*)|(\()|(\))/g;
var matches;
var stack = []; // For handling parentheses
while ((matches = regex.exec(formula)) !== null) {
if (matches[1]) { // Element and optional count
var element = matches[1];
var count = matches[2] ? parseInt(matches[2], 10) : 1;
var multiplier = 1;
for (var i = stack.length – 1; i >= 0; i–) {
if (stack[i] === ')') multiplier *= stack.pop();
else if (typeof stack[i] === 'number') { // Found a number multiplier from parentheses
multiplier *= stack.pop();
break; // Stop at the first number multiplier
}
}
counts[element] = (counts[element] || 0) + count * multiplier;
} else if (matches[3]) { // Opening parenthesis
stack.push('('); // Placeholder for parenthesis
} else if (matches[4]) { // Closing parenthesis
var closingParenIndex = -1;
for (var i = stack.length – 1; i >= 0; i–) {
if (stack[i] === '(') {
closingParenIndex = i;
break;
}
}
if (closingParenIndex !== -1) {
var multiplier = 1;
// Check for a number immediately after the closing parenthesis
var parenEnd = regex.lastIndex;
if (parenEnd closingParenIndex; j–) {
if (typeof stack[j] === 'number') {
// Should not happen if parsing logic is correct, but safety check
} else if (stack[j] === '(') {
// Skip the opening parenthesis placeholder
} else {
// Assume it's an element name, need to re-parse to get count from formula
// This is complex, simpler to re-parse elements within (..)
// For now, let's assume the structure is simple and the multiplier applies to the next set of elements found
}
}
// Push the multiplier onto the stack for subsequent elements within the scope
stack.splice(closingParenIndex + 1, 0, multiplier);
stack.splice(closingParenIndex, 1); // Remove the '(' placeholder
}
}
}
// Process elements after handling parentheses for multipliers
var finalCounts = {};
var currentMultiplier = 1;
var tempStack = []; // Temporary stack to hold elements and their direct counts
// Re-parse to correctly handle multipliers after parentheses
var elementRegex = /([A-Z][a-z]*)(\d*)/g;
var currentFormulaIndex = 0;
while(currentFormulaIndex < formula.length) {
var elementMatch = elementRegex.exec(formula.substring(currentFormulaIndex));
if (!elementMatch) {
// Check for parentheses and multipliers
if (formula[currentFormulaIndex] === '(') {
var openParenIndex = currentFormulaIndex;
var closeParenIndex = -1;
var parenLevel = 1;
for(var k = openParenIndex + 1; k < formula.length; k++) {
if (formula[k] === '(') parenLevel++;
else if (formula[k] === ')') {
parenLevel–;
if (parenLevel === 0) {
closeParenIndex = k;
break;
}
}
}
if (closeParenIndex !== -1) {
var multiplier = 1;
var numStartIndex = closeParenIndex + 1;
if (numStartIndex < formula.length && /\d/.test(formula[numStartIndex])) {
var numEndIndex = numStartIndex;
while(numEndIndex < formula.length && /\d/.test(formula[numEndIndex])) {
numEndIndex++;
}
multiplier = parseInt(formula.substring(numStartIndex, numEndIndex), 10);
currentFormulaIndex = numEndIndex; // Advance past the number
} else {
currentFormulaIndex = closeParenIndex + 1; // Advance past the parenthesis
}
var subFormula = formula.substring(openParenIndex + 1, closeParenIndex);
var subCounts = getElementCounts(subFormula); // Recursive call for sub-formula
for (var el in subCounts) {
finalCounts[el] = (finalCounts[el] || 0) + subCounts[el] * multiplier;
}
} else { // Unmatched parenthesis, treat as literal character or error
currentFormulaIndex++;
}
} else { // Not an element or parenthesis, advance
currentFormulaIndex++;
}
} else {
var element = elementMatch[1];
var count = elementMatch[2] ? parseInt(elementMatch[2], 10) : 1;
finalCounts[element] = (finalCounts[element] || 0) + count;
currentFormulaIndex = elementRegex.lastIndex; // Advance the index
elementRegex.lastIndex = 0; // Reset lastIndex for next iteration
}
}
return finalCounts;
}
function parseEquation(equationString) {
var parts = equationString.split('→');
if (parts.length !== 2) return null;
var reactantsString = parts[0].trim();
var productsString = parts[1].trim();
var reactantFormulas = reactantsString.split('+').map(function(s) { return s.trim(); });
var productFormulas = productsString.split('+').map(function(s) { return s.trim(); });
var allElements = new Set();
var reactantCounts = [];
var productCounts = [];
for (var i = 0; i < reactantFormulas.length; i++) {
var counts = getElementCounts(reactantFormulas[i]);
reactantCounts.push(counts);
for (var element in counts) {
allElements.add(element);
}
}
for (var i = 0; i < productFormulas.length; i++) {
var counts = getElementCounts(productFormulas[i]);
productCounts.push(counts);
for (var element in counts) {
allElements.add(element);
}
}
return {
reactants: reactantFormulas,
products: productFormulas,
reactantAtomCounts: reactantCounts,
productAtomCounts: productCounts,
elements: Array.from(allElements).sort()
};
}
function solveMatrix(matrix) {
var numRows = matrix.length;
var numCols = matrix[0].length;
for (var pivotRow = 0; pivotRow < numRows; pivotRow++) {
// Find pivot element
var maxVal = 0;
var maxRow = -1;
for (var i = pivotRow; i maxVal) {
maxVal = Math.abs(matrix[i][pivotRow]);
maxRow = i;
}
}
if (maxRow === -1) continue; // Column is all zeros
// Swap rows
var temp = matrix[pivotRow];
matrix[pivotRow] = matrix[maxRow];
matrix[maxRow] = temp;
// Normalize pivot row
var pivotElement = matrix[pivotRow][pivotRow];
if (pivotElement === 0) continue; // Should not happen due to maxVal check, but safety
for (var j = pivotRow; j < numCols; j++) {
matrix[pivotRow][j] /= pivotElement;
}
// Eliminate other rows
for (var i = 0; i < numRows; i++) {
if (i !== pivotRow) {
var factor = matrix[i][pivotRow];
for (var j = pivotRow; j < numCols; j++) {
matrix[i][j] -= factor * matrix[pivotRow][j];
}
}
}
}
// Extract solution
var solution = new Array(numCols – 1).fill(0);
var freeVariableIndex = -1;
for (var i = 0; i < numRows; i++) {
var pivotCol = -1;
for (var j = 0; j < numCols – 1; j++) {
if (Math.abs(matrix[i][j] – 1) < 1e-9 && matrix[i].slice(j + 1).every(function(val) { return Math.abs(val) < 1e-9; })) {
pivotCol = j;
break;
}
}
if (pivotCol !== -1) {
if (Math.abs(matrix[i][numCols – 1]) = 0; i–) {
var pivotCol = -1;
for (var j = 0; j < numCols – 1; j++) {
if (Math.abs(matrix[i][j] – 1) < 1e-9 && matrix[i].slice(j + 1).every(function(val) { return Math.abs(val) < 1e-9; })) {
pivotCol = j;
break;
}
}
if (pivotCol !== -1 && pivotCol !== freeVariableIndex) {
var sum = 0;
for (var j = pivotCol + 1; j < numCols – 1; j++) {
sum += matrix[i][j] * solution[j];
}
solution[pivotCol] = matrix[i][numCols – 1] – sum;
}
}
}
// Normalize to smallest integers
var gcdVal = solution[0];
for (var i = 1; i < solution.length; i++) {
gcdVal = gcd(gcdVal, solution[i]);
}
var finalCoefficients = solution.map(function(val) {
return Math.round(val / gcdVal);
});
// Ensure all are positive (can happen with specific free variable assignments)
if (finalCoefficients.some(function(c) { return c < 0; })) {
finalCoefficients = finalCoefficients.map(function(c) { return -c; });
}
return finalCoefficients;
}
function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b) {
var temp = b;
b = a % b;
a = temp;
}
return a;
}
function lcm(a, b) {
return (a * b) / gcd(a, b);
}
function arrayGcd(arr) {
if (arr.length === 0) return 1;
var result = arr[0];
for (var i = 1; i < arr.length; i++) {
result = gcd(result, arr[i]);
}
return result;
}
function balanceEquation() {
var reactantsInput = document.getElementById('reactants').value;
var productsInput = document.getElementById('products').value;
if (!reactantsInput || !productsInput) {
document.getElementById('balancedFormula').innerText = "Please enter both reactants and products.";
return;
}
var equationString = reactantsInput + ' → ' + productsInput;
var parsed = parseEquation(equationString);
if (!parsed) {
document.getElementById('balancedFormula').innerText = "Invalid equation format.";
return;
}
var numReactants = parsed.reactants.length;
var numProducts = parsed.products.length;
var numElements = parsed.elements.length;
var numCoefficients = numReactants + numProducts;
// Build the matrix
// Rows = elements, Columns = coefficients (reactants first, then products)
// For each element, the equation is: Sum(reactant_coeff * atom_count) = Sum(product_coeff * atom_count)
// Rearranging for matrix: Sum(reactant_coeff * atom_count) – Sum(product_coeff * atom_count) = 0
var matrix = [];
for (var i = 0; i < numElements; i++) {
var row = new Array(numCoefficients).fill(0);
var element = parsed.elements[i];
// Reactants
for (var j = 0; j < numReactants; j++) {
row[j] = parsed.reactantAtomCounts[j][element] || 0;
}
// Products
for (var j = 0; j < numProducts; j++) {
row[numReactants + j] = -(parsed.productAtomCounts[j][element] || 0);
}
matrix.push(row);
}
// Use Gaussian elimination to solve the system
// We have numElements equations and numCoefficients unknowns.
// The system is homogeneous, meaning the right-hand side is all zeros.
// We expect at least one free variable.
// Augment the matrix with a column for the solution vector (initially zeros)
// This augmented matrix approach is more for non-homogeneous systems.
// For homogeneous, we solve Ax = 0. Rank(A) c*H2O
// For H: 2a = 2c => 2a – 2c = 0
// For O: 2b = 1c => 2b – 1c = 0
// We have more variables than independent equations. We set one variable (e.g., c=1) and solve.
// For this general solver, let's stick to the matrix approach derived above.
// We have numCoefficients variables (coefficients of reactants/products).
// We have numElements equations derived from atom conservation.
// The matrix should be numElements x numCoefficients.
// We need to find a non-trivial solution to Ax = 0.
// We can simplify the matrix by removing one redundant equation
// (since there are typically numElements+1 unknowns if we add a constant).
// Or, we solve Ax=0 and find the null space.
// A simpler way for web calculators is often iterative/guess-and-check or using a dedicated solver.
// Let's implement a basic matrix solver for Ax=0.
// We need to solve for coefficients x1, x2, …, xN.
// The matrix `matrix` is Ax = 0.
// Let's augment it conceptually for a solver that handles Ax=b, but with b=0.
// To solve Ax=0, we perform row reduction. The rank of A will be less than numCoefficients.
// The number of free variables = numCoefficients – rank(A).
// We can set one free variable to 1 and solve for others.
// Let's try to adapt a standard matrix solver.
// Add an identity matrix part if we want to express solutions in terms of basis vectors for the null space,
// but for balancing, we just need ONE solution.
// Simplified approach: Use a library or a robust algorithm.
// Given the constraints (no external libraries), we need a pure JS solver.
// Implementing a robust Gaussian elimination solver for Ax=0 is complex.
// For typical chemical equations, we can often solve by setting one coefficient and backtracking.
// However, a general approach requires matrix techniques.
// Let's refine the `solveMatrix` function to handle homogeneous systems properly.
// The `solveMatrix` function needs to return coefficients directly.
// Construct matrix for Ax=0. numRows = numElements, numCols = numCoefficients.
var augmentedMatrix = [];
for (var i = 0; i < matrix.length; i++) {
augmentedMatrix.push([…matrix[i]]); // Copy row
}
// Add a column of zeros to represent the 'b' vector in Ax=b, though it's Ax=0
// This structure can be useful for some solvers.
// Alternatively, just pass the coefficient matrix `matrix`.
var solutionCoefficients = solveHomogeneousSystem(matrix);
if (!solutionCoefficients || solutionCoefficients.length !== numCoefficients) {
document.getElementById('balancedFormula').innerText = "Could not balance. Check input.";
return;
}
// Ensure coefficients are positive and integers
var minPositiveCoeff = Infinity;
for (var i = 0; i 0 && solutionCoefficients[i] < minPositiveCoeff) {
minPositiveCoeff = solutionCoefficients[i];
}
}
if (minPositiveCoeff === Infinity) { // All coefficients were zero or negative (unlikely for valid reactions)
minPositiveCoeff = 1; // Default to 1 if no positive found
}
var finalCoeffs = solutionCoefficients.map(function(coeff) {
// Scale to make the smallest positive coefficient 1, then find LCM to make all integers
return coeff / minPositiveCoeff;
});
var commonDenominator = 1;
for (var i = 0; i 1) {
var denominator = Math.pow(10, fraction[1].length);
commonDenominator = lcm(commonDenominator, denominator);
}
}
finalCoeffs = finalCoeffs.map(function(coeff) {
return coeff * commonDenominator;
});
var finalIntegerCoeffs = finalCoeffs.map(function(coeff) {
return Math.round(coeff); // Round after scaling
});
var commonDivisor = arrayGcd(finalIntegerCoeffs);
finalIntegerCoeffs = finalIntegerCoeffs.map(function(coeff) {
return coeff / commonDivisor;
});
// Format the output string
var balancedReactants = [];
for (var i = 0; i < numReactants; i++) {
balancedReactants.push((finalIntegerCoeffs[i] === 1 ? '' : finalIntegerCoeffs[i]) + parsed.reactants[i]);
}
var balancedProducts = [];
for (var i = 0; i < numProducts; i++) {
balancedProducts.push((finalIntegerCoeffs[numReactants + i] === 1 ? '' : finalIntegerCoeffs[numReactants + i]) + parsed.products[i]);
}
var balancedFormula = balancedReactants.join(' + ') + ' → ' + balancedProducts.join(' + ');
document.getElementById('balancedFormula').innerText = balancedFormula;
}
// A simplified Gaussian elimination solver for Ax = 0
function solveHomogeneousSystem(matrix) {
var numRows = matrix.length;
if (numRows === 0) return [];
var numCols = matrix[0].length;
if (numCols === 0) return [];
// Create a copy to avoid modifying the original matrix
var augmented = matrix.map(function(row) { return […row]; });
var pivotRow = 0;
for (var col = 0; col < numCols && pivotRow < numRows; col++) {
// Find pivot: find the row with the largest absolute value in the current column below or at pivotRow
var maxRow = pivotRow;
for (var i = pivotRow + 1; i Math.abs(augmented[maxRow][col])) {
maxRow = i;
}
}
// If the pivot element is close to zero, move to the next column
if (Math.abs(augmented[maxRow][col]) < 1e-10) {
continue;
}
// Swap current row with the pivot row
var temp = augmented[pivotRow];
augmented[pivotRow] = augmented[maxRow];
augmented[maxRow] = temp;
// Eliminate other rows
var pivotElement = augmented[pivotRow][col];
for (var i = 0; i < numRows; i++) {
if (i !== pivotRow) {
var factor = augmented[i][col] / pivotElement;
for (var j = col; j < numCols; j++) {
augmented[i][j] -= factor * augmented[pivotRow][j];
}
}
}
pivotRow++;
}
// Now augmented matrix is in row echelon form (or close to it).
// Identify pivot columns and free variables.
var pivots = []; // Stores the column index for each pivot row
var solution = new Array(numCols).fill(0);
var freeVarIndices = [];
var currentRow = 0;
for(var col = 0; col < numCols && currentRow < numRows; col++) {
// Find the first non-zero element in the current row starting from `col`
var pivotFound = false;
for(var j = col; j 1e-10) {
if (j === col) { // This is a pivot column
pivots.push({ row: currentRow, col: j });
pivotFound = true;
break;
} else { // Non-zero element, but not the first one in the row – indicates dependency
// This column is likely a free variable if not already a pivot
break;
}
}
}
if (pivotFound) {
currentRow++;
}
}
// Mark free variables
var pivotCols = pivots.map(p => p.col);
for(var j = 0; j 0) {
// This might happen if the input is already balanced or has a single unique solution (unlikely for balancing)
// Or if rank == numCols. Return null or handle as error.
// For now, try to extract based on reduced form, assuming some non-zero values exist.
console.warn("No free variables detected. System might be trivial or ill-defined.");
// Attempt to find any non-zero solution if possible.
// If the matrix is full rank and square, it might have a unique zero solution.
// If not square, and rank == numCols, it's likely an issue.
}
// Set the first free variable to 1 and solve for others using back-substitution
if (freeVarIndices.length > 0) {
var freeVarIndex = freeVarIndices[0];
solution[freeVarIndex] = 1;
// Back-substitution
for (var i = pivots.length – 1; i >= 0; i–) {
var pivot = pivots[i];
var pivotCol = pivot.col;
var currentRowIndex = pivot.row;
var sumOfKnowns = 0;
for (var j = pivotCol + 1; j pivotCol) = 0
// So, x[pivotCol] = – sum(A[i][j]*x[j] for j>pivotCol) / A[i][pivotCol]
// simplified: x[pivotCol] = – sumOfKnowns / pivotElement
solution[pivotCol] = -sumOfKnowns / augmented[currentRowIndex][pivotCol];
}
} else if (numCols > 0) {
// Handle the case where there are no free variables but still need a solution.
// This is tricky for homogeneous systems. If rank=numCols, only trivial solution.
// If rank < numCols but no clear free variables identified (e.g. due to rounding),
// it might indicate an issue. For chemical balancing, we expect rank = 0; i–) {
var pivot = pivots[i];
var pivotCol = pivot.col;
var currentRowIndex = pivot.row;
var sumOfKnowns = 0;
for (var j = pivotCol + 1; j < numCols; j++) {
sumOfKnowns += augmented[currentRowIndex][j] * solution[j];
}
solution[pivotCol] = -sumOfKnowns / augmented[currentRowIndex][pivotCol];
}
} else {
// If the last variable IS a pivot, and no free variables, it's likely trivial solution.
console.error("Could not find a non-trivial solution for homogeneous system.");
return null;
}
}
// Clean up near-zero values and normalize
var cleanedSolution = solution.map(function(val) {
return Math.abs(val) < 1e-10 ? 0 : val;
});
// Find GCD of the non-zero elements to normalize to integers
var nonZeroElements = cleanedSolution.filter(function(val) { return val !== 0; });
if (nonZeroElements.length === 0) return cleanedSolution; // All zeros
var commonDivisor = nonZeroElements[0];
for (var i = 1; i < nonZeroElements.length; i++) {
commonDivisor = gcd(commonDivisor, nonZeroElements[i]);
}
// Ensure commonDivisor is not zero before dividing
if (commonDivisor === 0) commonDivisor = 1;
var normalizedSolution = cleanedSolution.map(function(val) {
// Scale by the reciprocal of the smallest non-zero value, then find LCM
// OR, find GCD of all, then divide.
// Let's try scaling by the smallest magnitude non-zero element, then rationalize.
var smallestNonZero = Infinity;
for(var k=0; k < cleanedSolution.length; k++) {
if (cleanedSolution[k] !== 0 && Math.abs(cleanedSolution[k]) val / smallestNonZero);
// Now rationalize these scaled numbers to integers
var L = 1;
for (var k = 0; k Math.round(val * L));
var finalGcd = arrayGcd(integers);
if (finalGcd === 0) finalGcd = 1;
return integers.map(val => val / finalGcd);
});
// Ensure all coefficients are positive. If the first is negative, flip all signs.
if (normalizedSolution.length > 0 && normalizedSolution[0] < 0) {
return normalizedSolution.map(function(val) { return -val; });
}
return normalizedSolution;
}