Enter your unbalanced chemical equation below. Use element symbols (e.g., H, O, Na, Cl) and numbers for subscripts. Coefficients will be added automatically.
Understanding Chemical Equation Balancing
Chemical equations are a shorthand way of representing chemical reactions. They show the reactants (the substances that react) and the products (the substances that are formed). For a chemical equation to be scientifically accurate and useful, it must adhere to the Law of Conservation of Mass, which states that matter cannot be created or destroyed in a chemical reaction. This means the number of atoms of each element must be the same on both sides of the equation.
The Balancing Process
Balancing a chemical equation involves adjusting the coefficients (numbers placed in front of chemical formulas) to ensure that the number of atoms of each element is equal on the reactant side (left side of the arrow) and the product side (right side of the arrow).
Key Principles:
Conservation of Mass: The total number of atoms for each element must be identical on both sides.
Subscripts Cannot Be Changed: The chemical formula of a substance (e.g., H₂O) cannot be altered. Changing a subscript would change the identity of the substance.
Coefficients are Adjusted: Only the numbers in front of the chemical formulas (coefficients) can be changed to balance the equation.
How the Calculator Works (Conceptual Overview)
This calculator employs a mathematical approach, often using matrix methods or simultaneous equations, to determine the smallest whole-number coefficients that satisfy the conservation of atoms for each element. Here's a simplified conceptual breakdown:
Identify Elements: All unique elements present in the equation are identified.
Count Atoms: The number of atoms for each element is counted on both the reactant and product sides based on the given formulas.
Set up Equations: A system of linear equations is established where each equation represents the balance of one element. For example, for the reaction H₂ + O₂ → H₂O, we have:
Hydrogen (H): 2x = 2z (where x and z are coefficients for H₂ and H₂O respectively)
Oxygen (O): 2y = z (where y is the coefficient for O₂)
Solve the System: These equations are solved, typically yielding fractional or relative values.
Find Smallest Integers: The smallest set of whole numbers that satisfy these ratios is determined by finding a common denominator and multiplying accordingly. In the H₂ + O₂ → H₂O example, the solution might initially be H:O:H₂O = 1:0.5:1. Multiplying by 2 gives the balanced coefficients: 2H₂ + 1O₂ → 2H₂O.
Example: Synthesis of Water
Let's balance the synthesis of water, H₂ + O₂ → H₂O.
Oxygen is unbalanced. To balance oxygen, we need 2 oxygen atoms on the product side. We can't change the subscript in H₂O, so we add a coefficient of 2:
H₂ + O₂ → 2H₂O
Now, let's check the atoms again:
Reactants: H (2), O (2)
Products: H (4), O (2)
Hydrogen is now unbalanced. We need 4 hydrogen atoms on the reactant side. We add a coefficient of 2 to H₂:
2H₂ + O₂ → 2H₂O
Final Check:
Reactants: H (2 * 2 = 4), O (2)
Products: H (2 * 2 = 4), O (2 * 1 = 2)
The equation is now balanced!
Use Cases
Balancing chemical equations is a fundamental skill in chemistry, essential for:
Stoichiometry Calculations: Determining the quantitative relationships between reactants and products, crucial for predicting yields and reactant amounts.
Understanding Reaction Mechanisms: Visualizing the transformation of matter at an atomic level.
Environmental Science: Analyzing chemical processes in pollution control and remediation.
Industrial Chemistry: Designing and optimizing chemical manufacturing processes.
// Helper function to parse a chemical formula and count atoms
function parseFormula(formula) {
var elements = {};
// Regex to find element symbols (uppercase followed by optional lowercase) and their counts
var regex = /([A-Z][a-z]?)(\d*)/g;
var match;
while ((match = regex.exec(formula)) !== null) {
var element = match[1];
var count = match[2] === " ? 1 : parseInt(match[2]);
elements[element] = (elements[element] || 0) + count;
}
return elements;
}
// Function to get unique elements from a chemical equation string
function getUniqueElements(equation) {
var elements = new Set();
var parts = equation.split(/[+=]/); // Split by '+' and '='
for (var i = 0; i < parts.length; i++) {
var formula = parts[i].trim();
var regex = /([A-Z][a-z]?)(\d*)/g;
var match;
while ((match = regex.exec(formula)) !== null) {
elements.add(match[1]);
}
}
return Array.from(elements);
}
// Function to balance the chemical equation
function balanceEquation() {
var inputEquation = document.getElementById("equationInput").value.trim();
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
if (!inputEquation) {
resultDiv.innerHTML = "Please enter a chemical equation.";
return;
}
// Basic validation: Check for presence of '='
if (inputEquation.indexOf('=') === -1) {
resultDiv.innerHTML = "Invalid equation format. Please include an '=' sign.";
return;
}
var sides = inputEquation.split('=');
if (sides.length !== 2) {
resultDiv.innerHTML = "Invalid equation format. Should have one '=' sign.";
return;
}
var reactantsStr = sides[0].trim();
var productsStr = sides[1].trim();
var reactantFormulas = reactantsStr.split('+').map(function(f) { return f.trim(); });
var productFormulas = productsStr.split('+').map(function(f) { return f.trim(); });
var allFormulas = reactantFormulas.concat(productFormulas);
var numReactants = reactantFormulas.length;
var numProducts = productFormulas.length;
var numReactions = numReactants + numProducts;
var uniqueElements = getUniqueElements(inputEquation);
var numElements = uniqueElements.length;
// Create a matrix for solving the system of linear equations
// Rows = elements, Columns = coefficients (one for each formula)
var matrix = [];
for (var i = 0; i < numElements; i++) {
matrix.push(new Array(numReactions).fill(0));
}
// Populate the matrix
for (var j = 0; j = numReactants;
for (var k = 0; k […row]); // Deep copy
// Try to solve by setting the first coefficient to 1 and reducing the system
// This is a heuristic and might fail for complex cases.
// A robust solution would require a proper Gaussian elimination or similar algorithm.
// Let's simplify for educational purposes and common reactions.
// We'll represent it as a system Ax = 0 where x are coefficients.
// The null space of A contains the solutions.
// For a simplified implementation, we can try iterative balancing or
// a specific algorithm for chemical equations.
// Given the constraints, let's attempt a basic Gaussian elimination.
function solveLinearSystem(mat) {
var numRows = mat.length;
var numCols = mat[0].length;
var augmentedMatrix = mat.map(row => […row, 0]); // Make it Ax = 0 system
// Gaussian Elimination (Forward Elimination)
for (var i = 0, k = 0; i < numRows && k < numCols; i++, k++) {
// Find pivot row
var pivotRow = i;
for (var r = i + 1; r Math.abs(augmentedMatrix[pivotRow][k])) {
pivotRow = r;
}
}
// Swap rows
var temp = augmentedMatrix[i];
augmentedMatrix[i] = augmentedMatrix[pivotRow];
augmentedMatrix[pivotRow] = temp;
// Make pivot element 1
var pivot = augmentedMatrix[i][k];
if (Math.abs(pivot) < 1e-10) { // Pivot is essentially zero
k–; // Try next column without incrementing row
continue;
}
for (var j = k; j <= numCols; j++) {
augmentedMatrix[i][j] /= pivot;
}
// Eliminate other rows
for (var r = 0; r < numRows; r++) {
if (r !== i) {
var factor = augmentedMatrix[r][k];
for (var j = k; j <= numCols; j++) {
augmentedMatrix[r][j] -= factor * augmentedMatrix[i][j];
}
}
}
}
// After forward elimination, we should have a row echelon form or reduced row echelon form.
// For Ax = 0, the solution vector x lies in the null space.
// We often need to find the smallest integer solutions.
// This implementation is simplified and might not handle all cases perfectly.
// For a robust solver, consider dedicated libraries.
// Let's assume we get a solution vector, perhaps by setting the last variable
// to 1 and solving backwards, then scaling.
var solution = new Array(numCols).fill(0);
// Simplification: Assume the system is consistent and has a non-trivial solution.
// We need to find the null space. This requires more advanced techniques like SVD or
// finding a basis for the null space.
// For common chemical reactions, the matrix rank is usually numElements – 1.
// We can express numElements variables in terms of the last one.
var rank = 0;
for(var r=0; r<numRows; ++r) {
var nonZero = false;
for(var c=0; c 1e-10) {
nonZero = true;
break;
}
}
if (nonZero) rank++;
}
if (rank Array(m + 1).fill(0));
for (var r = 0; r < n; r++) {
for (var c = 0; c < m; c++) {
M[r][c] = matrix[r][c];
}
M[r][m] = 0; // RHS is 0 for Ax = 0
}
// Forward elimination
for (var pivot_row = 0, col = 0; pivot_row < n && col < m; col++) {
// Find pivot
var max_row = pivot_row;
for (var k = pivot_row + 1; k Math.abs(M[max_row][col])) {
max_row = k;
}
}
// Swap rows
[M[pivot_row], M[max_row]] = [M[max_row], M[pivot_row]];
// If pivot is zero, move to next column
if (Math.abs(M[pivot_row][col]) < 1e-10) {
continue;
}
// Normalize pivot row
var pivot_val = M[pivot_row][col];
for (var j = col; j <= m; j++) {
M[pivot_row][j] /= pivot_val;
}
// Eliminate other rows
for (var r = 0; r < n; r++) {
if (r !== pivot_row) {
var factor = M[r][col];
for (var j = col; j <= m; j++) {
M[r][j] -= factor * M[pivot_row][j];
}
}
}
pivot_row++;
}
// Now M is in Reduced Row Echelon Form (approximately)
// We need to find the null space. For chemical equations, rank is usually n-1.
// So, one variable is free. Let's assume the last one (m-1) is free.
var coeffs = new Array(m).fill(0);
var free_var_index = -1;
// Find the first column without a leading 1 (or is all zeros) – this indicates a free variable
for (var c = 0; c < m; c++) {
var is_pivot_col = false;
for (var r = 0; r < n; r++) {
if (Math.abs(M[r][c] – 1) < 1e-10) { // Check for leading 1
// Check if it's the first non-zero in its row
var is_leading = true;
for(var pc=0; pc 1e-10) {
is_leading = false;
break;
}
}
if (is_leading) {
is_pivot_col = true;
break;
}
}
}
if (!is_pivot_col) {
free_var_index = c;
break;
}
}
// If all columns are pivot columns, something is wrong or it's trivial (0=0)
if (free_var_index === -1) {
// Try assuming the last variable is free if rank is m-1
if (rank === m -1) {
free_var_index = m – 1;
} else {
// Fallback or error
resultDiv.innerHTML = "Could not determine coefficients (complex matrix).";
return;
}
}
// Set the free variable to 1 (temporarily)
coeffs[free_var_index] = 1;
// Back substitution to find other coefficients
for (var r = n – 1; r >= 0; r–) {
var pivot_col = -1;
for (var c = 0; c < m; c++) {
if (Math.abs(M[r][c] – 1) < 1e-10) { // Find leading 1
var is_leading = true;
for(var pc=0; pc 1e-10) {
is_leading = false;
break;
}
}
if (is_leading) {
pivot_col = c;
break;
}
}
}
if (pivot_col !== -1 && pivot_col !== free_var_index) {
var sum = 0;
for (var c = pivot_col + 1; c < m; c++) {
sum += M[r][c] * coeffs[c];
}
coeffs[pivot_col] = -sum; // Since M[r][m] is 0
}
}
// Now 'coeffs' contains a solution vector, possibly fractional.
// Find the least common multiple (LCM) of the denominators to get integers.
// Helper function for GCD
function gcd(a, b) {
return b === 0 ? a : gcd(b, a % b);
}
// Helper function for LCM of two numbers
function lcm(a, b) {
if (a === 0 || b === 0) return 0;
return Math.abs(a * b) / gcd(a, b);
}
// Convert coefficients to fractions and find LCM of denominators
var denominators = [];
for (var i = 0; i < m; i++) {
// Approximate fraction representation
var num = coeffs[i];
// Find denominator for num (e.g., if num is 0.5, denominator is 2)
// This part is tricky with floating point precision.
// A robust way is to use a fraction library.
// Simplified approach: check for common fractions or find denominator.
// Let's assume coefficients are reasonably simple fractions.
// A simple way to get denominators for numbers like 0.5, 0.333, 0.25, etc.
var approxNum = parseFloat(num.toFixed(6)); // limit precision issues
if (Math.abs(approxNum) epsilon) {
q++;
p = Math.round(approxNum * q);
if (q > 1000) break; // Avoid infinite loop for bad approximations
}
if (q > 1000) q = 1; // Default if approximation fails
denominators.push(q);
}
// Calculate LCM of all denominators
var final_lcm = 1;
for (var i = 0; i < denominators.length; i++) {
// Ensure we handle potential non-integer values from approximation
var denom = Math.round(denominators[i]);
if (denom === 0) denom = 1; // Avoid division by zero
final_lcm = lcm(final_lcm, denom);
}
// Scale coefficients by LCM to get smallest integers
var finalCoeffs = [];
for (var i = 0; i < m; i++) {
var scaledCoeff = coeffs[i] * final_lcm;
// Round to nearest integer, handling potential floating point inaccuracies
finalCoeffs.push(Math.round(scaledCoeff));
}
// Ensure all coefficients are positive (by flipping sign if necessary, though Ax=0 usually handles this implicitly)
// And find the smallest possible integers by dividing by their GCD if needed.
var finalGcd = finalCoeffs[0];
for(var i=1; i<m; ++i) {
finalGcd = gcd(finalGcd, finalCoeffs[i]);
}
for(var i=0; i<m; ++i) {
finalCoeffs[i] /= finalGcd;
// Ensure positivity if needed (usually handled by the solver setup)
if (finalCoeffs[i] < 0) {
for(var j=0; j<m; ++j) finalCoeffs[j] *= -1;
break;
}
}
// Construct the balanced equation string
var balancedEquation = "";
for (var i = 0; i < numReactants; i++) {
balancedEquation += (finalCoeffs[i] === 1 ? "" : finalCoeffs[i]) + reactantFormulas[i] + (i === numReactants – 1 ? "" : " + ");
}
balancedEquation += " = ";
for (var i = 0; i < numProducts; i++) {
balancedEquation += (finalCoeffs[numReactants + i] === 1 ? "" : finalCoeffs[numReactants + i]) + productFormulas[i] + (i === numProducts – 1 ? "" : " + ");
}
resultDiv.innerHTML = "Balanced Equation: " + balancedEquation + "";
} else {
// This case implies rank = numCols, which for Ax=0 means only the trivial solution (all zeros).
// Or rank > numCols, which indicates an inconsistent system (shouldn't happen for Ax=0).
resultDiv.innerHTML = "Could not balance the equation (likely trivial or inconsistent system).";
}
}
// Execute the solver
solveLinearSystem(matrix);
} // end of balanceEquation function