How to Calculate Rate Constant for Zero Order Reaction
In chemical kinetics, a zero order reaction is a unique type of chemical reaction where the rate is independent of the concentration of the reactants. This means that as the reaction proceeds and reactants are consumed, the speed of the reaction does not slow down; it remains constant until the limiting reactant is completely exhausted.
The Zero Order Integrated Rate Law
To determine the rate constant (k) for a zero order reaction, we use the integrated rate law formula. This linear equation relates concentration to time:
[A]ₜ = -kt + [A]₀
Where:
[A]ₜ = Concentration of the reactant at time t
[A]₀ = Initial concentration of the reactant
k = Rate constant (units of M/time)
t = Time elapsed
Formula for Calculating k
By rearranging the equation above, we can isolate the rate constant:
k = ([A]₀ – [A]ₜ) / t
This formula tells us that the rate constant is simply the change in concentration divided by the time it took for that change to occur. Because the rate is constant, the graph of Concentration vs. Time yields a straight line with a slope of -k.
Units of the Rate Constant
Unlike first or second order reactions, the units for the rate constant in a zero order reaction are the same as the reaction rate itself. The standard units are Molarity per unit time (e.g., M/s, M/min, or mol·L⁻¹·s⁻¹).
Half-Life of a Zero Order Reaction
This calculator also computes the half-life (t₁/₂), which is the time required for the concentration to reduce to half of its initial value. For zero order reactions, the half-life is dependent on the initial concentration:
t₁/₂ = [A]₀ / (2k)
Example Calculation
Imagine a decomposition reaction of gas on a hot metal surface that follows zero order kinetics.
Initial Concentration: 0.100 M
Concentration after 50 seconds: 0.085 M
Calculation: k = (0.100 – 0.085) / 50
Result: k = 0.0003 M/s
Common Applications
Zero order kinetics are often observed in enzyme-catalyzed reactions where the enzyme is saturated with substrate (Michaelis-Menten kinetics at high substrate concentration) or in surface-catalyzed reactions where the active sites on the metal surface are fully occupied.
function calculateRateConstant() {
// Get DOM elements
var initialConcInput = document.getElementById('initialConc');
var finalConcInput = document.getElementById('finalConc');
var timeInput = document.getElementById('timeElapsed');
var timeUnitSelect = document.getElementById('timeUnit');
var resultBox = document.getElementById('result');
var errorDisplay = document.getElementById('errorDisplay');
var concChangeDisplay = document.getElementById('concChangeResult');
var kDisplay = document.getElementById('rateConstantResult');
var halfLifeDisplay = document.getElementById('halfLifeResult');
// Get values
var a0 = parseFloat(initialConcInput.value);
var at = parseFloat(finalConcInput.value);
var t = parseFloat(timeInput.value);
var unit = timeUnitSelect.value;
// Reset functionality
resultBox.style.display = 'none';
errorDisplay.style.display = 'none';
// Validation logic
if (isNaN(a0) || isNaN(at) || isNaN(t)) {
errorDisplay.textContent = "Please enter valid numeric values for all fields.";
errorDisplay.style.display = 'block';
return;
}
if (t <= 0) {
errorDisplay.textContent = "Time elapsed must be greater than zero.";
errorDisplay.style.display = 'block';
return;
}
if (a0 < 0 || at a0 mathematically,
// though physically this implies a product or an error in input for a reactant.
// We will calculate magnitude of rate assuming reactant disappearance.
var deltaA = a0 – at;
// If deltaA is negative, it means concentration increased (Product logic?),
// but standard Zero Order formula usually assumes reactant: [A] = -kt + [A]0
// If the user inputs data where final > initial, k would be negative.
// We will display the raw calculation but alert if negative.
var k = deltaA / t;
// Calculate Half-life: t1/2 = [A]0 / 2k
// Note: If k is negative (impossible for standard rate constant), half life is undefined
var halfLife = 0;
var halfLifeText = "-";
if (k > 0) {
halfLife = a0 / (2 * k);
halfLifeText = halfLife.toPrecision(4) + " " + unit;
} else if (k === 0) {
halfLifeText = "Infinite (No reaction)";
} else {
halfLifeText = "N/A (Conc. Increased)";
}
// Formatting results
// Use scientific notation if numbers are very small ( 10000)
var kFormatted = Math.abs(k) 1e4 ? k.toExponential(4) : k.toPrecision(4);
var deltaFormatted = Math.abs(deltaA) < 1e-4 && deltaA !== 0 ? deltaA.toExponential(4) : deltaA.toPrecision(4);
// Update DOM
concChangeDisplay.innerHTML = deltaFormatted + " M";
kDisplay.innerHTML = kFormatted + " M/" + unit;
halfLifeDisplay.innerHTML = halfLifeText;
// Show Results
resultBox.style.display = 'block';
}