Calculate the rate constant (k) based on concentration changes over time.
Seconds (s)
Minutes (min)
Hours (h)
Days (d)
Rate Constant (k): –
Reaction Half-Life (t1/2): –
Note: For a first-order reaction, the units of k are time⁻¹ (e.g., s⁻¹, min⁻¹).
How to Calculate Rate Constant of First Order Reaction
In chemical kinetics, a first-order reaction is a reaction where the rate depends linearly on the concentration of only one reactant. Common examples include radioactive decay and the decomposition of hydrogen peroxide. Understanding how to calculate the rate constant (k) is crucial for predicting how fast a reaction will proceed or determining the shelf-life of pharmaceutical products.
The Integrated Rate Law
To calculate the rate constant from experimental data, we use the integrated rate law for first-order reactions. This equation relates the initial concentration of the reactant to its concentration at any specific time.
ln([A]₀ / [A]ₜ) = k × t
Where:
[A]₀ = Initial concentration (Molarity, mol/L, or pressure)
[A]ₜ = Concentration at time t
k = Rate constant (units of time⁻¹)
t = Time elapsed
Rearranging this formula to solve for the rate constant k gives:
k = (1 / t) × ln([A]₀ / [A]ₜ)
Understanding Half-Life
A unique characteristic of first-order reactions is that their half-life (t1/2) is constant. It does not depend on the initial concentration. This means it takes the same amount of time for the concentration to drop from 1.0M to 0.5M as it does from 0.5M to 0.25M.
The relationship between the rate constant and half-life is:
t1/2 = ln(2) / k ≈ 0.693 / k
Example Calculation
Suppose you have a reaction where the initial concentration of reactant A is 2.00 M. After 150 seconds, the concentration drops to 1.20 M.
function calculateRateConstant() {
// 1. Get DOM elements
var initialConcInput = document.getElementById('chemInitialConc');
var finalConcInput = document.getElementById('chemFinalConc');
var timeInput = document.getElementById('chemTime');
var unitSelect = document.getElementById('chemTimeUnits');
var resultBox = document.getElementById('chemResultBox');
var errorBox = document.getElementById('chemErrorMessage');
var resK = document.getElementById('resRateConstant');
var resHalf = document.getElementById('resHalfLife');
// 2. Get values
var a0 = parseFloat(initialConcInput.value);
var at = parseFloat(finalConcInput.value);
var t = parseFloat(timeInput.value);
var unit = unitSelect.value;
// 3. Validation Logic
errorBox.style.display = 'none';
resultBox.style.display = 'none';
if (isNaN(a0) || isNaN(at) || isNaN(t)) {
errorBox.textContent = "Please enter valid numerical values for all fields.";
errorBox.style.display = 'block';
return;
}
if (a0 <= 0 || at <= 0 || t = a0) {
errorBox.textContent = "For a reactant in a first-order reaction, final concentration must be lower than initial concentration.";
errorBox.style.display = 'block';
return;
}
// 4. Calculation Logic
// Formula: k = (1/t) * ln(a0 / at)
var ratio = a0 / at;
var lnRatio = Math.log(ratio);
var k = lnRatio / t;
// Calculate Half-life: t1/2 = 0.693 / k
var halfLife = Math.log(2) / k;
// 5. Formatting Results
// Determine precision based on magnitude
var kDisplay = k < 0.001 ? k.toExponential(4) : k.toPrecision(5);
var hlDisplay = halfLife < 0.001 ? halfLife.toExponential(4) : halfLife.toPrecision(5);
// 6. Output to DOM
resK.innerHTML = kDisplay + " " + unit + "-1";
resHalf.innerHTML = hlDisplay + " " + unit;
resultBox.style.display = 'block';
}