function calculateRateConstant() {
// Clear previous errors
var errorDiv = document.getElementById('errorMessage');
var resultDiv = document.getElementById('resultDisplay');
errorDiv.style.display = 'none';
resultDiv.style.display = 'none';
// Get Input Values
var a0 = parseFloat(document.getElementById('initConc').value);
var at = parseFloat(document.getElementById('finalConc').value);
var time = parseFloat(document.getElementById('timeElapsed').value);
// Validation Logic
if (isNaN(a0) || isNaN(at) || isNaN(time)) {
errorDiv.innerText = "Please enter valid numbers for all fields.";
errorDiv.style.display = 'block';
return;
}
if (a0 <= 0 || at <= 0 || time = a0) {
errorDiv.innerText = "Final concentration must be less than initial concentration for a reactant.";
errorDiv.style.display = 'block';
return;
}
// Calculation: Integrated Rate Law for Second Order: 1/[A]t – 1/[A]0 = kt
// Therefore, k = (1/[A]t – 1/[A]0) / t
var inverseAt = 1 / at;
var inverseA0 = 1 / a0;
var kVal = (inverseAt – inverseA0) / time;
// Calculate Half-Life: t1/2 = 1 / (k * [A]0)
var halfLife = 1 / (kVal * a0);
// Update Results
document.getElementById('kResult').innerHTML = kVal.toPrecision(4) + " M⁻¹s⁻¹";
document.getElementById('halfLifeResult').innerHTML = halfLife.toPrecision(4) + " s";
// Show result box
resultDiv.style.display = 'block';
}
How to Calculate Second Order Rate Constant
In chemical kinetics, determining the rate constant ($k$) is essential for understanding the speed at which a chemical reaction occurs. A second-order reaction implies that the rate of the reaction is proportional to the square of the concentration of a single reactant or the product of the concentrations of two reactants.
This calculator specifically addresses the most common second-order scenario involving a single reactant ($2A \rightarrow P$) or two reactants with equal initial concentrations using the Integrated Rate Law.
The Formula
To calculate the rate constant for a second-order reaction, we use the integrated rate law formula:
1/[A]ₜ – 1/[A]₀ = kt
Where:
[A]ₜ = The final concentration of the reactant at time $t$ (Molarity, M)
[A]₀ = The initial concentration of the reactant (Molarity, M)
k = The second-order rate constant ($M^{-1}s^{-1}$)
t = The time elapsed (seconds)
Rearranging this equation to solve for the rate constant ($k$), we get:
k = ( 1/[A]ₜ – 1/[A]₀ ) / t
Understanding Units
Unlike first-order reactions where $k$ has units of reciprocal time ($s^{-1}$), the units for a second-order rate constant depend on concentration. The standard unit is M⁻¹s⁻¹ (Inverse Molarity per Second), which can also be written as $L \cdot mol^{-1} \cdot s^{-1}$.
Example Calculation
Let's assume a dimerization reaction of butadiene ($2C_4H_6 \rightarrow C_8H_{12}$) follows second-order kinetics.
Given:
Initial Concentration ($[A]_0$): 0.100 M
Concentration after 500 seconds ($[A]_t$): 0.080 M
Time ($t$): 500 seconds
Step 1: Calculate the inverse of concentrations.
$1 / 0.080 = 12.5 M^{-1}$
$1 / 0.100 = 10.0 M^{-1}$
Step 3: Divide by time to find $k$.
$k = 2.5 / 500 = 0.005 M^{-1}s^{-1}$
Half-Life Calculation
For second-order reactions, the half-life ($t_{1/2}$) is not constant; it depends on the initial concentration. The calculator also computes this value using the formula:
t½ = 1 / (k[A]₀)
This means that as the reaction proceeds and the concentration decreases, the half-life increases—the reaction takes longer to consume half of the remaining reactant.