function calculateRateConstant() {
var initial = document.getElementById('initialConc').value;
var final = document.getElementById('finalConc').value;
var time = document.getElementById('timeElapsed').value;
var errorDiv = document.getElementById('errorMsg');
var resultDiv = document.getElementById('calcResult');
var kDisplay = document.getElementById('kResult');
var hlDisplay = document.getElementById('halfLifeResult');
// Reset display
errorDiv.style.display = 'none';
resultDiv.style.display = 'none';
errorDiv.innerHTML = ";
// Validation
if (initial === " || final === " || time === ") {
errorDiv.innerHTML = 'Please fill in all fields.';
errorDiv.style.display = 'block';
return;
}
var a0 = parseFloat(initial);
var at = parseFloat(final);
var t = parseFloat(time);
if (isNaN(a0) || isNaN(at) || isNaN(t)) {
errorDiv.innerHTML = 'Please enter valid numbers.';
errorDiv.style.display = 'block';
return;
}
if (a0 <= 0 || at <= 0 || t = a0) {
errorDiv.innerHTML = 'For a reaction to proceed, final concentration must be less than initial concentration.';
errorDiv.style.display = 'block';
return;
}
// Calculation Logic for Second Order: 1/[A]t – 1/[A]0 = kt
// Therefore k = (1/[A]t – 1/[A]0) / t
var inverseFinal = 1 / at;
var inverseInitial = 1 / a0;
var k = (inverseFinal – inverseInitial) / t;
// Half-life calculation for second order: t1/2 = 1 / (k * [A]0)
var halfLife = 1 / (k * a0);
// Formatting
// If numbers are very small or very large, use exponential notation
var kFormatted = k 1000 ? k.toExponential(4) : k.toFixed(5);
var hlFormatted = halfLife 1000 ? halfLife.toExponential(4) : halfLife.toFixed(2);
kDisplay.innerHTML = kFormatted + ' M⁻¹s⁻¹';
hlDisplay.innerHTML = hlFormatted + ' s';
resultDiv.style.display = 'block';
}
How to Calculate Rate Constant (k) for Second Order Reactions
Understanding chemical kinetics is essential for predicting how fast a reaction will occur. In a second-order reaction, the rate of reaction is proportional to either the square of the concentration of one reactant or the product of the concentrations of two different reactants. This calculator helps you determine the rate constant ($k$) and the half-life ($t_{1/2}$) based on the integrated rate law.
The Integrated Rate Law Formula
For a reaction of the type $2A \rightarrow P$ or $A + B \rightarrow P$ (where initial concentrations are equal), the integrated rate law is derived from the differential rate equation. The linear form used to solve for the rate constant $k$ is:
1 / [A]t = kt + 1 / [A]0
Where:
[A]t: The concentration of reactant A at time $t$ (Molarity, M).
[A]0: The initial concentration of reactant A (Molarity, M).
k: The rate constant (units: M-1s-1).
t: The time elapsed (seconds, minutes, etc.).
Rearranging to Find k
To use the calculator above manually, you can rearrange the formula to isolate $k$:
k = ( 1 / [A]t – 1 / [A]0 ) / t
Calculating Half-Life for Second Order Reactions
Unlike first-order reactions, the half-life of a second-order reaction depends on the initial concentration. As the reaction proceeds and concentration decreases, the half-life increases. The formula is:
t1/2 = 1 / (k × [A]0)
Example Calculation
Let's say we have a reaction where the initial concentration of reactant A is 0.100 M. After 60 seconds, the concentration drops to 0.050 M.
Calculate the inverse of the final concentration: $1 / 0.050 = 20$.
Calculate the inverse of the initial concentration: $1 / 0.100 = 10$.
Subtract the initial from the final: $20 – 10 = 10$.
Divide by the time: $10 / 60 = 0.1667$.
The rate constant $k$ is roughly 0.1667 M-1s-1.
Units of the Rate Constant
It is crucial to note that the units for the rate constant in a second-order reaction differ from zeroth or first-order reactions. Since Rate = k[A]2, and Rate is in M/s, $k$ must cancel out the M2 to result in M/s. Therefore, the units are inverse molarity times inverse time ($M^{-1}s^{-1}$ or $L/(mol \cdot s)$).