Calculate reaction rate constants based on integrated rate law plots.
Zero Order (0)
First Order (1)
Second Order (2)
Plot: [A] vs t | Slope = -k
Seconds (s)
Minutes (min)
Hours (h)
Days (d)
Molarity (M or mol/L)
Millimolar (mM)
Atmospheres (atm) [Gases]
Please enter a valid numeric slope.
Calculated Rate Constant (k):
Correct Units:
Relationship Used:
Half-Life Formula:
function updateFormulaHint() {
var order = document.getElementById('reactionOrder').value;
var hint = document.getElementById('formulaHint');
if (order === '0') {
hint.innerHTML = "Plot: [A] vs time | Slope = -k";
} else if (order === '1') {
hint.innerHTML = "Plot: ln[A] vs time | Slope = -k";
} else if (order === '2') {
hint.innerHTML = "Plot: 1/[A] vs time | Slope = k";
}
}
function calculateRateConstant() {
// Get Inputs
var orderStr = document.getElementById('reactionOrder').value;
var slopeInput = document.getElementById('slopeValue').value;
var tUnit = document.getElementById('timeUnit').value;
var cUnit = document.getElementById('concUnit').value;
var errorDiv = document.getElementById('errorMsg');
var resultBox = document.getElementById('resultBox');
// Validation
if (slopeInput === "" || isNaN(slopeInput)) {
errorDiv.style.display = 'block';
resultBox.style.display = 'none';
return;
}
errorDiv.style.display = 'none';
var slope = parseFloat(slopeInput);
var k = 0;
var unitString = "";
var formulaString = "";
var halfLifeString = "";
var order = parseInt(orderStr);
// Calculation Logic
if (order === 0) {
// Zero Order: Slope = -k, so k = -Slope
k = -slope;
// Note: If user entered positive slope by accident, k might be negative.
// Rate constants are typically positive. If k < 0, we assume user input absolute slope.
if (k < 0) k = Math.abs(k);
unitString = cUnit + "·" + tUnit + "⁻¹"; // e.g., M/s
formulaString = "k = -slope";
halfLifeString = "[A]₀ / 2k";
} else if (order === 1) {
// First Order: Slope = -k
k = -slope;
if (k < 0) k = Math.abs(k);
unitString = tUnit + "⁻¹"; // e.g., 1/s
formulaString = "k = -slope";
halfLifeString = "ln(2) / k ≈ 0.693 / k";
} else if (order === 2) {
// Second Order: Slope = k
k = slope;
// k should be positive
if (k < 0) k = Math.abs(k);
unitString = cUnit + "⁻¹·" + tUnit + "⁻¹"; // e.g., 1/(M·s)
formulaString = "k = slope";
halfLifeString = "1 / (k[A]₀)";
}
// formatting result
// Use scientific notation if number is very small or very large
var displayK = k;
if (k 10000) {
displayK = k.toExponential(4);
} else {
displayK = k.toFixed(5).replace(/\.?0+$/, ""); // Trim trailing zeros
}
// Display Results
document.getElementById('resultK').innerText = displayK;
document.getElementById('resultUnits').innerText = unitString;
document.getElementById('resultFormula').innerText = formulaString;
document.getElementById('halfLifeFormula').innerText = halfLifeString;
resultBox.style.display = 'block';
}
How to Calculate Rate Constant from Slope
In chemical kinetics, the rate constant ($k$) is a crucial value that quantifies the speed of a chemical reaction. One of the most accurate methods to determine $k$ is by analyzing experimental data using the Integrated Rate Laws. By plotting concentration data against time in specific ways, we can generate a linear graph ($y = mx + b$). The slope of this line allows us to calculate the rate constant directly.
1. Identifying the Reaction Order
Before calculating the rate constant, you must know the order of the reaction. This is determined by which plot yields a straight line:
Zero Order: Plot of Concentration $[A]$ vs. Time ($t$) is linear.
First Order: Plot of Natural Log $\ln[A]$ vs. Time ($t$) is linear.
Second Order: Plot of Inverse Concentration $1/[A]$ vs. Time ($t$) is linear.
2. Formulas for Calculating k from Slope
Once you have the slope ($m$) from your linear regression or graph, the relationship between the slope and the rate constant depends on the reaction order.
Order
Linear Plot (y vs x)
Relationship
Rate Constant Formula
Common Units
Zero (0)
$[A]$ vs $t$
Slope = $-k$
$k = -\text{slope}$
$M/s$
First (1)
$\ln[A]$ vs $t$
Slope = $-k$
$k = -\text{slope}$
$1/s$
Second (2)
$1/[A]$ vs $t$
Slope = $k$
$k = \text{slope}$
$1/(M\cdot s)$
3. Step-by-Step Calculation Example
Let's assume you are studying the decomposition of a reactant. You plot the natural logarithm of concentration ($\ln[A]$) against time in seconds and get a straight line.
Step 1: Determine Order. Since the plot of $\ln[A]$ vs. Time is linear, the reaction is First Order.
Step 2: Find the Slope. The equation of the line is $y = -0.0045x – 2.3$. Here, the slope ($m$) is $-0.0045$.
Step 3: Apply Formula. For a first-order reaction, Slope = $-k$. Therefore, $-0.0045 = -k$.
Step 4: Solve for k. $k = 0.0045 \text{ s}^{-1}$.
4. Why are Units Important?
The units of the rate constant change depending on the reaction order. This is a common point of confusion.
Zero Order: Rate is independent of concentration, so $k$ has units of concentration/time (e.g., $M \cdot s^{-1}$).
First Order: Rate is proportional to concentration. The units of concentration cancel out, leaving only reciprocal time (e.g., $s^{-1}$ or $min^{-1}$).
Second Order: Rate is proportional to the square of concentration. To balance the equation, $k$ must have units of inverse concentration-time (e.g., $M^{-1} \cdot s^{-1}$).
Frequently Asked Questions
Can the rate constant be negative?
No. The rate constant $k$ is always a positive value representing the speed of the reaction. If your slope calculation gives a negative $k$ (e.g., in a second-order plot where slope is positive but you calculated incorrectly), check your arithmetic. In Zero and First order plots, the slope is negative, so we multiply by -1 to get a positive $k$.
What if my time is in minutes?
The numeric value of $k$ depends on the time unit. If your graph uses minutes, your $k$ will be in $min^{-1}$. You can convert this to seconds by dividing by 60.