Rate Constant Calculation Examples

Rate Constant (k) Calculator

Zero Order (0) First Order (1) Second Order (2)

Resulting Rate Constant (k):

function calculateK() { var order = document.getElementById("reactionOrder").value; var a0 = parseFloat(document.getElementById("initialConc").value); var at = parseFloat(document.getElementById("finalConc").value); var t = parseFloat(document.getElementById("timeElapsed").value); var resultBox = document.getElementById("kResultDisplay"); var errorBox = document.getElementById("errorDisplay"); var kOutput = document.getElementById("kValueOutput"); var unitOutput = document.getElementById("kUnitOutput"); errorBox.style.display = "none"; resultBox.style.display = "none"; if (isNaN(a0) || isNaN(at) || isNaN(t) || t <= 0 || a0 <= 0 || at = a0) { errorBox.innerText = "Final concentration must be less than initial concentration for a reaction progress calculation."; errorBox.style.display = "block"; return; } var k = 0; var unit = ""; if (order === "0") { // Zero Order: k = ( [A]0 – [A]t ) / t k = (a0 – at) / t; unit = "Units: M · s⁻¹"; } else if (order === "1") { // First Order: k = ln([A]0 / [A]t) / t k = Math.log(a0 / at) / t; unit = "Units: s⁻¹"; } else if (order === "2") { // Second Order: k = (1/[A]t – 1/[A]0) / t k = ((1 / at) – (1 / a0)) / t; unit = "Units: M⁻¹ · s⁻¹"; } kOutput.innerText = k.toExponential(4); unitOutput.innerText = unit; resultBox.style.display = "block"; }

Understanding Rate Constant Calculation

In chemical kinetics, the rate constant (k) is a proportionality constant that links the rate of a chemical reaction to the concentrations of the reactants. Every reaction has a unique rate constant at a specific temperature, making it a critical value for chemists and engineers to predict how fast a reaction will proceed over time.

Core Formulas for Calculating 'k'

The mathematical approach to finding the rate constant depends entirely on the reaction order. Here are the integrated rate laws used in the calculator above:

• Zero Order: k = ([A]₀ – [A]ₜ) / t
  Rate is independent of concentration.
• First Order: k = ln([A]₀ / [A]ₜ) / t
  Rate is directly proportional to the concentration of one reactant.
• Second Order: k = (1/[A]ₜ – 1/[A]₀) / t
  Rate is proportional to the square of a concentration or the product of two.

Rate Constant Calculation Examples

Example 1: First-Order Decay

Suppose you have a reactant with an initial concentration of 0.800 M. After 300 seconds, the concentration drops to 0.200 M. What is the rate constant?

Step 1: Identify the formula for 1st order: k = ln([A]₀ / [A]ₜ) / t

Step 2: Plug in values: k = ln(0.800 / 0.200) / 300

Step 3: k = ln(4) / 300 = 1.386 / 300 ≈ 4.62 × 10⁻³ s⁻¹

Example 2: Zero-Order Reaction

A reaction starts at 1.0 M. After 50 seconds, the concentration is 0.5 M. What is k?

Step 1: Formula: k = (1.0 – 0.5) / 50

Step 2: k = 0.5 / 50 = 0.01 M · s⁻¹

Why Units Matter

A common mistake in kinetics is ignoring the units of k. Units change based on the reaction order to ensure the overall reaction rate is always in M/s:

Order	Typical Unit of k
0	M / s (or M·s⁻¹)
1	1 / s (or s⁻¹)
2	1 / (M·s) (or M⁻¹·s⁻¹)

Factors Affecting the Rate Constant

While this calculator solves for k based on concentration and time, remember that k itself is influenced by external factors:

• Temperature: Usually, increasing temperature increases k (governed by the Arrhenius Equation).
• Catalysts: Catalysts lower activation energy, effectively increasing the value of k.
• Surface Area: In heterogeneous reactions, more surface area allows for a higher effective rate constant.