Integrated Rate Law Calculations

Integrated Rate Law Calculator

Calculate the concentration of a reactant over time for zero, first, and second-order reactions.

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

Result:

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Understanding Integrated Rate Laws

In chemical kinetics, the integrated rate law relates the concentration of reactants to the elapsed time. While differential rate laws show how the rate changes with concentration, integrated rate laws allow chemists to predict how much reactant will remain after a specific period or how long it will take for a reaction to reach a certain level of completion.

Common Reaction Orders and Their Formulas

• Zero-Order Reaction: The rate is independent of the concentration.
  Formula: [A]ₜ = -kt + [A]₀
• First-Order Reaction: The rate depends on the concentration of one reactant raised to the first power.
  Formula: ln[A]ₜ = -kt + ln[A]₀ or [A]ₜ = [A]₀e⁻ᵏᵗ
• Second-Order Reaction: The rate depends either on the concentration of one reactant squared or two reactants to the first power.
  Formula: 1/[A]ₜ = kt + 1/[A]₀

Key Variables Explained

[A]₀: The initial concentration of the reactant (usually in Molarity, M).
[A]ₜ: The concentration of the reactant at time t.
k: The rate constant. Its units vary depending on the reaction order (e.g., M/s, 1/s, or 1/M·s).
t: The time elapsed since the start of the reaction.

Practical Example

Suppose you have a first-order reaction with an initial concentration of 0.800 M and a rate constant of 0.05 s⁻¹. To find the concentration after 20 seconds:

1. Identify the formula: [A]ₜ = [A]₀e⁻ᵏᵗ
2. Plug in values: [A]ₜ = 0.800 * e^(-0.05 * 20)
3. Calculate: [A]ₜ = 0.800 * e^(-1) ≈ 0.800 * 0.3679 = 0.294 M

function calculateRateLaw() { var order = document.getElementById("reactionOrder").value; var a0 = parseFloat(document.getElementById("initialConcentration").value); var k = parseFloat(document.getElementById("rateConstant").value); var t = parseFloat(document.getElementById("timeElapsed").value); var resultDiv = document.getElementById("rateResult"); var resultText = document.getElementById("resultText"); var formulaText = document.getElementById("formulaUsed"); if (isNaN(a0) || isNaN(k) || isNaN(t)) { alert("Please enter valid numeric values for all fields."); return; } var at = 0; var formula = ""; if (order === "0") { at = a0 – (k * t); formula = "Formula used: [A]t = -kt + [A]0 (Zero Order)"; } else if (order === "1") { at = a0 * Math.exp(-k * t); formula = "Formula used: [A]t = [A]0 * e^(-kt) (First Order)"; } else if (order === "2") { at = 1 / ( (k * t) + (1 / a0) ); formula = "Formula used: 1/[A]t = kt + 1/[A]0 (Second Order)"; } if (at < 0) { resultText.innerHTML = "Concentration: 0.000 M"; resultText.style.color = "#d73a49"; formulaText.innerHTML = formula + " – Note: Theoretical concentration reached zero."; } else { resultText.innerHTML = "Concentration [A]ₜ = " + at.toFixed(6) + " M"; resultText.style.color = "#004085"; formulaText.innerHTML = formula; } resultDiv.style.display = "block"; }