How to Calculate First Order Rate Constant

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First-Order Rate Constant Calculator

Determine the reaction rate constant (k) and half-life for first-order chemical kinetics.

Results:

Rate Constant (k): time⁻¹

Half-life (t₁/₂): units of time

Formula used: k = ln([A]₀ / [A]ₜ) / t

Understanding First-Order Kinetics

In chemical kinetics, a first-order reaction is a reaction in which the rate depends linearly on the concentration of only one reactant. This means if you double the concentration of the reactant, the rate of the reaction also doubles.

The First-Order Rate Constant Formula

The integrated rate law for a first-order reaction is expressed as:

ln([A]t) = -kt + ln([A]0)

To solve for the rate constant (k), we rearrange the formula to:

k = ln([A]0 / [A]t) / t
  • [A]₀: Initial concentration of the reactant.
  • [A]ₜ: Concentration of the reactant at time t.
  • t: The time interval elapsed.
  • k: The first-order rate constant (units: s⁻¹, min⁻¹, or hr⁻¹).

How to Calculate the Rate Constant Step-by-Step

  1. Identify Initial Concentration: Measure the starting amount of your reactant.
  2. Measure Final Concentration: Determine the remaining amount of reactant after a specific period.
  3. Determine Time: Record the duration (t) between the two measurements.
  4. Apply the Logarithm: Divide the initial concentration by the final concentration, then take the natural logarithm (ln) of that value.
  5. Divide by Time: Divide the result by the elapsed time to find k.

Example Calculation

Suppose a substance decomposes via a first-order reaction. If the initial concentration is 2.0 M and it drops to 0.5 M after 120 minutes, what is the rate constant?

  • Step 1: Ratio = 2.0 / 0.5 = 4
  • Step 2: ln(4) ≈ 1.386
  • Step 3: k = 1.386 / 120 minutes
  • Result: k ≈ 0.01155 min⁻¹

What is Half-Life?

The half-life (t₁/₂) is the time required for the concentration of a reactant to decrease to half of its initial value. For first-order reactions, the half-life is constant regardless of the starting concentration and is calculated as:

t₁/₂ = ln(2) / k ≈ 0.693 / k
function calculateFirstOrder() { var a0 = parseFloat(document.getElementById('initialConc').value); var at = parseFloat(document.getElementById('finalConc').value); var t = parseFloat(document.getElementById('timeVal').value); var errorDiv = document.getElementById('error-msg'); var resultBox = document.getElementById('result-box'); errorDiv.style.display = 'none'; resultBox.style.display = 'none'; if (isNaN(a0) || isNaN(at) || isNaN(t)) { errorDiv.innerText = "Please enter valid numeric values for all fields."; errorDiv.style.display = 'block'; return; } if (a0 <= 0 || at = a0) { errorDiv.innerText = "Final concentration must be less than initial concentration for a decay reaction."; errorDiv.style.display = 'block'; return; } if (t <= 0) { errorDiv.innerText = "Time elapsed must be greater than zero."; errorDiv.style.display = 'block'; return; } // Calculation: k = ln([A]0 / [A]t) / t var k = Math.log(a0 / at) / t; // Half life: t1/2 = ln(2) / k var hl = Math.log(2) / k; document.getElementById('rateK').innerText = k.toPrecision(5); document.getElementById('halfLife').innerText = hl.toPrecision(5); resultBox.style.display = 'block'; }

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