How do you calculate the rate constant k for a first order reaction?

Plot ln[A] versus time and measure the slope of the best-fit line — the rate constant k equals the negative of this slope. Alternatively, use two data points: k = ln([A]₁/[A]₂) / (t₂ − t₁). You can also calculate k from the half-life using k = 0.693/t₁/₂. Units of k for first order are always inverse time (s⁻¹, min⁻¹).

What are the units of the first order rate constant?

First order rate constant k has units of inverse time: s⁻¹, min⁻¹, or hr⁻¹. This contrasts with second-order (M⁻¹s⁻¹) and zero-order (M·s⁻¹) rate constants. The inverse-time unit reflects the fact that first-order rates depend only on concentration to the first power, so the rate constant alone determines the characteristic time scale.

What does a first order reaction graph look like?

A plot of [A] versus t shows a decaying exponential curve starting at [A]₀ and asymptotically approaching zero. A plot of ln[A] versus t yields a straight line with negative slope −k — this linearity is the diagnostic test for first-order behavior. If ln[A]-vs-t is curved, the reaction is not first order.

First Order Reaction

Quick Answer

A first order reaction follows the integrated rate law ln[A] = ln[A]₀ − kt, where [A] is concentration at time t, [A]₀ is initial concentration, and k is the rate constant in s⁻¹. The half-life t₁/₂ = ln(2)/k ≈ 0.693/k is independent of initial concentration. The Laplace transform of the first-order decay equation d[A]/dt = −k[A] yields [A](s) = [A]₀/(s + k), whose inverse is the exponential decay [A](t) = [A]₀·e⁻ᵏᵗ.

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What Is a First Order Reaction?

A first order reaction is a chemical process whose rate depends linearly on the concentration of a single reactant: rate = −d[A]/dt = k[A], where k is the rate constant with units of s⁻¹ (or min⁻¹, hr⁻¹). Doubling the reactant concentration exactly doubles the reaction rate. First order kinetics appear in radioactive decay (where k = λ, the decay constant), many drug metabolism processes in pharmacokinetics, unimolecular decomposition reactions, and pseudo-first-order approximations where one reactant is in large excess. The mathematical form is identical to the exponential decay observed in RC circuit discharge and thermal cooling, making the Laplace transform a universal tool for solving first-order rate problems — try it at www.lapcalc.com.

Key Formulas

Deriving the First Order Integrated Rate Law

Starting from the differential rate equation d[A]/dt = −k[A], separation of variables gives d[A]/[A] = −k·dt. Integrating both sides from t = 0 to t yields ln[A] − ln[A]₀ = −kt, or equivalently ln([A]/[A]₀) = −kt. Taking the exponential of both sides produces the explicit solution [A](t) = [A]₀·e⁻ᵏᵗ. Alternatively, applying the Laplace transform directly to the ODE gives s·A(s) − [A]₀ = −k·A(s), solving to A(s) = [A]₀/(s + k). The inverse Laplace transform of 1/(s + k) is e⁻ᵏᵗ, recovering the same exponential decay. This Laplace approach generalizes effortlessly to coupled first-order systems and higher-order kinetics, which is why engineering students learn it alongside separation of variables.

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Half-Life and Rate Constant for First Order Kinetics

The half-life t₁/₂ is the time for concentration to drop to half its initial value. Setting [A](t₁/₂) = [A]₀/2 in the integrated rate law gives ln(1/2) = −k·t₁/₂, so t₁/₂ = ln(2)/k ≈ 0.693/k. Crucially, the first-order half-life is independent of initial concentration — a defining characteristic that distinguishes it from zero-order (t₁/₂ = [A]₀/2k) and second-order (t₁/₂ = 1/(k[A]₀)) reactions. Carbon-14 dating exploits this property: with k = 1.21 × 10⁻⁴ yr⁻¹, the half-life is 5,730 years regardless of the initial ¹⁴C amount. In pharmacology, drug half-lives (typically 1–24 hours) determine dosing schedules to maintain therapeutic concentration windows.

First Order Reaction Graph: How to Identify the Order

Plotting concentration data in different forms reveals reaction order. For a first-order reaction, a plot of ln[A] versus time produces a straight line with slope = −k and y-intercept = ln[A]₀. If [A] versus t is plotted instead, the curve is a decaying exponential. A plot of 1/[A] versus t would be nonlinear for first order (this linear form applies to second-order reactions). To determine k experimentally, perform a linear regression on the ln[A]-vs-t data: the negative slope equals the rate constant, and R² > 0.99 confirms first-order behavior. MATLAB's polyfit() or Python's scipy.stats.linregress() perform this regression automatically. The time constant τ = 1/k represents the time for concentration to decay to 1/e ≈ 36.8% of its initial value, directly analogous to the RC time constant in electrical circuits.

First Order Kinetics in Engineering and Applied Sciences

First order kinetics extend far beyond chemistry. In electrical engineering, RC circuit discharge follows V(t) = V₀·e⁻ᵗ/ᴿᶜ with rate constant k = 1/RC. In heat transfer, Newton's law of cooling gives T(t) = T_env + (T₀ − T_env)·e⁻ʰᴬᵗ/ᵐᶜ. In population dynamics, exponential growth/decay follows N(t) = N₀·e^(rt). All share the same mathematical structure: dy/dt = −ky with solution y = y₀·e⁻ᵏᵗ. The Laplace transform unifies these diverse applications through the common transfer function 1/(s + k), enabling engineers to apply identical analytical techniques across disciplines. The LAPLACE Calculator at www.lapcalc.com computes these transforms instantly, showing the connection between chemical kinetics and system response theory.

Frequently Asked Questions

The integrated first order rate law is ln[A] = ln[A]₀ − kt, or equivalently [A](t) = [A]₀·e⁻ᵏᵗ. It relates reactant concentration [A] at any time t to the initial concentration [A]₀ and the rate constant k. A plot of ln[A] versus t gives a straight line with slope −k.

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