6.0653e-1
mol/L
1.3863e+1
s
60.65
%
0.606531
3.9347e-1
mol/L
6.0653e-1
mol/L
1.3863e+1
s
60.65
%
0.606531
3.9347e-1
mol/L
The Integrated Rate Law Calculator determines the concentration of a reactant at any point in time for zero-order, first-order, and second-order reactions. Unlike the differential rate law that gives the instantaneous rate, the integrated rate law provides a direct relationship between concentration and time, making it invaluable for predicting how much reactant remains after a given period. This tool is essential in pharmaceutical kinetics (drug shelf-life calculations), environmental chemistry (pollutant degradation), industrial processing (reactor design), and radioactive decay modeling. Simply select the reaction order, enter the initial concentration and rate constant, and the calculator provides the remaining concentration, half-life, and percentage of reactant remaining.
Each reaction order has a distinct integrated rate law relating concentration to time:
Zero Order (n = 0):
$$[A]_t = [A]_0 - kt$$
Concentration decreases linearly with time until it reaches zero.
First Order (n = 1):
$$[A]_t = [A]_0 \cdot e^{-kt} \quad \text{or equivalently} \quad \ln[A]_t = \ln[A]_0 - kt$$
Concentration decreases exponentially. A plot of ln[A] vs. time gives a straight line with slope −k.
Second Order (n = 2):
$$\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$$
The reciprocal of concentration increases linearly with time. A plot of 1/[A] vs. time gives a straight line with slope k.
The half-life formulas differ for each order:
$$t_{1/2}^{(0)} = \frac{[A]_0}{2k}, \quad t_{1/2}^{(1)} = \frac{\ln 2}{k} \approx \frac{0.693}{k}, \quad t_{1/2}^{(2)} = \frac{1}{k[A]_0}$$
Only the first-order half-life is independent of initial concentration, which is why radioactive decay (a first-order process) has a constant half-life.
The concentration at time t shows how much reactant remains after the specified time. The half-life is the time required for the concentration to drop to half its initial value. For first-order reactions, the half-life is constant regardless of starting concentration — a key diagnostic feature. For zero-order reactions, the half-life decreases as the initial concentration decreases. For second-order, the half-life also depends on the initial concentration and increases as the reaction progresses. The percent remaining gives you a quick measure of reaction progress.
Inputs
Results
For a first-order reaction with k = 0.05 s⁻¹ at t = 20 s: [A]ₜ = 1.0 × e^(−0.05×20) = 1.0 × e⁻¹ = 0.3679 mol/L. Half-life = 0.693/0.05 = 13.86 s. After 20 s, 36.79% remains.
Inputs
Results
For a second-order reaction: 1/[A]ₜ = 1/0.5 + 0.1×30 = 2 + 3 = 5, so [A]ₜ = 1/5 = 0.2 → actually 1/(2+3)=0.2. Half-life = 1/(0.1×0.5) = 20 s. After 30 s (1.5 half-lives), 25% remains.
Plot your experimental data three ways: [A] vs t (linear = zero-order), ln[A] vs t (linear = first-order), and 1/[A] vs t (linear = second-order). The plot that gives the best straight line reveals the reaction order.
In a first-order reaction, the rate is proportional to [A], so at half the concentration, the rate is also halved. The time to halve the concentration remains constant because the ratio of rate to concentration stays the same throughout the reaction.
The integrated rate law predicts negative concentrations after t > [A]₀/k, which is physically impossible. In reality, the reaction stops when [A] = 0. This calculator caps the concentration at zero for zero-order reactions.
This calculator applies to single-reactant systems or pseudo-order conditions where one reactant is in large excess. For true multi-reactant systems, more complex integrated expressions are needed.
For first-order: t½ = ln(2)/k. For zero-order: t½ = [A]₀/(2k). For second-order: t½ = 1/(k[A]₀). Only first-order has a direct, concentration-independent relationship.
For first-order reactions, after n half-lives, the fraction remaining is (1/2)ⁿ. To reach 1% remaining (99% consumed): (1/2)ⁿ = 0.01, so n = log(0.01)/log(0.5) ≈ 6.64 half-lives.
Rate constants span many orders of magnitude. Fast reactions may have k > 10⁶ s⁻¹, moderate reactions 10⁻³ to 10⁻¹ s⁻¹, and slow reactions (like geological processes) can be 10⁻¹⁰ s⁻¹ or smaller.
Pharmaceutical shelf life (t₉₀, time to 90% remaining) uses first-order kinetics: t₉₀ = 0.105/k. This is the standard approach for drug stability testing according to ICH guidelines.
Technically no — the order is fixed by the mechanism. However, apparent order can change if a reaction shifts from one regime to another (e.g., from zero-order enzyme saturation to first-order at low substrate).
Classic examples include: 2NO₂ → 2NO + O₂ (gas phase), the saponification of ethyl acetate with NaOH, and many bimolecular reactions where two molecules must collide for reaction.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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