First Order Kinetics Calculator

Calculate concentration, half-life, and rate constant for first-order reactions.

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Initial concentration [A]₀ (M)

Rate constant k (s⁻¹) (s⁻¹)

Time t (s) (s)

Target concentration [A] (M)

For time or k calculation

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The Signature of First-Order Decay

First-order kinetics means rate is proportional to concentration — the more there is, the faster it disappears. That single condition produces exponential decay and one defining consequence: half-life is independent of starting amount.

[A] = [A]₀e^−kt t_½ = 0.693/k

Half-lives	Fraction remaining	Percent eliminated
1	1/2	50%
2	1/4	75%
3	1/8	87.5%
4	1/16	93.75%
5	1/32	96.9%
7	1/128	99.2%

The five-half-life convention comes straight from this table. After five half-lives about 97% has gone, which is why drugs are considered effectively cleared — and equally why steady state on repeated dosing takes about five half-lives to reach.

Where First-Order Kinetics Appears

Radioactive decay is exactly first order, since each nucleus decays independently. Most drug elimination is first order because enzymes and transporters operate well below saturation at therapeutic concentrations. Notably, alcohol is the exception — alcohol dehydrogenase saturates at low blood concentrations, so ethanol clears at a constant rate (zero order) rather than proportionally.

Because the plot of ln[A] against time is linear with slope −k, taking logarithms is the standard test. Radioactive decay, unimolecular rearrangements and SN1 reactions all show this.

Worked Examples

Example 1: Drug elimination: A0=100mg, k=0.0693/hr, t=5hr

A=100×e^(-0.346)

Result: A=70.7mg remaining, t½=10hr

First-order drug clearance

Example 2: Find k: A0=0.80M → 0.20M in 20min

k=ln(0.80/0.20)/20=ln4/20

Result: k=0.0693 min⁻¹, t½=10 min

Constant half-life confirms 1st order

Example 3: Time to reach a target

[A]₀ = 100, target 5, k = 0.0693 hr⁻¹

Result: t = ln(20)/0.0693 = 43.2 hours

Roughly 4.3 half-lives. Rearranging to t = ln([A]₀/[A])/k answers 'how long until' questions directly.

Example 4: Why alcohol is different

Blood alcohol falls at about 0.015% per hour regardless of level

Result: Zero order, not first

Alcohol dehydrogenase is saturated at normal drinking concentrations, so elimination is constant in absolute terms. Doubling intake doubles sobering time.

Common Mistakes

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Applying constant half-life to other orders

Only first-order reactions have concentration-independent half-life. Zero-order half-life shortens as concentration falls; second-order half-life lengthens.

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Assuming all drug elimination is first order

Alcohol is the well-known exception, following zero-order kinetics because the metabolising enzyme saturates. Phenytoin and aspirin also shift to zero order at high doses.

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Confusing k with half-life

They are inversely related through 0.693. A large rate constant means a short half-life, and mixing them up inverts every answer.

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Mismatching time units

If k is in hours⁻¹, time must be in hours. The exponent must be dimensionless, so unit consistency is essential.

Frequently Asked Questions

Constant half-life test?

First-order: t½=ln2/k = constant regardless of concentration. Zero-order: t½=A0/2k (depends on A0). Second-order: t½=1/kA0 (depends on A0). Constant t½ = diagnostic for first-order.

Drug half-lives?

Most drugs: first-order elimination. t½ determines dosing frequency. After 5 half-lives: 97% eliminated. Warfarin t½=40hr: takes 8 days to clear. Aspirin t½=15-20min: rapid elimination.

Why is first-order half-life independent of concentration?

Because rate is proportional to concentration, so the time to halve is the same fraction regardless of starting point. The expression 0.693/k contains no concentration term.

Why five half-lives for drug clearance?

After five half-lives about 97% has been eliminated, which is generally considered complete. The same period is needed to reach steady state on repeated dosing.

Is all drug elimination first order?

Most is, at therapeutic doses. Alcohol is zero order because its enzyme saturates, and some drugs such as phenytoin shift to zero order in overdose.

How do I confirm a reaction is first order?

Plot ln[A] against time — it should be linear with slope −k. Alternatively check whether successive half-lives are equal.

Formula Explorer connections

Interpretation: This formula connects concentration, time, temperature or transport to the speed of a chemical process. Assumption: The reaction order and mechanism must match the model. Temperature, catalyst, mixing and mass-transfer limitations can alter the observed rate.

Half-Life Calculator (Chemistry) →Integrated Rate Law Calculator →Rate Constant Calculator →Chemistry Formula Explorer →