Half-Life (1st Order) Equation, Derivation & Rearrangements

Core idea

Overview

The half-life of a first-order reaction represents the time required for a reactant's concentration to decrease to half of its initial value. Uniquely, for first-order kinetics, this time interval remains constant regardless of the starting concentration, as it depends solely on the reaction's rate constant.

When to use: Apply this equation when analyzing radioactive decay or chemical reactions where the rate is directly proportional to the concentration of one reactant. It is specifically used for systems confirmed to follow first-order integrated rate laws where the concentration-time relationship is logarithmic.

Why it matters: This principle is critical for determining the shelf-life of pharmaceuticals and calculating the dosage intervals for medications in the body. It also forms the scientific basis for carbon dating and assessing the safety of nuclear waste storage over long durations.

Symbols

Variables

$t_{1/2}$ = Half-Life, k = Rate Constant

t_{1/2}

Half-Life

s

k

Rate Constant

s^{-} 1

Walkthrough

Derivation

Derivation of Half-Life (1st Order)

Shows first-order half-life is constant and independent of initial concentration.

Reaction follows first-order kinetics.

1

Use First-Order Integrated Law:

Equivalent form of the integrated first-order rate equation.

ln (\frac{[ A ] _{t}}{[ A ] _{0}}) = - k t

2

Apply the Half-Life Condition:

At half-life, concentration has halved.

[A]_{t} = \frac{[ A ] _{0}}{2}

3

Solve for t_{1/2}:

Since $ln (1/2) = - ln 2$ , the negatives cancel to give the constant half-life formula.

ln (\frac{1}{2}) = - k t_{1/2} ⟹ t_{1/2} = \frac{ln 2}{k}

Result

ln (\frac{1}{2}) = - k t_{1/2} ⟹ t_{1/2} = \frac{ln 2}{k}

Source: Edexcel A-Level Chemistry — Kinetics

Free formulas

Rearrangements

Solve for $k$

Make k the subject

k = \frac{0.693}{t _{1/2}}

To make $k$ the subject of the half-life equation for a first-order reaction, first multiply both sides by $k$ to clear the denominator, then divide by $t_{1/2}$ , and finally substitute the numerical value for $ln 2$ .

Difficulty: 2/5

Solve for $t_{1/2}$

Half-Life (1st Order): Make t1/2 the subject

t_{1/2} = \frac{0.693}{k}

This rearrangement simplifies the expression for the half-life of a first-order reaction by substituting the numerical value of the natural logarithm of 2.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

This graph is an inverse curve where the half-life decreases rapidly as the rate constant increases, with a vertical asymptote at zero. For a chemistry student, this means that reactions with a high rate constant proceed very quickly with a short half-life, while those with a small rate constant are slow and persist for a long time. The most important feature is that the curve never touches the axes, meaning that no matter how large the rate constant becomes, the half-life never truly reaches zero.

Graph type: inverse

Why it behaves this way

Intuition

An exponential decay curve where the horizontal distance required for the vertical value to drop by half remains constant, regardless of the starting height.

t_{1/2}

Half-life

The fixed duration required for any given quantity of reactant to be reduced by exactly 50%.

k

First-order rate constant

The intrinsic speed or probability per unit time that a specific reactant molecule will undergo the reaction.

ln 2

Natural logarithm of 2

A scaling constant (approximately 0.693) that maps the exponential decay rate to the specific threshold of 50% remaining concentration.

Signs and relationships

1/k: The inverse relationship indicates that a higher rate constant (faster reaction) results in a shorter time required to reach the half-concentration point.

Free study cues

Insight

Canonical usage

The units of the half-life must be the reciprocal of the units of the first-order rate constant k to ensure the product is dimensionless.

Common confusion

Students often attempt to use the same rate constant units for different reaction orders; however, only first-order rate constants have units of time^-1, making the half-life independent of concentration.

Dimension note

The term (ln 2) is a dimensionless scalar; therefore, the product of $t_{1}$ /2 and k must always be dimensionless.

Unit systems

$t_{1/2}$ s · Can be expressed in any time unit (min, h, yr) as long as it matches the reciprocal unit of k.

$k$ s^-1 · For a first-order reaction, the rate constant must have units of reciprocal time.

$ln 2$ dimensionless · Natural log of 2 is approximately 0.693147.

One free problem

Practice Problem

A radioactive isotope decays with a first-order rate constant of 0.0347 per year. Calculate the half-life of this isotope in years.

Rate Constant0.0347 s^-1

Solve for: $t$

Hint: Divide the natural log of 2 by the rate constant.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In radioactive decay and drug elimination in the body, Half-Life (1st Order) is used to calculate Half-Life from Rate Constant. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure the units of the rate constant (k) and time (t) are reciprocals of each other (e.g., s⁻¹ and s).
Recall that the natural log of 2 is approximately 0.693 for quick estimations.
If the half-life changes when you change the initial concentration, the reaction is not first-order.

Avoid these traps

Common Mistakes

Applying this formula to non-first-order reactions.
Confusing with 2nd order half-life.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Shows first-order half-life is constant and independent of initial concentration.

Apply this equation when analyzing radioactive decay or chemical reactions where the rate is directly proportional to the concentration of one reactant. It is specifically used for systems confirmed to follow first-order integrated rate laws where the concentration-time relationship is logarithmic.

This principle is critical for determining the shelf-life of pharmaceuticals and calculating the dosage intervals for medications in the body. It also forms the scientific basis for carbon dating and assessing the safety of nuclear waste storage over long durations.

Applying this formula to non-first-order reactions. Confusing with 2nd order half-life.

In radioactive decay and drug elimination in the body, Half-Life (1st Order) is used to calculate Half-Life from Rate Constant. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Ensure the units of the rate constant (k) and time (t) are reciprocals of each other (e.g., s⁻¹ and s). Recall that the natural log of 2 is approximately 0.693 for quick estimations. If the half-life changes when you change the initial concentration, the reaction is not first-order.

References

Sources

Atkins Physical Chemistry
IUPAC Gold Book
Wikipedia: First-order reaction
Atkins' Physical Chemistry
NIST Chemistry WebBook
Atkins' Physical Chemistry, 11th Edition
IUPAC Gold Book: Half-life (t1/2)
Wikipedia: Half-life

Half-Life (1st Order)

Overview

Variables

Derivation

Use First-Order Integrated Law:

Apply the Half-Life Condition:

Solve for t_{1/2}:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Integrated Rate Law (1st Order)

Frequently Asked Questions

Sources