Integrated rate laws — half-life t₁ - ₂ for each order: Foundations

Before you can read a single line of the parent note, you need to own every symbol it throws at you. This page builds them one at a time, from nothing. We assume you have seen fractions and basic algebra — nothing else.

1. What is a "reactant"? — the thing being used up

Picture a jar full of blue marbles ( $A$ ). As time passes, marbles vanish (they become "products"). The jar slowly empties.

Figure — Integrated rate laws — half-life t₁ - ₂ for each order

Why the topic needs it: half-life is about how fast this jar empties. No reactant, nothing to measure.

2. Concentration and the bracket notation $[A]$

Picture: not how many marbles total, but how packed they are in the jar. A tightly packed jar has high $[A]$ ; a nearly empty one has low $[A]$ .

Why the topic needs it: half-life watches a number — $[A]$ — fall to half its start.

3. Subscripts: $[A]_{0}$ versus $[A]_{t}$ — snapshots in time

Picture two photographs of the same jar: one at the beginning (label it $[A]_{0}$ ), one taken later (label it $[A]_{t}$ ). The subscript is just the timestamp on the photo.

Why the topic needs it: half-life is defined by comparing a later photo to the first photo — specifically when the later one shows exactly half.

4. Time $t$ and the special time $t_{1/2}$

The condition is written in symbols as: $[A]_{t} = \frac{[ A ] _{0}}{2} happens exactly at t = t_{1/2}$

Picture: on a graph of concentration versus time (curve falling from left to right), draw a horizontal line at half the starting height. Where the curve crosses it, drop straight down to the time axis. That landing point is $t_{1/2}$ .

Why the topic needs it: this is the topic. Every formula in the parent note answers "what is $t_{1/2}$ ?"

5. Rate and the rate constant $k$

Picture: two jars emptying. The one with a bigger $k$ empties faster (steeper curve). Everything else being equal, $k$ is the "speed dial."

See Rate constant k — units and temperature dependence for the full story on $k$ .

6. The three integrated rate laws — what they are

An integrated rate law is a ready-made formula that tells you $[A]_{t}$ at any time $t$ , given $[A]_{0}$ and $k$ . The parent note gives one per order — we just need to read them here (their derivation lives in Integrated rate laws — derivation and graphical analysis).

Each is just " $[A]_{t}$ at time $t$ " written for a different shape of falling curve. The parent note plugs the half-life condition $[A]_{t} = [A]_{0} /2$ into each and solves for $t_{1/2}$ .

7. Reaction "order" — the shape of the fall

Picture three jars draining differently: one drips at a fixed rate no matter how full (zero), one drains fast while full and lazily when nearly empty (first), one drains furiously when packed but crawls at the end (second).

Determining order from data is its own skill — see Reaction order determination from experimental data.

8. The natural log $ln$ and $ln 2$ — the tool first-order needs

Recall Why does

$ln 2$ pop out of the half-life? Because halving means $[A]_{t} / [A]_{0} = 1/2$ , and $ln (1/2) = - ln 2$ . The minus sign cancels with the $- k t$ , leaving $t_{1/2} = ln 2/ k$ . ::: Halving turns into $- ln 2$ under the logarithm, giving the $0.693$ .

9. Proportionality words: proportional vs inversely proportional

Why the topic needs it: the whole diagnostic idea — "watch how $t_{1/2}$ changes as concentration changes to identify the order" — is built on telling these three patterns apart.

How the foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer aloud.

What does the bracket notation

[A]

mean?

The concentration of reactant

A

(how crowded it is), in molar (M) — a single quantity, not a multiplication.

What is the difference between

[A]_{0}

and

[A]_{t}

[A]_{0}

is concentration at the start (

t = 0

);

[A]_{t}

is concentration at a later time

t

State the half-life condition in symbols.

[A]_{t} = [A]_{0} /2

, which occurs exactly at

t = t_{1/2}

What does the rate constant

k

control, and why must you check its units?

It sets the reaction speed; its units depend on the order (M s⁻¹, s⁻¹, M⁻¹s⁻¹), so a wrong unit means wrong order.

Why does

ln

appear only in the first-order case?

First-order decay is exponential;

ln

is the inverse that undoes

e^{- k t}

to free the time from the exponent.

What is the numerical value of

ln 2

About

0.693

Name the three ways

t_{1/2}

can depend on

[A]_{0}

Directly proportional (zero-order), independent (first-order), inversely proportional (second-order).

Which order has a constant half-life?

First-order — the fingerprint of radioactive decay (see Radioactive decay — first-order kinetics).

Ready? Return to the parent topic and every symbol will now read as plain English.