= First-order reaction
A <first-order reaction> has a <reaction rate> proportional to one <reactant> <concentration> in the simple single-species case. For an irreversible $\mathrm A\longrightarrow\mathrm B$ in a <closed system> at fixed <volume>, take $[\mathrm A](0)=A_0$, $[\mathrm B](0)=0$, and constant <rate constant> $k>0$. Its <rate equation> and solution are
$$
\frac{d[\mathrm A]}{dt}=-k[\mathrm A],\qquad
[\mathrm A](t)=A_0e^{-kt},\qquad
[\mathrm B](t)=A_0(1-e^{-kt}),\qquad
v(t)=kA_0e^{-kt}.
$$
A short check is enough: differentiating $A_0e^{-kt}$ gives $-kA_0e^{-kt}$, and the two <concentrations> sum to $A_0$, as required by the one-to-one <stoichiometry>. Thus the same <exponential decay> describes both remaining <reactant> and decreasing <reaction rate>. Fixed $k$ means a fixed fractional loss per unit time, not a fixed absolute consumption rate.
Choose illustrative values $A_0=1\,\mathrm{mol\,L^{-1}}$ and $k=0.2\,\mathrm{s^{-1}}$, with $t$ measured in seconds. Then $[\mathrm A]=e^{-0.2t}\,\mathrm{mol\,L^{-1}}$, $[\mathrm B]=(1-e^{-0.2t})\,\mathrm{mol\,L^{-1}}$, and $v=0.2e^{-0.2t}\,\mathrm{mol\,L^{-1}\,s^{-1}}$. The <half-life> is
$$
t_{1/2}=\frac{\ln2}{k}=\frac{\ln2}{0.2}\,\mathrm s\approx3.4657\,\mathrm s.
$$
At each successive <half-life>, half the remaining <reactant> is consumed; $t_{1/2}$ is independent of $A_0$ for this <first-order reaction>. At $t=10\,\mathrm s$, the calculated values are $[\mathrm A]\approx0.135335\,\mathrm{mol\,L^{-1}}$, $[\mathrm B]\approx0.864665\,\mathrm{mol\,L^{-1}}$, and $v\approx0.027067\,\mathrm{mol\,L^{-1}\,s^{-1}}$. These are model predictions for chosen parameters, rather than experimental measurements.
\Image[/chemical-first-order-reaction.png]
{title=Illustrative <first-order reaction>: <reactant> loss, <reaction product> formation, and declining <reaction rate> for $k=0.2\,\mathrm{s^{-1}}$}
The left panel tracks the two <concentrations>; the right tracks their instantaneous conversion <reaction rate>. Their different vertical units matter: $v=k[\mathrm A]$, so the numerical curves need not have the same height. This gives a practical way to translate a measured <concentration> into a forecast of both conversion and waiting time.
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