Integrated Rate Law of First‑Order Reaction
Understanding how concentration changes over time is fundamental to chemical kinetics. In real terms, for a reaction that follows first‑order kinetics, the integrated rate law of first‑order reaction provides a simple mathematical relationship between the reactant concentration and elapsed time. This law allows chemists to predict how fast a substance will disappear, determine half‑lives, and extract rate constants from experimental data.
Honestly, this part trips people up more than it should Not complicated — just consistent..
Introduction
In chemical kinetics, reactions are classified by their order, which describes how the rate depends on the concentration of reactants. A first‑order reaction has a rate that is directly proportional to the concentration of a single reactant. When the rate law is expressed as
This changes depending on context. Keep that in mind.
[ \text{rate} = k[A] ]
where k is the first‑order rate constant and [A] is the instantaneous concentration of reactant A, integrating this differential expression yields the integrated rate law of first‑order reaction:
[ \ln[A] = \ln[A]_0 - kt ]
or equivalently
[ [A] = [A]_0 e^{-kt} ]
This equation shows an exponential decay of concentration with time, a hallmark of first‑order processes such as radioactive decay, certain enzyme‑catalyzed reactions, and the decomposition of N₂O₅.
Derivation Steps
To see how the integrated form arises, follow these steps:
-
Write the differential rate law
[ -\frac{d[A]}{dt}=k[A] ] The negative sign indicates that [A] decreases as time progresses. -
Separate variables
[ \frac{d[A]}{[A]} = -k,dt ] -
Integrate both sides from the initial time (t = 0) to an arbitrary time t, and from the initial concentration [A]₀ to [A] at time t:
[ \int_{[A]0}^{[A]} \frac{d[A]}{[A]} = -k \int{0}^{t} dt ] -
Perform the integration
[ \ln[A] - \ln[A]_0 = -kt ] -
Re‑arrange to isolate [A]
[ \ln[A] = \ln[A]_0 - kt ] Exponentiating both sides gives the exponential form:
[ [A] = [A]_0 e^{-kt} ]
These steps illustrate why a plot of (\ln[A]) versus t yields a straight line with slope (-k) and intercept (\ln[A]_0). This linearization is a practical tool for determining the rate constant from experimental data.
Scientific Explanation
Why First‑Order Kinetics Produce Exponential Decay
In a first‑order process, each molecule of reactant has a constant probability per unit time of reacting, independent of how many other molecules are present. This situation is analogous to radioactive decay, where each nucleus decays with a fixed probability. This means the number of reacting molecules drops by a constant fraction in each equal time interval, leading to the exponential term (e^{-kt}) Most people skip this — try not to. Practical, not theoretical..
Half‑Life ((t_{1/2}))
A useful characteristic derived from the integrated rate law is the half‑life, the time required for the concentration to fall to half its initial value. Setting ([A] = \frac{[A]_0}{2}) in the integrated expression:
[ \frac{[A]0}{2} = [A]0 e^{-kt{1/2}} ;\Longrightarrow; e^{-kt{1/2}} = \frac{1}{2} ]
Taking natural logs:
[ -kt_{1/2} = \ln\left(\frac{1}{2}\right) = -\ln 2 ]
Thus,
[ t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k} ]
Notice that the half‑life of a first‑order reaction is independent of the initial concentration, a distinctive feature that helps identify first‑order behavior experimentally Small thing, real impact..
Determining the Rate Constant
Experimentalists often measure concentration at various times, plot (\ln[A]) versus t, and fit a straight line. The slope ((-k)) gives the rate constant directly. Alternatively, if half‑life data are available, the constant can be calculated from (k = \ln 2 / t_{1/2}).
Applications
- Radioactive decay: The decay of isotopes like ({}^{14})C follows first‑order kinetics, enabling radiocarbon dating.
- Pharmacokinetics: Elimination of many drugs from the bloodstream approximates first‑order processes, guiding dosing regimens.
- Atmospheric chemistry: Photolysis of trace gases often exhibits first‑order behavior under constant light intensity.
Frequently Asked Questions
Q1: How can I tell if a reaction is truly first‑order?
A: Plot (\ln[A]) versus time. If the data fall on a straight line (high correlation coefficient), the reaction follows first‑order kinetics. Additionally, the half‑life should remain constant across different starting concentrations But it adds up..
Q2: What units does the first‑order rate constant k have?
A: Since the rate law is (\text{rate}=k[A]) and rate has units of concentration · time(^{-1}) (e.g., M s(^{-1})), k must have units of time(^{-1}) (e.g., s(^{-1}), min(^{-1}), h(^{-1})).
Q3: Can a reaction be first‑order with respect to more than one reactant?
A: The overall order is the sum of the exponents in the rate law. A reaction can be first‑order overall while being first‑order in each of two reactants only if one reactant is present in large excess (pseudo‑first‑order conditions). In such cases, the excess reactant’s concentration is effectively constant, and the observed rate law reduces to (\text{rate}=k'[A]) where (k' = k[B]_{\text{excess}}) Easy to understand, harder to ignore..
Q4: What happens if the temperature changes?
A: The rate constant k is temperature‑dependent, typically described by the Arrhenius equation (k = A e^{-E_a/(RT)}). Increasing temperature raises k, shortening the half‑life, while the functional form of the integrated rate law remains unchanged.
Q5: Is the integrated rate law applicable to reversible reactions?
A: For a simple reversible first‑order reaction (A \rightleftharpoons B), the net rate law becomes more complex. Even so, if the reverse reaction is negligible or if one monitors the approach to equilibrium, the integrated expression can still describe the decay of A toward its equilibrium concentration.
Conclusion
The integrated rate law of first‑order reaction—expressed as ([A] = [A]_0 e^{-kt}) or (\ln[A] = \ln[A]_0 - kt)—is a cornerstone of chemical kinetics. Its derivation from the differential rate law showcases the power of calculus in linking microscopic reaction probabilities to macroscopic observable changes. The law’s exponential form provides immediate insight into half‑life behavior, enables straightforward determination of the rate constant from linear plots, and finds wide application across fields such as nuclear chemistry, pharmacology
