Integrated Rate Law For First Order Reaction

Integrated rate law for first orderreaction is a fundamental concept in chemical kinetics that describes how the concentration of a reactant decreases exponentially over time when the reaction rate depends linearly on that reactant’s concentration. Understanding this relationship allows chemists to predict reaction progress, calculate rate constants, and determine half‑lives from experimental data.

Introduction to First‑Order Kinetics

A reaction is classified as first order when its rate is directly proportional to the concentration of a single reactant. Mathematically, the differential rate law is expressed as [ \text{rate} = -\frac{d[A]}{dt}=k[A] ]

where ([A]) is the instantaneous concentration of reactant A, (t) is time, and (k) is the first‑order rate constant (units of s⁻¹, min⁻¹, etc.). The negative sign indicates that ([A]) decreases as the reaction proceeds.

Derivation of the Integrated Rate Law

Starting from the differential form, we separate variables and integrate:

[-\frac{d[A]}{[A]} = k,dt ]

Integrate from the initial concentration ([A]_0) at time (t=0) to ([A]) at time (t):

[ \int_{[A]0}^{[A]} -\frac{d[A]}{[A]} = \int{0}^{t} k,dt ]

[ -\ln[A] + \ln[A]_0 = kt ]

Re‑arranging gives the integrated rate law for a first‑order reaction:

[ \boxed{\ln[A] = \ln[A]_0 - kt} ]

or equivalently,

[ [A] = [A]_0 e^{-kt} ]

This exponential decay equation shows that a plot of (\ln[A]) versus time yields a straight line with slope (-k) and intercept (\ln[A]_0).

Graphical Representation

Experimentally, measuring ([A]) at several time points and plotting (\ln[A]) versus (t) provides a quick test for first‑order behavior. Deviations from linearity suggest a different reaction order or complex mechanisms.

Half‑Life of a First‑Order Reaction

The half‑life ((t_{1/2})) is the time required for the reactant concentration to fall to one‑half its initial value. For a first‑order process, substituting ([A] = \frac{[A]_0}{2}) into the integrated law gives:

[ \ln\left(\frac{[A]_0}{2}\right) = \ln[A]0 - kt{1/2} ]

[ -\ln 2 = -kt_{1/2} ]

[ \boxed{t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}} ]

Key point: The half‑life is independent of the initial concentration ([A]0); it depends solely on the rate constant (k). This property distinguishes first‑order reactions from zero‑ or second‑order processes, where (t{1/2}) varies with ([A]_0).

Applications in Chemistry and Beyond

1. Radioactive decay – Nuclei disintegrate via first‑order kinetics; the integrated law predicts activity over time and enables dating techniques (e.g., carbon‑14). 2. Pharmacokinetics – Drug elimination often follows first‑order elimination; clinicians use the half‑life to design dosing intervals.
2. Atmospheric chemistry – Trace gases such as OH radicals react with pollutants in a first‑order fashion, influencing air‑quality models. 4. Industrial reactors – Design of batch reactors for first‑order reactions relies on the integrated law to size vessels and predict conversion.

Worked Example

Problem: A gaseous reactant A decomposes with a rate constant (k = 2.5 \times 10^{-3},\text{s}^{-1}). If the initial concentration is ([A]_0 = 0.80,\text{M}), what is ([A]) after 15 minutes?

Solution:

1. Convert time to seconds: (t = 15,\text{min} \times 60,\text{s/min} = 900,\text{s}).
2. Apply the integrated law: ([A] = [A]_0 e^{-kt}).
3. Compute exponent: (-kt = -(2.5 \times 10^{-3},\text{s}^{-1})(900,\text{s}) = -2.25).
4. Evaluate: ([A] = 0.80,\text{M} \times e^{-2.25} \approx 0.80,\text{M} \times 0.105 = 0.084,\text{M}).

Thus, after 15 minutes the concentration of A has dropped to approximately 0.084 M.

Half‑life check:

[ t_{1/2} = \frac{0.693}{2.5 \times 10^{-3},\text{s}^{-1}} \approx 277,\text{s} \approx 4.6,\text{min} ]

After about 4.6 min the concentration halves (0.40 M), after another 4.6 min it quarters (0.20 M), and so on—consistent with the exponential decay observed.

Frequently Asked Questions

Q: How can I tell experimentally if a reaction is truly first order?
A: Plot (\ln[A]) versus time. A linear relationship with a constant slope confirms first‑order kinetics. Additionally, the half‑life should remain unchanged when you vary the initial concentration.

Q: What units should the rate constant (k) have for a first‑order reaction?
A: Since the rate law is (\text{rate}=k[A]) and rate has units of concentration · time⁻¹, (k) must have units of time⁻¹ (e.g., s⁻¹, min⁻¹, h⁻¹).

Q: Does temperature affect the integrated rate law?
A: Temperature influences the magnitude of (k) (via the Arrhenius equation), but the functional form (\ln[A] = \ln[A]_0 - kt) remains unchanged; only the slope of the (\ln[A]) vs. (t) line changes.

Q: Can the integrated law be used for reversible reactions?
A: For a reversible first‑order process (A \rightleftharpoons B), the net rate becomes more complex, but each direction still follows first‑order kinetics. The integrated expression for ([A]) includes terms for both forward and reverse rate constants.

Q: Is the exponential decay model applicable to biological systems?
A: Yes, many biological degradation processes (e.g

DNA decay, drug metabolism, radioactive decay) exhibit first-order kinetics. The exponential decay model provides a useful framework for understanding and predicting the rates of these processes, allowing for quantitative analysis and modeling. However, it's crucial to remember that real-world systems often deviate from ideal first-order behavior. Factors like complex reaction mechanisms, inhibition, and changes in environmental conditions can introduce non-first-order kinetics. Therefore, careful experimental design and kinetic analysis are essential for accurately determining the order of a reaction and applying the integrated rate law appropriately.

Conclusion:

The integrated rate law is a powerful tool for analyzing and predicting the behavior of first-order reactions. Its simplicity and versatility make it applicable to a wide range of scientific and engineering problems, from atmospheric chemistry and industrial process design to biological kinetics and pharmaceutical analysis. Understanding the underlying principles of first-order kinetics, including the concept of the rate constant and half-life, is fundamental to interpreting experimental data and developing accurate models of chemical and biological systems. While the assumptions of first-order kinetics must be carefully considered, the integrated rate law provides a valuable foundation for understanding exponential decay processes and making quantitative predictions.