Reaction Order / Half-Life
Kinetic context
Reaction order defines the concentration-time profile and therefore directly controls half-life behavior.
\[ -\frac{dc}{dt}=k\,c^n \]
Only first-order kinetics yields a concentration-independent half-life. For zero- and second-order reactions, \(t_{1/2}\) changes with \(c_0\).
- Extract kinetic parameters from lab data
- Estimate process and hold times
- Support stability and degradation modeling
Formulas (MathJax)
\[ t_{1/2}^{(0)}=\frac{c_0}{2k},\quad t_{1/2}^{(1)}=\frac{\ln 2}{k},\quad t_{1/2}^{(2)}=\frac{1}{k\,c_0} \]
\[ c(t)_{(0)}=c_0-k t,\quad c(t)_{(1)}=c_0 e^{-kt},\quad c(t)_{(2)}=\frac{c_0}{1+k c_0 t} \]
\[ k_{(1)}=\frac{\ln 2}{t_{1/2}},\quad k_{(0)}=\frac{c_0}{2t_{1/2}},\quad k_{(2)}=\frac{1}{c_0 t_{1/2}} \]
General Symbols
- \(v\) = Reaction rate
- \(k\) = Rate constant
- \([A]\) = Concentration of substance A at any given time
- \([A]_0\) = Initial concentration of substance A
- \( t_{1/2}\) = Half-life (time until the concentration has dropped to half of the initial value)
- \(ln 2\) = Natural logarithm of 2 ≈ 0.693
Detailed examples
Example 1 (1st order):
k = 0.15 min⁻¹
\(t_{1/2}=\ln(2)/0.15=4.62\,\text{min}\)
k = 0.15 min⁻¹
\(t_{1/2}=\ln(2)/0.15=4.62\,\text{min}\)
Example 2 (0th order):
\(c_0=1.2\), \(k=0.08\)
\(t_{1/2}=1.2/(2\cdot0.08)=7.5\)
\(c_0=1.2\), \(k=0.08\)
\(t_{1/2}=1.2/(2\cdot0.08)=7.5\)
Example 3 (2nd order):
\(c_0=1.0\), \(k=0.25\)
\(t_{1/2}=1/(0.25\cdot1.0)=4\)
\(c_0=1.0\), \(k=0.25\)
\(t_{1/2}=1/(0.25\cdot1.0)=4\)
Interpretation:
For second-order reactions, lower \(c_0\) increases \(t_{1/2}\). For zero-order kinetics, it decreases \(t_{1/2}\).
For second-order reactions, lower \(c_0\) increases \(t_{1/2}\). For zero-order kinetics, it decreases \(t_{1/2}\).
Practical depth
How to identify order experimentally
Check linearized plots: \(c\) vs \(t\) (0th), \(\ln c\) vs \(t\) (1st), \(1/c\) vs \(t\) (2nd). The most linear relation indicates the likely order.
Why it matters
Order and rate constant selection controls time predictions, reactor design, and shelf-life estimates. Small parameter errors can produce large timing deviations.
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