Reference ID: MET-9499 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
A first-order reaction is one whose rate is directly proportional to the concentration of a single reactant. In process engineering this behaviour is encountered in thermal degradation of vitamins, radioactive decay, microbial inactivation, and many homogeneous liquid-phase decompositions. Knowing the residual concentration after a given residence time at a fixed temperature is essential for reactor sizing, sterilisation validation, and nutrient-loss estimation in high-temperature short-time (HTST) processes.
Methodology & Formulas
Temperature conversion
\[ T = T_{\text{°C}} + 273.15 \]
Arrhenius rate constant
\[ k = A\,\exp\left(\frac{-E}{R\,T}\right) \]
where
\(A\) = pre-exponential factor (s−1)
\(E\) = activation energy (kJ kmol−1)
\(R\) = 8.314 kJ kmol−1 K−1
First-order integrated rate law
\[ \frac{C}{C_0} = \exp(-k\,t) \]
Residual concentration
\[ C = C_0\,\exp(-k\,t) \]
Percent loss
\[ \text{loss} = \left(1 - \frac{C}{C_0}\right)\,100\% \]
Validity regime for Arrhenius correlation
Variable
Range
Extrapolation tolerance
Temperature
100–150 °C
±5 °C (<5 % error)
Concentration
≤ 0.2 Cwater (≈ 200 kg m−3)
First-order kinetics required
Use the integrated first-order law:
Measure the initial concentration C₀ at t = 0 and the concentration C at time t.
Re-arrange ln(C₀/C) = kt to solve for k = (1/t) ln(C₀/C).
Report k with time units consistent with t (e.g., h⁻¹ if t is in hours).
Plot the natural logarithm of concentration versus time.
A straight line confirms first-order kinetics.
The slope equals –k; the intercept is ln C₀.
Use linear regression on ln C vs t to obtain k with statistical confidence limits.
Temperature dependence is captured by the Arrhenius equation:
k(T) = A e^(–Ea/RT) where A is the pre-exponential factor, Ea is activation energy, R is 8.314 J mol⁻¹ K⁻¹, and T is absolute temperature in kelvin.
Store A and Ea as constants in the DCS; read T from the reactor thermocouple and compute k in real time for conversion predictions.
The half-life t½ is independent of initial concentration:
t½ = ln 2 / k ≈ 0.693 / k.
If you know t½ from lab data, invert to find k = 0.693 / t½.
Use t½ to estimate batch time needed for 99% conversion: roughly 6.9 half-lives.
Worked Example: Chloramine Decay in a Hot Process Line
A small plug-flow reactor is used to study the thermal decomposition of chloramine in boiler feed-water. Engineers need to know how much chloramine remains after 30 s at 95 °C so they can adjust the injection rate upstream.
Knowns
Initial concentration, \(C_0 = 5.000\) mg L-1
Residence time, \(t = 30.000\) s
Process temperature, \(T = 95.000\) °C (368.150 K)
Universal gas constant, \(R = 8.314\) kJ kmol-1 K-1
Step-by-Step Calculation
Convert temperature to absolute units: \(T = 95 + 273.15 = 368.150\) K.
Compute the rate constant via the Arrhenius equation:
\[
k = A \exp\left(-\frac{E}{RT}\right)
= 1.600 \times 10^{13} \exp\left(-\frac{112000}{8.314 \times 368.15}\right)
= 0.002\ \text{s}^{-1}.
\]
Use the first-order integrated rate law to find the concentration ratio:
\[
\frac{C}{C_0} = \exp(-kt)
= \exp(-0.002 \times 30)
= 0.940.
\]
Calculate the remaining concentration:
\[
C = C_0 \times 0.940
= 5.000 \times 0.940
= 4.701\ \text{mg L}^{-1}.
\]
