Batch reactor data analysis: integral and differential methods

Why it matters

Rate laws are not looked up; they are extracted from laboratory data, almost always from a constant-volume batch reactor in which concentration is measured against time. Choosing the right analysis method, and recognising when the data do not fit an assumed order, decides whether the reactor you later design is the right size.

Key ideas

The batch reactor as a kinetics tool. In a well-mixed, constant-volume batch reactor the mole balance reduces to −rA = −dCA/dt. Measuring CA(t) therefore gives the rate directly. For gas-phase reactions at constant volume one often measures total pressure instead and converts it to partial pressure of A through stoichiometry.

Two families of analysis.

• Integral method: guess a rate form, integrate it, and test whether the data fall on the predicted straight line. Zero order: CA vs t is linear. First order: ln(CA0/CA) vs t is linear through the origin. Second order (−rA = k·CA²): 1/CA vs t is linear. If the guessed line is curved, guess again. It is simple, smooths noise because it uses concentrations directly, and is best when you already suspect the mechanism or the order is an integer.
• Differential method: obtain −rA = −dCA/dt from the data (slopes of a smoothed curve, finite differences or equal-area graphical differentiation), then fit ln(−rA) = ln k + n·ln CA. The slope of this log-log plot is the order n and the intercept gives k. It handles fractional or unknown orders but differentiation amplifies scatter, so it needs more and better data.

Special techniques.

• Half-life method: for −rA = k·CA^n with n ≠ 1, t1/2 = (2^(n−1) − 1)·CA0^(1−n)/[k·(n−1)]. A log-log plot of t1/2 against CA0 has slope (1 − n). For n = 1, t1/2 = ln 2/k, independent of CA0. Fractional-life methods (t1/3, t3/4) generalise the idea.
• Initial-rate method: run several experiments at different CA0, measure the initial slope each time; avoids complications from products or reverse reaction.
• Method of excess (isolation): with B in large excess, CB stays nearly constant and −rA = k'·CA^α with k' = k·CB0^β — a pseudo-order with respect to A alone.

Conversion form. With XA = (CA0 − CA)/CA0 at constant density, first order becomes −ln(1 − XA) = k·t, and second order (equal-molar or single reactant) becomes XA/(1 − XA) = k·CA0·t.

Variable-volume systems. If a gas-phase batch reactor runs at constant pressure and the number of moles changes, V = V0·(1 + εA·XA), where εA is the fractional volume change at complete conversion. The integrated first-order law then becomes −ln(1 − XA) = k·t still, but written through the volume: −ln(1 − ΔV/(εA·V0)) = k·t.

Temperature. Repeat the analysis at several temperatures and use an Arrhenius plot to get E.

Formulas

−rA = −dCA/dt (constant-volume batch)

• CA: concentration (kmol/m³ = mol/L); t: time (s or min).

CA0 − CA = k·t (zero order, valid until CA = 0)

• k in kmol/m³·s.

ln(CA0/CA) = −ln(1 − XA) = k·t (first order)

• k in s⁻¹.

1/CA − 1/CA0 = k·t, XA/(1 − XA) = k·CA0·t (second order, −rA = k·CA²)

• k in m³/kmol·s.

CA^(1−n) − CA0^(1−n) = (n − 1)·k·t (n-th order, n ≠ 1)

t1/2 = (2^(n−1) − 1)·CA0^(1−n) / [k·(n − 1)] (n ≠ 1); t1/2 = ln 2 / k (n = 1)

ln(−rA) = ln k + n·ln CA (differential method; slope n)

pA = CA·R·T = pA0 − (a/Δn)·(π − π0) (constant-volume gas phase, aA + … → products with Δn = change in moles per reaction)

• π: total pressure (Pa); pA: partial pressure of A (Pa).

Worked examples

Example 1 (standard, integral method). A liquid-phase reaction A → products in a batch reactor gives CA0 = 1.00 kmol/m³ and CA = 0.61, 0.37, 0.135 kmol/m³ at t = 5, 10, 20 min. Find the order and k.

1. Test first order: k = ln(CA0/CA)/t.
   • t = 5: ln(1/0.61)/5 = 0.494/5 = 0.0989 min⁻¹
   • t = 10: 0.994/10 = 0.0994 min⁻¹
   • t = 20: 2.002/20 = 0.1001 min⁻¹
2. Test second order: k = (1/CA − 1/CA0)/t gives 0.128, 0.170, 0.320 m³/kmol·min — rising, so rejected.
3. The first-order k is constant within scatter.

First order, k ≈ 0.100 min⁻¹.

Example 2 (GATE level, half-life method). For A → products, the half-life is 10 min when CA0 = 1.0 kmol/m³ and 5 min when CA0 = 2.0 kmol/m³. Find n and k.

1. t1/2 ∝ CA0^(1−n), so 5/10 = (2/1)^(1−n) → 0.5 = 2^(1−n) → 1 − n = −1 → n = 2.
2. For n = 2: t1/2 = 1/(k·CA0).
3. k = 1/(t1/2·CA0) = 1/(10 × 1.0) = 0.1 m³/kmol·min.

Second order, k = 0.1 m³/kmol·min.

Example 3 (differential / initial rates). Initial rates are 0.003 kmol/m³·min at CA0 = 1.0 kmol/m³ and 0.012 kmol/m³·min at CA0 = 2.0 kmol/m³.

1. n = ln(0.012/0.003)/ln(2.0/1.0) = ln 4/ln 2 = 2.
2. k = 0.003/1.0² = 0.003 m³/kmol·min, order 2.

Common mistakes

• Using a first-order plot of ln CA vs t and reading the slope as +k (the slope is −k).
• Concluding "first order" from just two points — any two points fit any line; use the whole data set.
• Applying the half-life formula t1/2 = ln 2/k to a second-order reaction.
• Mixing minutes and seconds between data and the answer's unit.
• Forgetting that in a constant-volume gas reactor total pressure changes only if moles change; for A → R (Δn = 0) the total pressure stays constant and gives no information.
• Differentiating noisy raw data point-to-point instead of smoothing first.

For GATE CH

Typical questions give concentration–time or half-life data and ask for the order or k (NAT), the time to reach a conversion for a given order, or the order from the units of k. Total-pressure data for gas-phase batch reactions also appear. Practise all integrated forms from memory, the half-life ratio trick, and the log-log slope for the differential method.

Quick check

1. Which plot is linear for a first-order reaction in a batch reactor?
2. Half-life is independent of CA0 for which order?
3. If t1/2 doubles when CA0 doubles, what is the order?
4. Why is the differential method sensitive to experimental scatter?

Answers: 1. ln(CA0/CA) versus t; 2. first order; 3. zero order (t1/2 ∝ CA0^(1−n) → 2 = 2^(1−n) → n = 0); 4. because it requires taking slopes of the data, which amplifies noise.