Batch Time for a First-Order Reaction

For a constant-volume batch reactor with first-order kinetics:

$-\frac{dC_A}{dt} = kC_A \quad\Longrightarrow\quad C_A = C_{A0}e^{-kt}$

With $C_A = C_{A0}(1-X)$:

$t = \frac{1}{k}\ln\!\left(\frac{1}{1-X}\right) = \frac{1}{0.050}\ln\!\left(\frac{1}{0.05}\right)$

$= 20\ln(20) = 20(2.9957) = 59.9\ \text{min}$

$\boxed{t = 59.9\ \text{min}}$

Half-life check. For first order, $t_{1/2} = \ln 2/k = 0.693/0.05 = 13.9$ min, and 95% conversion is between 4 and 5 half-lives ($1/2^4 = 6.25\%$ remaining, $1/2^5 = 3.1\%$). Four to five half-lives is 55-69 min — our 59.9 min sits right in that band ✓

Note the residence-time parallel. This is the same expression as the PFR result, with $t$ replacing $V/v_0$. A batch reactor and a PFR are mathematically equivalent for constant-density kinetics.