1d:I[72985,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],"default"] 1e:I[63508,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],"default"] 1f:T248d,

In a chemical engineering process, reactant $A$ decomposes via an isothermal, liquid-phase homogeneous reaction in a batch reactor according to the chemical equation:

$A \longrightarrow \text{Products}$

This reaction follows first-order kinetics with a specific reaction rate constant ( $k$ ) of $0.050\text{ s}^{-1}$ at the operating temperature. If the reactor is initially charged with reactant $A$ at an initial concentration ( $[A]_0$ ) of $2.00\text{ mol/L}$ (or M), calculate:

The remaining concentration of reactant $A$ in the reactor ( $[A]_t$ ) after elapsed reaction time $t = 10\text{ s}$ .
The percentage conversion of reactant $A$ (Conversion, $X_A$ ) at the same elapsed time ( $10\text{ s}$ ). (Note: You can modify the variables in the input data section)

20:Tf3b,

Click to view integrated rate laws and units of k summary table

Fundamental chemical kinetics relationships categorized by reaction order (n):

Order (n)	Differential Rate Law	Integrated Rate Law	Half-Life (t_1/2)	Units of k
0 (Zero-order)	$-r_A = k$	$[A]_t = [A]_0 - kt$	$\frac{[A]_0}{2k}$	$\text{M}\cdot\text{s}^{-1}$ ($\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$)
1 (First-order)	$-r_A = k[A]$	$[A]_t = [A]_0 e^{-kt}$	$\frac{\ln 2}{k} \approx \frac{0.693}{k}$	$\text{s}^{-1}$
2 (Second-order)	$-r_A = k[A]^2$	$\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$	$\frac{1}{k[A]_0}$	$\text{M}^{-1}\cdot\text{s}^{-1}$ ($\text{L}\cdot\text{mol}^{-1}\cdot\text{s}^{-1}$)

21:T23ec,

Step 1: Calculate the remaining reactant concentration ( $[A]_t$ ) (Using the first-order integrated rate law, which characterizes the exponential decay of concentration over time)

💡 Calculator Tip: Calculate the exponential term first: $e^{-kt} = e^{-(0.05 \times 10)} = e^{-0.5} \approx 0.60653$ . Then multiply by the initial concentration $[A]_0 = 2.00\text{ M}$ to obtain $[A]_t \approx 1.213\text{ M}$ (or $\text{mol/L}$ ).

Step 2: Calculate the reactant conversion percentage ( $X_A$ ) (Evaluating the fraction of the initial reactant that has been converted into products)

22:T38c5,

From the first step calculation, after $10\text{ s}$ of elapsed reaction time, the concentration of reactant $A$ decreases from $2.00\text{ M}$ to M (or $\text{mol/L}$ ), corresponding to a reactant conversion ( $X_A$ ) of %.

From a chemical reaction engineering and reactor design perspective, these results provide critical practical insights:

Residence Time and Reactor Sizing: Achieving % conversion in just $10\text{ s}$ demonstrates that the reaction proceeds rapidly. The characteristic half-life ( $t_{1/2} = \frac{\ln 2}{k} = \frac{0.6931}{0.05}$ ) remains constant at approximately s, which means that every ~14 seconds, half of the remaining reactant is consumed, completely independent of the starting concentration (a unique property of first-order kinetics). When designing a continuous plug flow reactor (PFR), this characteristic reaction time directly determines the required tubular volume for a specified volumetric throughput.
Thermal & Safety Management: If this reaction is exothermic, the maximum rate of heat generation occurs at the start of the reaction because the rate is directly proportional to concentration ( $-r_A = k[A]$ ). During this initial $10$ -second period when concentration drops fastest, the reactor cooling jacket or internal coils must be capable of removing the peak heat duty to prevent thermal runaway.
Economic Cycle Time vs Diminishing Returns: For first-order reactions, the absolute reaction rate decelerates exponentially as reactant is consumed. While achieving $\approx 39.35\%$ conversion takes only $10\text{ s}$ , reaching $90\%$ requires $46.1\text{ s}$ , and $99\%$ requires $92.1\text{ s}$ . In commercial chemical manufacture, operating to near-complete conversion is often uneconomical due to excessive reactor cycle time; engineers typically choose an optimal economic conversion point and separate unreacted $A$ in downstream distillation or stripping units for recycling.

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This example demonstrates how to calculate the remaining concentration of a reactant and its percentage conversion for a first-order chemical reaction, providing fundamental insights for reactor sizing and residence time determination in chemical engineering.

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