Determine Reaction Order From Reactant Concentration Time Data

To determine the order of the reaction $\text{R} \rightarrow \text{P}$, we analyze the relationship between the concentration of reactant $[\text{R}]$ and time $t$.

For an $n^{\text{th}}$-order reaction, the rate law is expressed as:
$\text{Rate} = -\frac{d[\text{R}]}{dt} = k[\text{R}]^n$

Let us test for a zero-order reaction ($n = 0$). The integrated rate law for a zero-order reaction is given by:
$[\text{R}]_t = [\text{R}]_0 - kt \implies k = \frac{[\text{R}]_0 - [\text{R}]_t}{t}$

Now, we calculate the value of $k$ at different time intervals from the given data:

1. At $t_1 = 0.05\text{ min}$ and $[\text{R}]_1 = 0.75\text{ M}$:
   $k_1 = \frac{1.0 - 0.75}{0.05 - 0.0} = \frac{0.25}{0.05} = 5.0\text{ M}\cdot\text{min}^{-1}$

2. At $t_2 = 0.12\text{ min}$ and $[\text{R}]_2 = 0.40\text{ M}$:
   $k_2 = \frac{1.0 - 0.40}{0.12 - 0.0} = \frac{0.60}{0.12} = 5.0\text{ M}\cdot\text{min}^{-1}$

3. At $t_3 = 0.18\text{ min}$ and $[\text{R}]_3 = 0.10\text{ M}$:
   $k_3 = \frac{1.0 - 0.10}{0.18 - 0.0} = \frac{0.90}{0.18} = 5.0\text{ M}\cdot\text{min}^{-1}$

Since the rate constant $k$ remains constant ($k = 5.0\text{ M}\cdot\text{min}^{-1}$) across all given time intervals, the reaction follows zero-order kinetics.

Thus, the order of the reaction is 0.