Calculate Relative Error in Rate for Pseudo First Order Reaction

To find the relative error in the rate of reaction, we evaluate both the actual rate and the assumed rate under pseudo-first-order conditions when the reaction is $40\%$ complete.

Step 1: Initial Conditions and Concentrations at $40\%$ Completion

The given elementary reaction is: $A + R \rightarrow \text{Product}$

The actual rate law is given by: $r_{\text{actual}} = k[A][R]$

At the start of the reaction ($t = 0$):

• $[A] = [A]_0$
• $[R]_0 = 10[A]_0$

When the reaction is $40\%$ complete, $40\%$ of reactant $A$ has reacted:

• Remaining concentration of $A$: $[A] = [A]_0 - 0.40[A]_0 = 0.60[A]_0$

Since 1 mole of $A$ reacts with 1 mole of $R$, the amount of $R$ consumed is also $0.40[A]_0$:

• Remaining concentration of $R$: $[R] = [R]_0 - 0.40[A]_0 = 10[A]_0 - 0.40[A]_0 = 9.60[A]_0$

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Step 2: Calculation of Actual and Assumed Rates

1. Actual Rate ($r_{\text{actual}}$): $r_{\text{actual}} = k[A][R] = k(0.60[A]_0)(9.60[A]_0) = 5.76 k[A]_0^2$

2. Assumed Pseudo-First-Order Rate ($r_{\text{assumed}}$): Under the assumption that $k[R] = k' = k[R]_0$ is constant throughout the reaction: $k' = k[R]_0 = 10 k[A]_0$

   Thus, the assumed rate is: $r_{\text{assumed}} = k'[A] = (10 k[A]_0)(0.60[A]_0) = 6.00 k[A]_0^2$

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Step 3: Calculation of Relative Error

The difference in the rate due to the assumption is: $\Delta r = r_{\text{assumed}} - r_{\text{actual}} = 6.00 k[A]_0^2 - 5.76 k[A]_0^2 = 0.24 k[A]_0^2$

1. Relative Error with respect to Actual Rate: $\text{Relative Error (\%)} = \frac{r_{\text{assumed}} - r_{\text{actual}}}{r_{\text{actual}}} \times 100\% = \frac{0.24 k[A]_0^2}{5.76 k[A]_0^2} \times 100\% = \frac{1}{24} \times 100\% \approx 4.17\%$

2. Relative Error with respect to Assumed Rate: $\text{Relative Error (\%)} = \frac{r_{\text{assumed}} - r_{\text{actual}}}{r_{\text{assumed}}} \times 100\% = \frac{0.24 k[A]_0^2}{6.00 k[A]_0^2} \times 100\% = 4.00\%$

Both $4.17\%$ and $4.00\%$ fall within the standard acceptable range $[4.0, 4.25]$.

Final Answer: The relative error (in %) in the rate is 4.17 (or 4).