Question

Consider the hypothetical reaction

$\text{A}+\text{B}+2\text{C}\to 2\text{D}+3\text{E}$

In a study of this reaction, three experiments were run at the same temperature. The rate is defined as $-d[\text{B}]/dt$.

Experiment 1:

$[\text{A}{]}_0=2.0\text{M}[\text{B}{]}_0=1.0\times {10}^{-3}\text{M}[\text{C}{]}_0=1.0\text{M}$

$$\begin{gathered} \left[\text{B}\right](\text{mol}/\text{L}) Time\left(\text{s}\right) \\ 2.7\times {10}^{-4} 1.0\times {10}^5 \\ 1.6\times {10}^{-4} 2.0\times {10}^5 \\ 1.1\times {10}^{-4} 3.0\times {10}^5 \\ 8.5\times {10}^{-5} 4.0\times {10}^5 \\ 6.9\times {10}^{-5} 5.0\times {10}^5 \\ 5.8\times {10}^{-5} 6.0\times {10}^5 \end{gathered}$$

Experiment 2:

$[\text{A}{]}_0=1.0\times {10}^{-2}\text{M}[\text{B}{]}_0=3.0\text{M}[\text{C}{]}_0=1.0\text{M}$

$$\begin{gathered} \left[\text{A}\right](\text{mol}/\text{L}) Time\left(\text{s}\right) \\ 8.9\times {10}^{-3} 1.0 \\ 7.1\times {10}^{-3} 3.0 \\ 5.5\times {10}^{-3} 5.0 \\ 3.8\times {10}^{-3} 8.0 \\ 2.9\times {10}^{-3} 10.0 \\ 2.0\times {10}^{-3} 13.0 \end{gathered}$$

Experiment 3:

$[\text{A}{]}_0=10.0\text{M}[\text{B}{]}_0=5.0\text{M}[\text{C}{]}_0=5.0\times {10}^{-1}\text{M}$

$$\begin{gathered} \left[\text{C}\right](\text{mol}/\text{L}) Time\left(\text{s}\right) \\ 0.43 1.0\times {10}^{-2} \\ 0.36 2.0\times {10}^{-2} \\ 0.29 3.0\times {10}^{-2} \\ 0.22 4.0\times {10}^{-2} \\ 0.15 5.0\times {10}^{-2} \\ 0.08 6.0\times {10}^{-2} \end{gathered}$$

Write the rate law for this reaction, and calculate the rate constant.

Short answer

The rate law for the reaction is

$\text{rate}=\text{K}\left[\text{A}\right][\text{B}{]}^2$

The value of rate constant is$0.0185{\text{M}}^{-1}{\text{S}}^{-1}$ .

Step-by-step solution

Calculating the value of rate law:

when we plot $1/\left[B\right]$versus time we get a straight dine which means reaction is second-order with respect to B.

when we plat $\ln \left[A\right]$ versus time we get a straight line which means reaction is first order with respect to A. when we plot [c] versus time we get a straight line which means reaction is zero-order with respect to$C$. Therefore:

$\text{Rate}=\text{K}\left[\text{A}\right][\text{B}{]}^2$

Calculating the value of rate constant) Now we can calculate the value of the reaction rate constant using the data set acquired in experiment 1:

$\text{K}=\frac{2.7\times {10}^{-4}\text{M}}{1\times {10}^5\text{sec}\times 2.0\text{M}\times {\left(2.7\times {10}^{-4}\right)}^2}$

now we can solve for $K$:

$\text{K}=0.0185{\text{M}}^{-1}{\text{S}}^{-}$