Q16.114 CP - Chemistry: Molecular Nature Of Matter And Change

Question

Carbon disulfide, a poisonous, flammable liquid, is an excellent solvent for phosphorus, sulfur, and some other nonmetals. A kinetic study of its gaseous decomposition reveals these data: 

Experiment	Initial $\left[C{S}_2\right]$ mol/L	Initial rate ($\mathit{m}\mathit{o}\mathit{l}\mathbf{/}\mathit{L}{\mathit{s}}^{\mathbf{-}\mathbf{1}}$ )
1	0.100	$\mathbf{2}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$
2	0.080	2.2$\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$
3	0.055	1.5 $\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$
4	0.044	1.2$\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$

 

(a) Write the rate law for the decomposition of $\mathit{C}{\mathit{S}}_{\mathbf{2}}$ . 

(b) Calculate the average value of the rate constant

Step-by-Step Solution

Answer

1. Rate law is expressed as:  $R=K{\left[C{S}_2\right]}^2$
2. Average rate constant $=1.9\times {10}^{-7}mol/L{s}^{-1}$ 

Step 1: Rate law

The rate law is the expression of the reaction used to tell how the reaction rate depends upon the reactant concentration and informs about the order of the reaction.

 $\mathit{R}\mathbf{=}\mathit{K}{\left[A\right]}^{\mathbf{X}}{\left[B\right]}^{\mathbf{Y}}$

Step 2: Rate law for carbon disulfide

As per the given experiments:

Experiment	Initial $\left[C{S}_2\right]$mol/L	Initial rate ( $mol/L{s}^{-1}$)
1	0.100	$2.7\times {10}^{-7}$
2	0.080	2.2 $\times {10}^{-7}$
3	0.055	1.5 $\times {10}^{-7}$
4	0.044	1.2 $\times {10}^{-7}$

 

The reaction rate is given with the initial concentration of the reactant. 

Rate law expresses as:

 $R=K{\left[A\right]}^X$

To find the order (x), compare the rate of experiments 1 and 3:

$$\begin{gathered} \frac{ R_1}{R_3}=\frac{K{\left[C{S}_2\right]}^X}{K{\left[C{S}_2\right]}^X} \\ \frac{2.7\times {10}^{-7}}{1.5\times {10}^{-7}}=\frac{K{[0.100]}^X}{K{[0.055]}^X} \\ Hence, k is same for the reaction \\ so, \\ x=2 \end{gathered}$$ 

The rate law is expressed as: $R=K{\left[C{S}_2\right]}^2$ 

Step 3: Average rate constant

The average rate constant is defined as the sum of the rate constant divided by the experiments.

$$\begin{gathered} Average rate constant=\frac{\left(2.7+2.2+1.5+1.2\right)\times {10}^{-7}}{4} \\ =\frac{7.6\times {10}^{-7}}{4} \\ =1.9\times {10}^{-7}mol/L{s}^{-1} \end{gathered}$$