Question
Consider the two hypothetical reactions given below :
$$\left( {\text{i}} \right)\,\,\,\,\,aA \to {\text{Products,}}$$ \[k=x\,mo{{l}^{-1}}L\,{{\min }^{-1}}\]
$$\left( {{\text{ii}}} \right)\,\,bB \to {\text{Products,}}$$ \[k=y\,{{\min }^{-1}}\]
The half-lives of both the reactions are the same, equal to $$1\,hr$$ when molar concentration of the reactant is $$1.0\,M$$ in each case. If these reactions are started at the same time taking $$1M$$ of the reactant in each case, the ratio $$\frac{{\left[ A \right]}}{{\left[ B \right]}}$$ after $$3\,hr$$
will be :
A.
0.5
B.
4
C.
1
D.
2
Answer :
2
Solution :
Units of $$k$$ indicate that reaction (i) is of second order and reaction (ii) is first order.
For (i) reaction, $${t_{\frac{1}{2}}} \propto \frac{1}{a},$$
$$\eqalign{
& {\rm{first}}\,\,{t_{{1 \over 2}}} = 1\,hr,\,{\rm{second}}\,{t_{{1 \over 2}}} = 2\,hr \cr
& \left[ A \right] = 1M\buildrel {1\,hr} \over
\longrightarrow 0.5M\buildrel {2\,hr} \over
\longrightarrow 0.25M \cr
& \left[ B \right] = 1M\buildrel {1\,hr} \over
\longrightarrow 0.5M\buildrel {1\,hr} \over
\longrightarrow 0.25M\buildrel {1\,hr} \over
\longrightarrow 0.12M \cr
& {{\left[ A \right]} \over {\left[ B \right]}} = {{0.25M} \over {0.125M}} = 2 \cr} $$