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Question: During the kinetic study of the reaction, \[2{\text{A + B }} \to {\text{ C + D}}\] the following res...

During the kinetic study of the reaction, 2A + B → C + D2{\text{A + B }} \to {\text{ C + D}} the following results were obtained.

S.No[A]\left[ {\text{A}} \right][B]\left[ B \right]Rate of Formation of D in M s−1{\text{M }}{{\text{s}}^{ - 1}}
a.0.10.16.0 × 10−36.0{\text{ }} \times {\text{ 1}}{{\text{0}}^{ - 3}}
b.0.30.27.2 × 10−27.2{\text{ }} \times {\text{ 1}}{{\text{0}}^{ - 2}}
c.0.30.42.88 × 10−1{\text{2}}{\text{.88 }} \times {\text{ 1}}{{\text{0}}^{ - 1}}
d.0.40.12.40 × 10−2{\text{2}}{\text{.40 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}

On the basis of above which of the following is true.
A. r = k[A]2[B]r{\text{ = k}}{\left[ {\text{A}} \right]^2}\left[ {\text{B}} \right]
B. r = k[A][B]r{\text{ = k}}\left[ {\text{A}} \right]\left[ {\text{B}} \right]
C. r = k[A]2[B]2r{\text{ = k}}{\left[ {\text{A}} \right]^2}{\left[ {\text{B}} \right]^2}
D. r = k[A][B]2r{\text{ = k}}\left[ {\text{A}} \right]{\left[ {\text{B}} \right]^2}

Explanation

Solution

We will write the rate of reactions in terms of order of reaction. Then we will find the ratio of rate of formation of D at multiple instant of concentration values given in the table. Thus after getting the ratio we are able to find the order of the reaction.

Complete answer: For the given reaction, 2A + B → C + D2{\text{A + B }} \to {\text{ C + D}} the rate of reaction can be written as,
r = k[A]a[B]br{\text{ = k}}{\left[ {\text{A}} \right]^a}{\left[ {\text{B}} \right]^b}
We have to find the value of powers of concentration of A and B. To find the values of powers we will take help from the data given in the table. From a. we can see that when concentration of A and B is equal to 0.10.1 then rate of formation of D is 6.0 × 10−36.0{\text{ }} \times {\text{ 1}}{{\text{0}}^{ - 3}} which can be represented as,
6.0 × 10−3 = k[0.1]a[0.1]b{\text{6}}{\text{.0 }} \times {\text{ 1}}{{\text{0}}^{ - 3}}{\text{ = k}}{\left[ {0.1} \right]^a}{\left[ {0.1} \right]^b} _______(i)(i)
Similarly we can write for d. as,
r = k[A]a[B]br{\text{ = k}}{\left[ {\text{A}} \right]^a}{\left[ {\text{B}} \right]^b}
2.40 × 10−2 = k[0.4]a[0.1]b{\text{2}}{\text{.40 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}{\text{ = k}}{\left[ {0.4} \right]^a}{\left[ {0.1} \right]^b} ________(ii)(ii)
On dividing both the equations we get the result as,
2.40 × 10−26.0 × 10−3 = k[0.4]a[0.1]bk[0.1]a[0.1]b\dfrac{{{\text{2}}{\text{.40 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}}}{{6.0{\text{ }} \times {\text{ 1}}{{\text{0}}^{ - 3}}}}{\text{ = }}\dfrac{{{\text{k}}{{\left[ {0.4} \right]}^a}{{\left[ {0.1} \right]}^b}}}{{{\text{k}}{{\left[ {0.1} \right]}^a}{{\left[ {0.1} \right]}^b}}}
4 = 4a{\text{4 = }}{{\text{4}}^a}
a = 1a{\text{ = 1}}
Therefore the value of a is one. Now we will apply the same for b. and c. For serial number b. we can write the rate of reaction as,
r = k[A]a[B]br{\text{ = k}}{\left[ {\text{A}} \right]^a}{\left[ {\text{B}} \right]^b}
7.2 × 10−2 = k[0.3]a[0.2]b{\text{7}}{\text{.2 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}{\text{ = k}}{\left[ {0.3} \right]^a}{\left[ {0.2} \right]^b} _________(iii)(iii)
For serial number c. it can also be written as,
r = k[A]a[B]br{\text{ = k}}{\left[ {\text{A}} \right]^a}{\left[ {\text{B}} \right]^b}
2.88 × 10−1 = k[0.3]a[0.4]b{\text{2}}{\text{.88 }} \times {\text{ 1}}{{\text{0}}^{ - 1}}{\text{ = k}}{\left[ {0.3} \right]^a}{\left[ {0.4} \right]^b} __________(iv)(iv)
On diving both equations we get the result as,
2.88 × 10−17.2 × 10−2 = k[0.3]a[0.4]bk[0.3]a[0.2]b\dfrac{{{\text{2}}{\text{.88 }} \times {\text{ 1}}{{\text{0}}^{ - 1}}}}{{7.2{\text{ }} \times {\text{ 1}}{{\text{0}}^{ - 2}}}}{\text{ = }}\dfrac{{{\text{k}}{{\left[ {0.3} \right]}^a}{{\left[ {0.4} \right]}^b}}}{{{\text{k}}{{\left[ {0.3} \right]}^a}{{\left[ {0.2} \right]}^b}}}
288 72  = 2b\dfrac{{{\text{288 }}}}{{72{\text{ }}}}{\text{ = }}{{\text{2}}^b}
4 = 2b{\text{4 = }}{{\text{2}}^b}
22 = 2b{{\text{2}}^2}{\text{ = }}{{\text{2}}^b}
b = 2b{\text{ = 2}}
Therefore we get the values of powers of coefficient of concentration. Thus we can represent the rate of reaction as,
r = k[A]a[B]br{\text{ = k}}{\left[ {\text{A}} \right]^a}{\left[ {\text{B}} \right]^b}
r = k[A][B]2r{\text{ = k}}\left[ {\text{A}} \right]{\left[ {\text{B}} \right]^2}
Hence the correct option is D.

Note:
We can also find the rate of reaction by observing the value of the rate of formation of D. When concentration of A is quadrupled the rate also gets quadrupled. So the order of reaction is four with respect to A. It is preferred to find the order of reaction in the above way to avoid minor problems with calculations.