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Question: The rate of a gaseous reaction is given by the expression \(k\left[ A \right]\left[ B \right]\). If ...

The rate of a gaseous reaction is given by the expression k[A][B]k\left[ A \right]\left[ B \right]. If the volume of the reaction vessel is suddenly reduced to 1/4  th{{{1}/{4}\;}^{th}}of the initial volume, the reaction rate relating to the original rate will be:
(a) 1/10  {1}/{10}\;
(b) 1/8  {1}/{8}\;
(c) 8
(d)16

Explanation

Solution

The concentration of a substance can be calculated by dividing the number of moles of the substance by the volume of the substance.
The equation is as follows:Concentration=No. of molesVolumeConcentration=\dfrac{No.\,of\,moles}{Volume}

Complete step-by-step answer: In the question, the rate expression of a gaseous reaction is given and we have to find the new reaction rate when the volume is reduced by 1/4  th{{{1}/{4}\;}^{th}}of the initial volume.
We know that the concentration of a substance can be calculated by dividing the number of moles of the substance by the given volume of the substance.
And by correlating the terms, we get an equation for concentration as,
Concentration=No. of molesVolumeConcentration=\dfrac{No.\,of\,moles}{Volume}
Now we know how concentration and volume are related to each other, concentration is inversely proportional to the volume of the substance.
The rate equation given is :k[A][B]k\left[ A \right]\left[ B \right], where k is the rate constant,[A]\left[ A \right] is the concentration of A, and [B]\left[ B \right] is the concentration of B.
Now the new volume is reduced to 1/4  th{{{1}/{4}\;}^{th}}of the initial volume.
So as volume and concentration parameters are inversely proportional, the concentration of the reagents will increase by four times.
This can be represented as : Concentration=Number of molesVolume4Concentration=\dfrac{Number\,of\,moles}{\dfrac{Volume}{4}}
4×concentration=Number of molesVolume4\times concentration=\dfrac{Number\,of\,moles}{Volume}
So the concentration of A and B increases, hence the new rate equation can be written as,
New rate equation=k[4A][4B]=16k[A][B]=k\left[ 4A \right]\left[ 4B \right]=16k\left[ A \right]\left[ B \right]
Hence the new rate of the reaction is 16 times greater than the old rate of the reaction

Therefore the correct answer for the above-given question is option (d).

Note: Always consider various parameters and compare them and find the relationship between the parameters for solving these types of problems in which the dimensions of the parameters associated are changed.
And in the rate equation, k is a proportionality constant and it does not have anything to do with concentration and volume. Some get confused and change the value of k concerning the change in volume terms or concentration.