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Question: For a given reaction \[3A + B \to C + D\] the rate of reaction can be represented by: A. \[ - \dfr...

For a given reaction 3A+B→C+D3A + B \to C + D the rate of reaction can be represented by:
A. −13d[A]dt=+d[C]dt=+d[D]dt - \dfrac{1}{3}\dfrac{{d[A]}}{{dt}} = \dfrac{{ + d[C]}}{{dt}} = \dfrac{{ + d[D]}}{{dt}}
B. −13d[A]dt=−d[C]dt=k[A]m[B]n - \dfrac{1}{3}\dfrac{{d[A]}}{{dt}} = - \dfrac{{d[C]}}{{dt}} = k{[A]^m}{[B]^n}
C. +13d[A]dt=−d[C]dt=k[A]m[B]n + \dfrac{1}{3}\dfrac{{d[A]}}{{dt}} = \dfrac{{ - d[C]}}{{dt}} = k{[A]^m}{[B]^n}
D.All of these

Explanation

Solution

Rate of reaction for any given reaction depends on change in concentration of reactants or products with respect to time multiplied by reciprocal of their respective stoichiometric coefficients. Reactants are written with negative signs because in the reaction, reactants are decreasing and products are written with positive signs because in the reaction products are increasing.

Complete step by step answer:
Let us see how to calculate the rate of reaction for the given reaction:
3A+B→C+D3A + B \to C + D
We know that the rate of reaction is equal to the decrease in the concentration of the reactants, such as A and B here, divided by the time interval or it is equal to the increase in the concentration of the products which are c and d here, divided by the time interval. So, we can write a general formula for rate of reaction as:
rate  of  reaction=decrease  in  concentration  of A  or Btime  int⁡erval=Increase  in  concentration  of C or Dtime  int⁡ervalrate\,\,of\,\,reaction = \dfrac{{decrease\,\,in\,\,concentration\,\,of\,A\,\,or\,B}}{{time\,\,\operatorname{int} erval}} = \dfrac{{Increase\,\,in\,\,concentration\,\,of\,C\,or\,D}}{{time\,\,\operatorname{int} erval}}
We can now define the rate of reaction as a change in concentration of any one of the reactants or products per unit time. It can also be written as
rate  of  reaction=−ΔAΔt=+ΔCΔtrate\,\,of\,\,reaction = \dfrac{{ - \Delta A}}{{\Delta t}} = + \dfrac{{\Delta C}}{{\Delta t}}
The negative sign of reactant implies that the reactant concentration is decreasing with time and the positive sign of product implies that the product concentration is increasing with time. We can calculate the rate of reaction at any instant of time and this can be referred to as instantaneous rate of reaction. The stoichiometric coefficients are added as prefixes in fraction form.
So, for the given chemical equation, it can be written as
−13d[A]dt=−d[B]dt=+d[C]dt=+d[D]dt- \dfrac{1}{3}\dfrac{{d[A]}}{{dt}} = - \dfrac{{d[B]}}{{dt}} = \dfrac{{ + d[C]}}{{dt}} = \dfrac{{ + d[D]}}{{dt}}

Hence, the correct option is (A).

Note:
The unit of concentration can be expressed as moles/litre and time can be expressed in either seconds or minutes. From these two units, we get the unit of rate of reaction as moles litre−1sec⁡−1moles\,litr{e^{ - 1}}{\sec ^{ - 1}} or moles litre−1min⁡−1moles\,litr{e^{ - 1}}{\min ^{ - 1}} . There are many factors which affect the rate of reaction such as concentration, rate law and order of a reaction.