Question

Write differential rate equation for the reaction $2A+B\to $ products.

Answer 1

Hint: The rate of reaction of a third order reaction depends on three concentration terms. Differential rate equation describes the rate of the reaction in terms of changes in reactant concentrations over an interval of time. For a general reaction given below:
\[aA+bB\to cC+dD\]
Differential rate equation can be given as $-\dfrac{dx}{dt}=k{{\left[ A \right]}^{a}}{{\left[ B \right]}^{b}}$

Complete step by step solution:
The given reaction is $2A+B\to $ products.
We can write the rate of the reaction as the change in concentration of any one of the reactants with the change in time. For the given reaction, the rate of the disappearance of A is twice than that of B, so $-\dfrac{d\left[ A \right]}{dt}$ is divided by 2 to make the rate of disappearance of both A and B equal, i.e.
\[-\dfrac{1}{2}\dfrac{d\left[ A \right]}{dt}=-\dfrac{d\left[ B \right]}{dt}\]
Here, the negative sign represents that the concentration of reactants is decreasing.
According to the law of mass action, the rate of the reaction depends on the concentration of reactants in which each term is raised to the power equal to its stoichiometric coefficient. For the given reaction, we can write
\[Rate=k{{\left[ A \right]}^{2}}\left[ B \right]\]
Let ‘$a$’ moles/litre be the initial concentration of A and ‘$b$’ moles/liter be the initial concentration of B.
If the decrease in the concentration of the reactants A and B in time ‘t’ is ‘$x$’ moles/liter, then the rate of the reaction at any instant of time t can be given by
\[Rate=\dfrac{dx}{dt}\]
 Comparing the above two equation, we get
\[\dfrac{dx}{dt}=k{{\left[ A \right]}^{2}}\left[ B \right]\]
Further, at any instant of time t, then we have
Concentration of reactant A, $\left[ A \right]$ = ($a-x$) moles/liter
And, concentration of B, $\left[ B \right]$ = ($b-x$) moles/liter
We can, thus, write
\[\begin{align}
  & \dfrac{dx}{dt}=k{{\left[ A \right]}^{2}}\left[ B \right] \\
 & \dfrac{dx}{dt}=k{{(a-x)}^{2}}(b-x) \\
\end{align}\]
The above equation is the differential rate equation for the given reaction.

Note: It is to be noted here that we are considering the Law of mass action of the given reaction. However, it is not necessary the rate of the reaction depends on all the concentration terms of the reactants and the order is equal to the sum of stoichiometric coefficients of the reactants, i.e. 3. Actual order and rate equation (or rate law) for a reaction are determined experimentally.