Question

The degree of dissociation of a first order reaction is:
[A] ${{e}^{-kt}}$
[B] ${{a}_{\circ }}{{e}^{-kt}}$
[C] $1-{{e}^{+kt}}$
[D] $1-{{e}^{-kt}}$

Answer 1

Hint: To answer this, consider a first order reaction and write down the number of moles before and after dissociation. Do not forget to assume the degree of dissociation ($\alpha $). Then, you can use the Arrhenius equation for a first order reaction i.e. \[A={{A}_{\circ }}{{e}^{-kt}}\] and solve for $\alpha $.

Complete answer:
To answer this firstly we should know that degree of dissociation is the extent of the molecules breaking into ions in a solution.
We know that order of a reaction is the relation between the rate of the reaction and the concentration of the reactants in it.
Order of the reaction tells us how the reaction rate is affected by the concentration of the reactants that are involved in the reaction. To answer this, firstly let us discuss what a first order reaction is.
For a first order reaction, the concentration of only one of the reactant species will affect the reaction rate.
We know that for a first order reaction, Arrhenius equation is given as -
     \[A={{A}_{\circ }}{{e}^{-kt}}\]
Now, let us consider a first order reaction - $A\to B$. Let us consider the initial amount of A to be ${{a}_{\circ }}$ and the degree of dissociation to be $\alpha $. Therefore, we can write that-

	A	B
Before dissociation	${{a}_{\circ }}$	0
After dissociation	${{a}_{\circ }}$(1-$\alpha $)	${{a}_{\circ }}$$\alpha $

Now, if we put the values in the Arrhenius equation, we will get-
     \[{{a}_{\circ }}\left( 1-\alpha \right)={{a}_{\circ }}{{e}^{-kt}}\]
From here, if we solve for alpha, we will get -
     \[\begin{align}
  & \left( 1-\alpha \right)={{e}^{-kt}} \\
 & or,\alpha ={{e}^{1-kt}} \\
\end{align}\]

Therefore, the correct answer is option [D] $1-{{e}^{-kt}}$ .

Note:
We should remember that the rate of a reaction depends upon factors like the presence or absence of a catalyst, nature of the reactants and also on the temperature at which the reaction is taking place. Here, as the reaction is first ordered, the rate is only dependent upon one reactant. However, other reactants can be present but they will be zero – ordered.