Pseudo first-order reactions are reactions that are not first-order but appear to be first order due to larger concentrations of one or more reactants than the other reactants. Some of the examples of pseudo first-order reactions involve Acid-catalysed hydrolysis of ethyl acetate, and inversion of sugarcane. The unit of the rate of the reaction \(\left( {\rm{k}} \right)\) is \({\left( {{\rm{mol}}\,{{\rm{L}}^{ – 1}}} \right)^{1 – {\rm{n}}}}\,{{\rm{S}}^{ – 1}}\).

In this article, we will provide detailed information on Pseudo First-order reactions. This article will give one example of pseudo-first-order reaction and thus explain pseudo-first-order reaction. Scroll down to learn more about pseudo-first-order reactions and pseudo first-order reaction examples.

Pseudo First-order Reaction: Overview

We know that the order of the reaction is determined by the rate of reaction’s dependency on the concentration of reactants. The order of the reaction is zero if the rate is independent of the concentrations of reactants. Similarly, the order of the reaction is one if the rate of reaction is proportional to the first power of the reactant concentration.

However, changing the concentration of the reactants, that is, increasing or decreasing the concentration of one or the other reactant, can sometimes change the order of a reaction.

Pseudo first-order reaction is a reaction that is not inherently first-order but is made by increasing or decreasing the concentration of one or the other components. The word ‘pseudo’ denotes ‘fake.’

As a result, it is evident from the name that it is not a first-order reaction. Certain conditions are changed to change the reaction sequence. Let’s look at some examples to learn more about pseudo first-order reactions.

What is a Pseudo First-order Reaction?

Consider the general reaction, \({\rm{A}} + {\rm{B}} \to \) products, in which order with respect to each reactant is \(1\) so that the overall order of the reaction is \(2.\) The rate law equation is:

Rate \( = {\rm{k}}\left[ {\rm{A}} \right]\left[ {\rm{B}} \right]\)

However, if one of the reactants is present in a high concentration (solvent), then there is very little change in its concentration during the reaction.

In other words, the concentration of that reactant remains practically constant during the reaction. For example, if \(\left[ {\rm{A}} \right] = 0.01\,{\rm{M}}\) and that of solvent water \(\left[ {\rm{B}} \right] = 55.5\,{\rm{M,}}\) the concentration of \({\rm{B}}\) changes only from \(55.50\) to \(55.49\,{\rm{M}}\) even after the completion of the reaction. Under such conditions, we may write,

Rate \({\rm{ = }}{{\rm{k}}_0}\left[ {\rm{A}} \right]\) where \({{\rm{k}}_0} = {\rm{k}}\left[ {\rm{B}} \right]\)

The reaction, therefore, behaves as a first-order reaction in \({\rm{A}}.\) Such reactions are called pseudo-first-order reactions.

Such reactions which are not truly of the first order but under certain conditions become reactions of the first order are called pseudo-first-order reactions.

Examples of Pseudo First-order Reaction

Acid-catalysed hydrolysis of ethyl acetate: \({\text{C}}{{\text{H}}_{\text{3}}}{\text{COO}}{{\text{C}}_{\text{2}}}{{\text{H}}_{\text{5}}}{\text{ + }}{{\text{H}}_{\text{2}}}{\text{O}}\xrightarrow{{{{\text{H}}^{\text{ + }}}}}{\text{C}}{{\text{H}}_{\text{3}}}{\text{COOH + }}{{\text{C}}_{\text{2}}}{{\text{H}}_{\text{5}}}{\text{OH}}\)

Acid-catalysed inversion of cane-sugar:\({{\text{C}}_{{\text{12}}}}{{\text{H}}_{{\text{22}}}}{{\text{O}}_{{\text{11}}}}{\text{ + }}{{\text{H}}_{\text{2}}}{\text{O}}\xrightarrow{{{{\text{H}}^{\text{ + }}}}}\mathop {{{\text{C}}_{\text{6}}}{{\text{H}}_{{\text{12}}}}{{\text{O}}_{\text{6}}}}\limits_{{\text{Glucose}}} {\text{ + }}\mathop {{{\text{C}}_{\text{6}}}{{\text{H}}_{{\text{12}}}}{{\text{O}}_{\text{6}}}}\limits_{{\text{Fructose}}} \)

Both the above reactions are bimolecular but are found to be of the first order; we can observe these reactions as

For the first reaction, rate of reaction \({\rm{\alpha }}\left[ {{\rm{C}}{{\rm{H}}_3}{\rm{COO}}{{\rm{C}}_2}{{\rm{H}}_5}} \right]\) only

and for the second reaction, rate of reaction \({\rm{\alpha }}\left[ {{{\rm{C}}_{12}}{{\rm{H}}_{22}}{{\rm{O}}_{11}}} \right]\) only.

The reason for such a behaviour is obvious from the fact that water is present in such a large excess that its concentration remains almost constant during the reaction.

The kinetics of the above reactions have been studied as follows:

Hydrolysis of Ethyl Acetate

\({\text{C}}{{\text{H}}_{\text{3}}}{\text{COO}}{{\text{C}}_{\text{2}}}{{\text{H}}_{\text{5}}}{\text{ + }}{{\text{H}}_{\text{2}}}{\text{O}}\xrightarrow{{{{\text{H}}^{\text{ + }}}}}{\text{C}}{{\text{H}}_{\text{3}}}{\text{COOH + }}{{\text{C}}_{\text{2}}}{{\text{H}}_{\text{5}}}{\text{OH}}\)

In this reaction, acetic acid is one of the products, the amount of which can be found by titration against standard NaOH solution. But being an acid-catalyzed reaction, the acid present originally as a catalyst also reacts with \({\rm{NaOH}}\) solution. Hence, a little careful thought reveals that for the same value of reaction mixture withdrawn at different times,

Combining results \(\left( {\rm{i}} \right)\) and \(\left( {\rm{ii}} \right)\), we find that

Combining results \(\left( {\rm{i}} \right)\) and \(\left( {\rm{v}} \right)\), we find that

The maximum amount of \({\rm{C}}{{\rm{H}}_3}{\rm{COOH}}\) produced \({\rm{\alpha }}\left( {{{\rm{V}}_\infty } – {{\rm{V}}_0}} \right)\)

But the maximum amount of \({\rm{C}}{{\rm{H}}_3}{\rm{COOH}}\) produced α Initial concentration of \({\rm{C}}{{\rm{H}}_3}{\rm{COO}}{{\rm{C}}_2}{{\rm{H}}_5}\left( {\rm{a}} \right)…..\left( {{\rm{vi}}} \right)\)

Hence, \({\rm{a}}{\mkern 1mu} {\rm{\alpha }}\left( {{{\rm{V}}_\infty } – {{\rm{V}}_0}} \right)\)

From equations \(\left( {{\rm{iv}}} \right)\) and \(\left( {{\rm{vi}}} \right)\), we have

\(\left( {{\rm{a}} – {\rm{X}}} \right)\,{\rm{\alpha }}\left( {{{\rm{V}}_\infty } – {{\rm{V}}_0}} \right) – \left( {{{\rm{V}}_{\rm{t}}} – {{\rm{V}}_0}} \right)\) or \(\left( {{\rm{a}} – {\rm{X}}} \right)\alpha \left( {{\rm{V}}\infty  – {{\rm{V}}_{\rm{t}}}} \right)…..\left( {{\rm{vii}}} \right)\)

Substituting the values of a and \(\left( {{\rm{a}} – {\rm{X}}} \right)\) from equations \(\left( {{\rm{vi}}} \right)\) and \(\left( {{\rm{vii}}} \right)\) in the first-order equation, we get

\({\rm{k}} = \frac{{2.303}}{{\rm{t}}}\log \frac{{\rm{a}}}{{{\rm{a}} – {\rm{X}}}}\) or \({\rm{k}} = \frac{{2.303}}{{\rm{t}}}\log \frac{{{{\rm{V}}_\infty } – {{\rm{V}}_0}}}{{{{\rm{V}}_\infty } – {{\rm{V}}_{\rm{t}}}}}\)

Inversion of Cane sugar (Sucrose)

The hydrolysis of sucrose in the presence of a mineral acid takes place according to the equation:

An important characteristic of the reaction is that sucrose is dextro-rotatory, whereas the products glucose and fructose are dextro-rotatory and laevorotatory, respectively. Further, the laevo-rotation of fructose is more (being – \({92^{\rm{o}}}\)) than the dextro-rotation of glucose (being \( + {52.5^{\rm{o}}}\)) so that the mixture as a whole is laevorotatory. Thus, on hydrolysis, the dextro-rotatory sucrose gradually changes into the laevorotatory mixture. It is for this reason that the reaction is called the inversion of sucrose.

The kinetics of the above reaction is studied by noting the angle of rotation at different intervals of time with the help of a polarimeter. Suppose

1. Reading of the polarimeter at zero time \( = {{\rm{r}}_0}\)
2. Reading of the polarimeter at any time \({\rm{t}} = {{\rm{r}}_{\rm{t}}}\)
3. Reading of the polarimeter at infinite time \( = {{\rm{r}}_\infty }\) (i.e., after \(24\) hours or more)

It is evident that the reading at zero time will be positive and would decrease with the passage of time, pass through zero and ultimately become negative.

A little careful consideration shows that

Angle of rotation at any instant of time,

i.e., \(\left( {{{\rm{r}}_0} – {{\rm{r}}_{\rm{t}}}} \right){\rm{\alpha }}\) amount of sucrose hydrolysed \(\left( {\rm{x}} \right),\)

i.e., \({\rm{x}}{\mkern 1mu} \alpha \left( {{{\rm{r}}_0} – {{\rm{r}}_{\rm{t}}}} \right)….\left( {\rm{i}} \right)\)

Angle of rotation at infinite time,

i.e., \(\left( {{{\rm{r}}_0} – {{\rm{r}}_\infty }} \right){\rm{\alpha }}\) the initial concentration of sucrose \(\left( {\rm{a}} \right),\)

i.e., \({\rm{a}}{\mkern 1mu} \alpha \left( {{{\rm{r}}_0} – {{\rm{r}}_\infty }} \right)….\left( {{\rm{ii}}} \right)\)

From equation \(\left( {\rm{i}} \right)\) and \(\left( {\rm{ii}} \right)\), we have

\(\left( {{\rm{a – x}}} \right)\,{\rm{\alpha }}\left( {{{\rm{r}}_0} – {{\rm{r}}_\infty }} \right) – \left( {{{\rm{r}}_0} – {{\rm{r}}_{\rm{t}}}} \right)\)

i.e., \(\left( {{\rm{a – x}}} \right){\mkern 1mu} \alpha \left( {{{\rm{r}}_{\rm{t}}} – {{\rm{r}}_\infty }} \right)….\left( {{\rm{iii}}} \right)\)

Substituting the values of \({\rm{a}}\) and \(\left( {{\rm{a – x}}} \right)\) from equations \(\left( {\rm{ii}} \right)\) and \(\left( {\rm{iii}} \right)\) in the first-order equation,

\({\rm{k}} = \frac{{2.303}}{{\rm{t}}}\log \frac{{\rm{a}}}{{{\rm{a}} – {\rm{x}}}},\) we get \({\rm{k}} = \frac{{2.303}}{{\rm{t}}}\log \frac{{{{\rm{r}}_0} – {{\rm{r}}_\infty }}}{{{{\rm{r}}_{\rm{t}}} – {{\rm{r}}_\infty }}}.\)

The applicability of this equation for the inversion of sucrose was first shown by Wilhelm \((1850).\) It should be noted that the actual value of \({\rm{k}},\) of course, depends upon the concentration of \({{\rm{H}}^ + }\) ions.

We are isolating a reactant in pseudo-first-order reactions by increasing the concentration of the other reactants. Changes in the concentrations of the other reactants do not affect the reaction when they are in excess; therefore, the reaction now just depends on the concentration of the isolated reactant. In the rate law, the concentrations of all other reactants are assumed to be constant. As a result, the order of reaction becomes one.

Units of the Rate Constant of a Pseudo Unimolecular Reaction

The unit of the rate of the reaction \(\left( {\rm{k}} \right)\) is \({\left( {{\rm{mol}}\,{{\rm{L}}^{ – 1}}} \right)^{1 – {\rm{n}}}}\,{{\rm{S}}^{ – 1}},\) where n is the order of the reaction.

For a pseudo first-order reaction \({\rm{n}} = 1.\)

So, \({\left( {{\rm{mol}}\,{{\rm{L}}^{ – 1}}} \right)^{1 – {\rm{n}}}}\,{{\rm{S}}^{ – 1}} = {\left( {{\rm{mol}}\,{{\rm{L}}^{ – 1}}} \right)^{1 – 1}}\,{{\rm{S}}^{ – 1}} = {{\rm{S}}^{ – 1}}.\)

Summary

Pseudo-first-order reactions are described as reactions that are in the higher order but under particular conditions behave as first-order reactions. Pseudo-first order kinetics is an important example. When a reaction is a \(2\)and order overall but first order with regard to two reactants, it is said to be first order. The initial rate is influenced by both \({\rm{A}}\) and \({\rm{B}}\) and as the reaction progresses, both \({\rm{A}}\) and \({\rm{B}}\)’s concentrations change, affecting the rate.

FAQs on Pseudo First Order Reaction

Q.1: What is the difference between first-order and pseudo-first-order reactions?
Ans: A first-order reaction is when the concentration of only one reactant changes or in which the reaction rate is proportional to the first power of the concentration of the reactant.
The reactions which are not truly of the first order but under certain conditions become reactions of the first order are called pseudo-first-order reactions.

Q.2: How do you solve a pseudo-first-order reaction?
Ans: In any reaction, if one of the components is in excess quantity, then the concentration change will be constant. The reaction is belonging to a pseudo-first-order reaction.

Q.3: What is the pseudo-first-order reaction?
Ans: The reactions apparently seem to be of higher order but under certain circumstances behave as first-order reactions referred to as pseudo-first-order reactions.

Q.4: What is the pseudo rate constant?
Ans: The unit of the rate of the reaction \(\left( {\rm{k}} \right)\) is \({\left( {{\rm{mol}}\,{{\rm{L}}^{ – 1}}} \right)^{1 – {\rm{n}}}}\,{{\rm{S}}^{ – 1}},\)
where \({\rm{n}}\) is the order of the reaction.
For a pseudo first-order reaction \({\rm{n}} = 1.\)
So, \({\left( {{\rm{mol}}\,{{\rm{L}}^{ – 1}}} \right)^{1 – {\rm{n}}}}\,{{\rm{S}}^{ – 1}} = {\left( {{\rm{mol}}\,{{\rm{L}}^{ – 1}}} \right)^{1 – 1}}\,{{\rm{S}}^{ – 1}} = {{\rm{S}}^{ – 1}}.\)

Q.5: Does temperature affect the pseudo rate constant?
Ans: The reaction rate was found to increase with increasing temperature, indicating that a highly solvated transition state was formed during the reaction.

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