What is the order of chemical reactions?
The sum of the power of concentration of reactants in the rate law expression is called the order of that chemical reaction. Reactions can be a first-order reaction, second-order reaction, pseudo-first-order reaction, etc. depending on the concentration of the reactants. In this article, we will discuss second-order reactions in detail.
Suppose a reaction is \[ – aA + bB \rightarrow cC + dD \]
Rate according to rate law expression \[ = kA^{x}B^{y}\]
Where x and y are concentrations of A and B, respectively.
So, order of reaction will be = x + y
We can say x is the order of reaction with respect to A and y is the order of reaction with respect to B.
Now if suppose x=1 and y = 1 then the reaction will be a 2nd order reaction. Reactions in which reactants are identical and form a product can also be second-order reactions.
Many reactions such as decomposition of nitrogen dioxide, alkaline hydrolysis of ethyl acetate, decomposition of hydrogen iodide, formation of double-stranded DNA from two strands, etc. can be explained by second-order kinetics.
What is Second Order Reaction?
A reaction is called a second-order reaction when the overall order is two. Suppose if the reaction is as follows:
\[ A + A \rightarrow P \]
Or \[ 2A \rightarrow P \]
In these reactions, the rate is proportional to the square of the concentration of one reactant. The differential rate law for the above second-order reaction can be written as follows:
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The rate of such reactions can also be written as \[ r = kA^{2} \]
Here k is the rate constant for the second-order reaction. Unit of reaction rate (r) is moles per liter per second (\[mol.L^{-1}.s^{-1}\]) and the unit of second-order rate constant is \[ M^{-1}.s^{-1} \] (M is molarity which can be expressed as mol/L).
If both the reactants are different in the reaction:
\[ A + B \rightarrow P\]
The rate for the above reaction can be written as follows:
\[ R = kA^{x}B^{y}\]
Where the sum of x and y is equal to two.
Examples of Second-Order Reactions
A few examples of second-order reaction are given below:
Nitrogen dioxide decomposes into nitrogen monoxide and oxygen. Reaction is given below:
\[ 2NO_{2} \rightarrow 2NO + O_{2} \]
Decomposition of hydrogen iodide – Hydrogen iodide breaks down into iodine and hydrogen. The reaction is given below :
\[ 2HI \rightarrow I_{2} + H_{2} \]
Decomposition of nitrosyl bromide:
\[2NOBr \rightarrow 2NO + Br_{2}\]
Hydrolysis of an ester in presence of a base –
\[ CH_{3}COOC_{2}H_{5} + NaOH \rightarrow CH_{3}COONa + C_{2}H_{5}OH \]
Combustion Reaction –
\[ O_{2} + C \rightarrow O + CO \]
Integrated and differential Rate Equation for Second-Order Reactions
We are considering here that equation where chemical reaction can be represented as follows –
\[ A + A \rightarrow P ....(1) \]
Generally, polymerization reactions follow the same as in the two monomer units combine and form a polymer.
The differential rate law equation for the chemical equation (1) can be written as follows –
(Image to be added soon)_ _ _ _ _ (2)
On rearranging the above equation (2), we get –
(Image to be added soon) _ _ _ _ _ (3)
On integrating the above equation (3) considering that concentration of the reactant changes between time 0 and time t, we get –
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Applying the power rule of integration in equation (4), we get –
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On simplifying equation (5), we get –
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Equation (6) is the required integrated rate expression of second-order reactions.
Second-Order Reaction Graph
On rearranging the equation (6), we get –
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On comparing equation (7) with straight-line equation or linear equation y = mx + c, we can write –
Y = 1/At (on y-axis)
X = t (on x-axis)
m = k (Slope)
c = 1/A0 (Intercept)
So, the graph can be drawn as follows –
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It is clear from the graph that the slope is equal to the value of rate constant k.
The Half-life of Second-Order Reactions
The amount of time required by reactant/s in a reaction for undergoing decay by half is called the half-life of that reaction. In the same way, the amount of time required by reactant/s to undergo decay by half in the second-order reaction is called the half-life of the second-order reaction. So, while calculating the half-life of a reaction t becomes \[t_{\frac{1}{2}}\] and as \[t = t_{\frac{1}{2}}\] then At becomes \[ \frac{A_{0}}{2}\].
Now putting the values of t and A in equation (6), we get –
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On solving equation (8), we get –
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On simplifying equation (9), we get –
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Equation (11) is the equation for the half-life of a second-order reaction.
As we can see t1/2 is inversely proportional to the concentration of the reactant in second-order reactions. The graph is given below for the half-life of second-order reactions which is drawn between A and t.
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Although the graph looks very similar to first-order plots it decreases at a much faster rate as the graph shows above and the length of half-life increases while the concentration of the reactant decreases. This is the reason generally students find the concept of half-life for second-order reactions more difficult than first and zero-order reactions. The value of the rate constant of second-order reactions cannot be calculated directly from the half-life equation unless the initial concentration is known.
Determination of the Half-life of reactions is largely used in the pharma field. For example, drug dosage interval is determined on the basis of the half-life period of the reaction of the drug. When chemical kinetics is used in pharma, it is called pharmacokinetics. It can also be defined as the branch of pharmacology concerned with the movement of drugs within the body. Another vital application of half-life in pharmacokinetics is that half-life for the drug reaction shows how tightly drugs bind to each ligand before it is undergoing decay. It is very important for drug design to know how tightly it binds with ligands.
Conclusion
This was all about second-order reactions. You can get articles on related topics such as pseudo-first-order reactions, zero-order reactions, etc. as well on our website. If you want to get free PDFs of NCERT Solutions of Chemistry (for all classes), then register yourself on Vedantu or download the Vedantu learning app for Class 6-10, IITJEE and NEET.
FAQs on Second Order Reaction
1. What is a second-order reaction in Chemistry?
A second-order reaction is a type of chemical reaction where the rate of the reaction is proportional to the square of the concentration of a single reactant or to the product of the concentrations of two reactants. In simple terms, the overall order of the reaction is two. For a reaction 2A → P, the rate law would be expressed as Rate = k[A]², where 'k' is the rate constant.
2. What are some common examples of second-order reactions studied in Class 12?
According to the CBSE syllabus, some typical examples of second-order reactions include:
- Decomposition of Nitrogen Dioxide: 2NO₂(g) → 2NO(g) + O₂(g)
- Decomposition of Hydrogen Iodide: 2HI(g) → H₂(g) + I₂(g)
- Alkaline Hydrolysis of an Ester: CH₃COOC₂H₅ + NaOH → CH₃COONa + C₂H₅OH. In this case, the rate depends on the concentration of both the ester and the NaOH.
3. What is the main difference between a first-order and a second-order reaction?
The primary difference lies in how the reaction rate depends on reactant concentration. A first-order reaction's rate is directly proportional to the concentration of one reactant ([A]¹), while a second-order reaction's rate is proportional to the square of one reactant's concentration ([A]²) or the product of two ([A][B]). This also leads to a key distinction in their half-lives: the half-life of a first-order reaction is constant, whereas the half-life of a second-order reaction is inversely proportional to the initial reactant concentration.
4. What is the integrated rate equation for a second-order reaction?
For a simple second-order reaction of the type 2A → Products, the integrated rate equation is: 1/[A]t - 1/[A]₀ = kt. In this formula:
- [A]t is the concentration of reactant A at time 't'.
- [A]₀ is the initial concentration of reactant A.
- k is the rate constant for the reaction.
5. What is the unit of the rate constant (k) for a second-order reaction?
The unit for the rate constant (k) in a second-order reaction is L mol⁻¹ s⁻¹ or M⁻¹ s⁻¹. This unit ensures that the overall rate equation (Rate = k[A]²) is dimensionally consistent, where the rate is in mol L⁻¹ s⁻¹ and concentration is in mol L⁻¹.
6. How do you interpret the graph for a second-order reaction?
For a second-order reaction, a plot of 1/[A]t on the y-axis versus time (t) on the x-axis yields a straight line. This linear plot is a key identifier for second-order kinetics. The slope of this line is equal to the rate constant (k), and the y-intercept gives the value of 1/[A]₀ (the reciprocal of the initial concentration).
7. What is the formula for the half-life of a second-order reaction, and how does it relate to concentration?
The half-life (t₁/₂) of a second-order reaction is given by the formula: t₁/₂ = 1 / (k[A]₀). This equation shows that the half-life is inversely proportional to the initial concentration of the reactant ([A]₀). This means if you start with a higher concentration of reactants, the reaction will reach its half-life faster.
8. Why does the half-life of a second-order reaction get longer as the reaction proceeds?
This happens because a second-order reaction's rate depends on the collision of two reactant molecules. Initially, when the concentration is high, molecules are crowded, leading to frequent collisions and a fast reaction (short half-life). As reactants are consumed, their concentration drops. The remaining molecules are farther apart, so collisions become less frequent. This slows down the reaction rate, causing it to take a progressively longer time to halve the remaining concentration.
9. Can a reaction with two different reactants, like A + B → Products, be a second-order reaction?
Yes, absolutely. A reaction involving two different reactants can be second-order if its rate law is found to be Rate = k[A][B]. In this case, the reaction is first-order with respect to reactant A and first-order with respect to reactant B. The overall order of the reaction is the sum of the individual orders (1 + 1 = 2), making it a second-order reaction.
10. How can a student experimentally determine if a reaction follows second-order kinetics?
There are two primary methods to determine this experimentally:
- Graphical Method: Measure the reactant concentration [A] at different times (t) during the reaction. Then, plot a graph of 1/[A] versus t. If the resulting plot is a straight line, the reaction is confirmed to be second-order.
- Half-Life Method: Perform the experiment multiple times with different initial concentrations ([A]₀). Calculate the half-life (t₁/₂) for each experiment. If you observe that the half-life is inversely proportional to the initial concentration (e.g., doubling the concentration halves the half-life), the reaction follows second-order kinetics.
