What is an Integrated Rate Equation Mean?
An equation that represents the dependence of the reaction rate on the concentration of reacting species is called the differential rate equation. The instantaneous rate of reaction can be expressed as the tangent slope at any instant of time in the graph of concentration-time type. Therefore, it is more difficult to define the rate of reaction from the concentration-time graph. Hence, we integrate the differential rate equation to get a relation between the rate constant and the concentration at various points. This resultant equation is called the integrated rate equation. For different order reactions, we can notice different integrated rate equations.
Integrated Rate Law for a Zero-order Reaction
In a zero-order reaction, the rate of reaction completely depends upon the zeroth power of the concentration of reactants. The zero-order reactions are noticed very rarely. A few examples of zero-order reactions can be given as decomposition of gaseous ammonia on a hot platinum surface, thermal decomposition of HI on a gold surface, and more. A general equation for a zero-order reaction including the rate constant k is derived below.
A → B
Rate is given by = - \[\frac{d[A]}{d}\] = k[A]⁰
⇒ - \[\frac{d[A]}{d}\] = k
⇒ - d[A] = -k dt
Integrating on both sides, we get:
⇒ [A] = - kt + c - (1)
Where c is given as the constant of integration,
At time t=0, [A] = [A]₀
Substituting the limits in equation (1) we get the value of c as follows,
⇒ [A]₀ = c
Using the resultant value of c in the equation (1), we get as follows,
⇒ [A] = - kt + [A]₀
The above-derived equation is referred to as an integrated rate equation for the zero-order reactions. We can also observe the above equation as a straight line with the concentration of reactant on the y-axis and time on the x-axis. And, the slope of the straight line signifies the value of the rate constant, k.
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Integrated Rate Law for a First-order Reaction
In the first-order reaction, the rate of reaction depends on the first power of the reactant’s concentration. Artificial and Natural radioactive decay of the unstable nuclei is a few examples of the first-order reaction. A general equation for a first-order reaction including the rate constant k is derived below:
A → B
Rate is given by = - \[\frac{d[A]}{dt}\] = k[A]
⇒ \[\frac{d[A]}{[A]}\] = - k dt
Integrating on both sides:
⇒ ln[A] = - kt + c ----(2)
Where c is given as the constant of integration,
At time t=0, [A] = [A]₀
Substituting the limits in equation (2) we get the value of c, as given below.
⇒ ln[A]₀ = c
By using the value of c in the above equation we get,
⇒ ln[A] = - kt + ln [A]₀
We can notice that the above-derived equation can be plotted as a straight line including the ln[A] on the y-axis and time (t) on the x-axis. The negative slope of this straight line provides us with the value of the rate constant, k.
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We can also define the value of the rate constant, k from the above-given equation as:
ln \[\frac{[A]}{[A]_{0}}\] = -kt
⇒ k = - \[\frac{ln\frac{[A]}{[A]_{0}}}{t}\]
So, the concentration at any time moment can be given as,
[A] = [A]0\[^{e^{-kt}}\]
Hence, we can now define the concentration and the rate of reaction at any moment by the help of the integrated rate equation for zero and the first-order reaction.
Integrated Rate Law
The mathematical relationship of the reaction rate including reactant concentrations is referred to as the rate law. This relationship can rely more heavily on the concentration of one specific reactant, whereas, the resulting rate law can include either none, some, or all of the reactant species that are involved in the reaction.
Consider the following hypothetical reaction:
a A + b B → c C
For this, the rate law can be expressed as:
Rate = k[A]y[B]z
Here, ‘k’ is The proportionality constant, which is known as the rate constant and also specific for the reaction, represented at a specific temperature. And, the rate constant changes with the temperature, whereas, its units depends on the sum of the concentration term exponents present in the rate law. The exponents of y and z must be experimentally determined and they do not correspond require mentally to the coefficients in the balanced chemical equation.
Factors Affecting the Rate of Reaction
There are primarily 5 factors that affect the rate of reaction, which are listed as follows:
Temperature
Pressure
Presence of catalyst
Concentration of mixture
The surface area of mixture molecules
For a reaction to take place/occur, as per the Collision Theory, the collisions between the 2 molecules of the 2 various mixtures must possess a degree of energy, which is called the ACTIVATION ENERGY. Only when the energy reaches this threshold can new bonds be formed after their original bonds have been broken.
FAQs on Integrated Rate Equation
1. What is an integrated rate equation and why is it important in chemical kinetics?
An integrated rate equation is a mathematical expression that relates the concentration of a reactant to time. Unlike a differential rate law, which describes the instantaneous rate, the integrated rate equation allows us to predict the amount of reactant remaining or product formed after a specific period. Its importance lies in determining the rate constant (k) and the half-life of a reaction directly from experimental data.
2. What is the integrated rate equation for a zero-order reaction as per the CBSE syllabus?
For a zero-order reaction, the rate is independent of the reactant's concentration. The integrated rate equation is derived from the differential rate law and is given by:
[R] = -kt + [R]₀
Where:
- [R] is the concentration of the reactant at time 't'.
- [R]₀ is the initial concentration of the reactant.
- k is the rate constant.
3. What is the formula for the integrated rate equation of a first-order reaction?
For a first-order reaction, the rate is directly proportional to the concentration of one reactant. The integrated rate equation is expressed in two common forms as per the NCERT curriculum for the 2025-26 session:
- In natural logarithm form: ln[R] = -kt + ln[R]₀
- In base-10 logarithm form: k = (2.303/t) log([R]₀ / [R])
Here, [R] is the concentration at time t, [R]₀ is the initial concentration, and k is the rate constant.
4. How can you graphically determine the order of a reaction using integrated rate equations?
You can determine the reaction order by plotting concentration data against time and checking for a straight line. The type of plot that yields a straight line indicates the order:
- Zero-Order: A plot of concentration [R] vs. time (t) gives a straight line with a slope of -k.
- First-Order: A plot of the natural logarithm of concentration ln[R] vs. time (t) gives a straight line with a slope of -k.
This graphical method is a key technique used to analyse kinetic data experimentally.
5. What is the significance of the half-life (t₁/₂) in a reaction, and how does it differ for zero and first-order reactions?
The half-life (t₁/₂) of a reaction is the time required for the concentration of a reactant to reduce to half its initial value. Its significance lies in indicating the speed of a reaction. The key difference is its dependence on concentration:
- For a zero-order reaction, the half-life is directly proportional to the initial concentration (t₁/₂ = [R]₀ / 2k).
- For a first-order reaction, the half-life is constant and independent of the initial concentration (t₁/₂ = 0.693 / k). This is a defining characteristic, often seen in radioactive decay.
6. Can you provide an example of a zero-order reaction?
A classic example of a zero-order reaction is the decomposition of gaseous ammonia (NH₃) on a hot platinum surface. The chemical equation is 2NH₃(g) → N₂(g) + 3H₂(g). The rate of this reaction is independent of the ammonia concentration because the platinum surface gets saturated, making the surface area the limiting factor, not the reactant concentration. Thus, Rate = k[NH₃]⁰ = k.
7. What is a pseudo-first-order reaction and how does it relate to integrated rate equations?
A pseudo-first-order reaction is a bimolecular reaction that is made to behave like a first-order reaction. This occurs when one of the reactants is present in such a large excess that its concentration remains practically constant throughout the reaction. A common example is the hydrolysis of ethyl acetate: CH₃COOC₂H₅ + H₂O → CH₃COOH + C₂H₅OH. Here, water is in large excess, so the rate depends only on the concentration of ethyl acetate, allowing us to apply the first-order integrated rate equation for analysis.
8. What is the primary difference between a differential rate law and an integrated rate equation?
The primary difference lies in what they describe:
- A differential rate law (or rate expression) shows how the rate of a reaction depends on the instantaneous concentration of reactants. For example, Rate = k[A].
- An integrated rate equation shows how the concentration of reactants changes over a period of time. For example, ln[A] = -kt + ln[A]₀.
In essence, the differential form describes the rate at one moment, while the integrated form predicts concentration across a time interval.
