Integrated Rate Equation

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.