Integrated Rate Equations

The differential rate equation is an equation that represents the reaction rate’s dependence on the concentration of reacting species. The tangent slope at any point in time in a graph of concentration-time type can be used to express the instantaneous rate of reaction. As a result, determining the reaction rate from a concentration-time graph is more complicated. 

The differential rate equation is integrated to obtain a relationship between the concentration at different points and the rate constant. This equation is an  integrated rate equation. Other integrated rate equations are seen for reactions of different orders. 

Factors influencing the rate of reaction

The rate of reaction is primarily influenced by five factors:

• Temperature
• Pressure
• Catalyst presence
• Mixture concentration
• The surface area of mixture molecules

 According to Collision Theory, for a reaction to take place/occur, the collisions between the two molecules of the two different mixtures must have a degree of energy, known as the ‘Activation energy’. After the original bonds have been broken, new bonds can only be formed when the energy reaches this threshold.

The integrated rate equation for the zero-order reaction

The Integrated rate equation for the zero-order reaction is determined by the concentration of reactants to the zeroth power. Zero-order reactions are incredibly uncommon. Examples are:

• The thermal decomposition of HI on a gold surface.
• Gaseous ammonia on a hot platinum surface.

The following is a general equation for a zero-order reaction with constant rate k:

When both sides are integrated, we get:

⇒ A = – kt + c …………………..(1)

Where c denotes the integration constant,

At time t=0, A = A₀

By substituting the limits in equation (1), we obtain the following value for c:

⇒ A₀ = c 

We get the following when we use the resultant value of c in equation (1):

⇒ A = – kt + A₀  

For zero-order reactions, the above-derived equation is an integrated rate equation. The above equation can also be visualized as a straight line with the reactant concentration on the y-axis and time on the x-axis. The rate constant, k, is represented by the slope of the straight line. 

The integrated rate equation for a first-order reaction

The integrated rate equation for a first-order reaction is determined by the first power of the reactant’s concentration. The first-order reaction is exemplified by unstable nuclei’s artificial and natural radioactive decay. The following is a general equation for a first-order reaction that includes the rate constant k:

A → B

 On both sides, integrating:

⇒ ln A = – kt + c —-(2)

 Where c denotes the integration constant,

At time t=0, A = A₀

 We get the value of c by substituting the limits in equation (2), as shown below.

⇒ ln A₀ = c

 We get by plugging in the value of c into the above equation:

⇒ ln A = – kt + ln A₀ 

We can see that the above-derived equation, with the ln A on the y-axis and time (t) on the x-axis, can be plotted as a straight line. The rate constant, k, is determined by the negative slope of this straight line.

At any given time, the concentration and rate of reaction can be determined with the help of the integrated rate equations for zero and first-order reactions.

 Integrated rate law equation

The integrated rate law equation is a mathematical relationship between the reaction rate and the reactant concentrations. This relationship may depend more on the concentration of a single reactant, whereas the rate law results may include none, some, or all of the reactant species involved in the reaction.

Consider this hypothetical reaction:

A + b B → c C 

The rate law can be further expressed as follows:

Rate = k [A]y [B]z 

The proportionality constant, also known as the rate constant, is also specific for the reaction and is represented at a particular temperature in this case. While the rate constant varies with temperature, the units of the rate law are determined by the sum of the concentration term exponents. The exponents of y and z must be determined experimentally, as they do not match the balanced chemical equation’s coefficients. 

Conclusion : 

Integrated rate equations are used to find various information about a chemical reaction

Such as rate constant , time of completion of reaction , half life time of reaction etc.

It also gives us details of reaction in progress as how much of the reaction has progressed at a certain time and to estimate the concentrations of reactants consumed in particular time