Rate Constant and its Units

What is the rate constant?

 The constant that relates the rate of a chemical reaction at a given temperature to the concentration of reactant (in a unimolecular reaction) or the product of the concentrations of reactants is termed as rate constant. The proportionality constant in the rate law expression is known as the rate constant of the reaction. 

Let us consider some general expressions of an equation: 

Aa+bB → cC+dD 

Here a,b,c, d are the stoichiometric coefficients of the corresponding reactants and products. As per the study of chemical kinetics, we can write the rate expression of the reaction as follows: 

Rate proportional to [ A ]x[B]y

Here the x and y are the corresponding constants that assume some real value that may be both negative and 0. 

We know that to remove the proportionality sign, we need to add a constant k. Now, this equation becomes: 

Rate=k* [ A ]x[B]y

This k in the language of chemical kinetics is known as the rate constant. The value of the rate constant and its unit depends on the type of reaction. 

Note: Although the formula is derived for that reaction only, this is generally valid. 

Key features of the rate constant: 

➢ It is the measure of the reaction rate—the greater the value of k, the greater the reaction rate and vice versa. 

➢ The value of k depends on the type of reaction. 

➢ If the reaction temperature remains constant, then the value of k also remains the same, considering that the reaction takes place with the same mechanism. This indicates that temperature has a huge impact on chemical reactivity.

➢ The rate constant is independent of the concentration for any particular reaction.

Different types of rate constants and their units: 

Zero-order reaction: 

Consider a typical zero-order reaction: 

A → B (Here A is the reactant and B is the product of the reaction) 

Let x be the concentration of reactants consumed at time t. 

Rate = −dA/dt=k[A]0

d[A]=-k*dt 

Now, after integrating both sides of the reaction, we get: 

[A]=-k*t+C 

Putting t=0 and [A]=[A]0 in the equation we have:

[A]0=C

Further substituting the value in the integrated equation, we have 

[A]=-K*t+[A]0

K=1/t{[A]-[A]0}

This is the required expression of the rate constant of the zero-order reaction. 

Units of k: from the equation, we can write:

K=molar concentration/time taken

K=mol L-1/time taken

K=mol L-1 t-1

First-order reaction: 

Consider the first-order reaction: 

                           A →   B (Here A is the reactant and B is the product of the reaction)

At t=0,               a        0 

At t=t,             (a-x)     x 

Let x be the concentration of reactants consumed at time t; a be the initial concentration. 

We can rearrange this equation to combine our variables, and integrate both sides to get our integrated rate law: 

dx/dt=k(a-x) 

Now we integrate both side: 

dx/(a-x)=k*dt

-ln(a-x) =k*t+c 

Here c is some integration constant. 

Putting the value t=0 and x=0 in the above equation we have: 

C=-ln(a) 

Putting the value of c in the integrated rate equation we have: 

ln(a/(a-x)) =kt 

So, k=2.303/t * log(a/(a-x)) 

This is the required integrated equation of the first-order reaction. 

Unit of rate constant: 

From the reaction, we can see that the rate constant here does not depend upon the concentration; instead, it depends only on time (inversely proportional). So, the unit of k is s-1. 

Second-order reaction: 

Let us consider a typical second-order reaction: 

                  2A → B (Here A is the reactant and B is the product of the reaction) 

At t=0,        a      0 

At t=t,    (a-x)    x 

Let x be the concentration of reactants consumed at time t; a be the initial concentration. 

rate=−d[A]/dt=k[A]2

On Integrating both side we get:

1/(a-x) =k*t+c 

Where c is some integration constant. 

Now putting the value t=0 and x=0 in the given reaction we have: 

C=1/a 

Now substituting the values in the integrated rate expression, we have: 

1/(a-x) =kt+1/a 

K*t=1/(a-x)-1/a 

k=1/t*x/a(a-x) 

This is the required rate constant expression of the reaction. 

Units: from the equation, it can be seen that:

Unit of k=1/time taken * concentration

Unit of k=1/t*mol L-1

Unit of k=mol -1 L t-1

General method /Alternative method to find the units of the rate constant: 

Let us consider any nth order reaction. 

 n*A → B (Here A is the reactant and B is the product of the reaction) 

Now we can write rate=k[A]n

K=rate/[A]n

So, the dimension of constant rate k can be written as: 

K=concentration of the reactant/time consuming 

K=[concentration]1-n[time]-1 

Where n is the order of the reaction. 

Now taking the SI unit of the corresponding measurements, we have: 

K= [mol L-1]1-n [s]-1

Putting the values n=0,1,2, etc., we can easily find the units of the rate constants of the corresponding reactions.

Conclusion: 

The rate constant represents the relationship between the chemical reaction rate and the concentrations of reacting substances. Its value depends on the nature of the reaction. Besides, temperature plays a vital role in the constant rate value. The calculation of the rate of any chemical reaction is very much essential to determine whether a given chemical reaction is feasible or not.

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