Differential and integral forms of zero , their characteristics and half-lives

Introduction

Order of reaction is defined as the sum of exponents of concentration in a rate law.It refers to the power dependence of the rate on the concentration of each reactant.

Zero-order reactions are typically found when a material necessary to proceed with the reaction, such as a surface or a catalyst, is saturated with the reactants.

The chemical mechanisms of concentrated acids and concentrated solutions are studied under a branch of chemistry called chemical kinetics. This branch deals with studying rates of chemical reactions and the factors affecting these rates. 

The sections below will give you an insight into the rates and order of a reaction in concentrated solutions with the detailed study of zero-order reactions.

What is the rate of a reaction?

The reaction rate in chemistry is determined by the change in the concentration of reactants and products in a unit time. This change in concentration taken over a while gives us the average rate of a reaction. Therefore, the rate of a reaction can only be positive. 

Unit of rate :

Rate = concentration/time

         =mol L-1 s-1

Factors that influence the rate of a reaction are:

Concentration and pressure in case of gaseous reactions

The more the concentration, the higher the rate of reaction.

Temperature increases the average kinetic energy of the concentrated solution particles. This means the rate of the reaction will increase with the increase in temperature.

Nature of reactants

The state and nature of the reactants play an essential role in determining the rate of a reaction. For example, if both the reactants are in a liquid state, the rate will be faster than that of a heterogeneous reaction. On the other hand, for a heterogeneous reaction, the rate depends on the surface area of the reactant in a more condensed state.

Presence of a catalyst

A catalyst helps initiate the reaction at a much lower temperature. This way, it provides an alternate path for the reaction to happen faster. 

What is the order of a reaction?

The order of a reaction defines the relationship between the rate and concentration of elements of a concentrated solution in a chemical reaction. It is given by the sum of powers of reactants in a rate law expression. 

Let us consider a general reaction:

 aA + bB → cC + dD

(where A and B are reactants and C and D are products and a, b, c, d represent the balancing factor)

The rate law expression will be:-

Rate = k [A]x [B]y

(where k is rate constant; x and y are partial order of reactions)

The order of this reaction will be given by x+y.

The order of a reaction can be a whole number like 0, 1, 2, 3, or even a fraction.

What is a zero-order reaction?

A zero-order reaction is when the concentration rate does not change throughout the reaction. For example, the rate of a concentrated solution reaction is equal to the rate constant of that particular reaction. 

Zero-order reactions are also called pseudo-zero-order reactions, carried out in specially designed conditions. 

What are the characteristics of a zero-order reaction?

The rate constant is directly proportional to the rate.

The concentration on the side of the reactant decreases linearly over time.

Completion = initial concentration/ rate constant

Differential form of zero order reaction

In a concentrated solution, the rate of a reaction depends on the concentration of reactants. The equation which represents this is termed the differential rate equation. 

In the case of a zero-order reaction, the reaction rate depends on the constant concentration of the reactants. Take, for example, a general reaction:

         A → B 

(A=reactant and B=product)

Rate of consumption of A = -d [A]/dT

         = k   (k is rate constant)

For a zero-order reaction, the unit of the rate constant is always M/s. In a higher-order reaction, the rate may have different units.Here ‘-’ sign indicates that the concentration of A is decreasing with time.

Integral form of a zero-order reaction

The tangent slope gives us the instantaneous rate of reaction at any instant of time in a concentration-time graph. It is difficult to determine the overall rate of the reaction from this graph. Hence, we integrate the differential equation to find the relationship between the rate constant and concentration. The integral equation will allow you to calculate the concentration of the reactant at any given time after the reaction has started.

The differential equation can be rearranged to find the integral equation. This integral equation is different for every order of the reaction.

Rate of consumption of A= -dA/dt

For a zero order reaction rate is directly proportional to initial concentration power 0.

-dA/dt = k 

d[A] = -kdt

now integrating both sides

∫[A0][A] d[A] = -∫0t kdt

( the integral A0 to A refers to the integral from an initial concentration of the reactant A at time t=0 to time t)

Solving further for A, we get

[A] = [A0] –kt

We can use the above derived integral equation to draw the graph of zero-order reaction. To draw a concentration-time graph, we can compare this equation to a straight line: y = mx + c.

Half-life of a zero-order reaction?

The half-life of concentrated acid in a chemical reaction can be defined as the specific amount of time in which the concentration of the reactant reaches fifty percent of its initial concentration. Half-life can be written as t1/2. The SI unit of half-life is seconds. Again, note that the formula of the half-life of a reaction varies with the order of a reaction.

Use the above derived integrated equation 

A = A0 – kt

Replace t with t1/2

=>   A1/2 = A0– kt1/2

=>   k t1/2 = ½ A0

=>   t1/2 = A0 / 2k

This shows that the half-life of a reaction can be determined by the rate constant and the initial concentration of the reactant. 

Conclusion

The kinetic principles are important to understand the basic mechanisms of a reaction. It helps you understand how a reaction works when more than just pressure and temperature influence the rate. The kinetics of any reaction depends on the reaction mechanism or rate law and the initial conditions. However, the kinetics rate of the zero system in chemistry always corresponds to the constant rate in all concentrations. This article has shown how the lives and differentials can be determined per unit time.