Rate Law

The chemical response is the process of conversion of one or further reactants to one or further products. In the chemical response, we need to consider the response rate, the medium, and the equilibrium in which it’s progressed. Rate law plays a major part in chemical responses.

The speed at which reactants are converted into products is called the rate of the responses. The factors that affect the response rate are the type of response, attention, pressure, temperature, the face area of the reactants, catalyst, and light intensity. In any chemical reaction, the rate law is an important property that shows how the reaction rate depends on the concentration of the reactants.

Rate law- Expression, Integrated rate law, and Rate Constant

The relationship between the response rate of the chemical response and the attention of one or further reactants is called “Rate law”. It’s also called a rate equation.

The rate law for the general reaction

 aA + bB → cC

is denoted by the following expression,

rate = k [A]x [B]y

In the above expression,

[A], [B] – molar attention of reactants A, B

k- rate constant of the response

 x, y- orders of responses with respect to a and b.

 We can determine the expression of the rate law for a specific response experimentally only. We can not use the balanced chemical equation to gain rate law expression since the partial orders of the reactants aren’t inescapably equal to the stoichiometric portions.

The Half-life of response rates

 Response rates are grounded on time needed for the attention of a reactant to drop to one-half its original value. It’s called the half-life of the response, written as t1/ 2. Therefore the half-life of a response is the time needed for the reactant’s attention to drop from [A]0 to [A]0/2.

Still, the brisk response will have a shorter half-life, and the slower response will have a longer half-life, If two same order responses.

Integrated Rate Law

There is the second form of rate law that relates to the concentration of reactants and time. These are called integrated rate laws. This can be used to determine the amount of reactant or product present after the time. Reactions that have different orders have different integrated rate equations.

For example, we can take the time taken for the radioactive decay of radioactive material.

Now, we perform the integrated rate law calculations on zero, first, and second-order reactions.

Zero-order reactions

For zero-order reactions, the rate is equal to k. So, the integrated rate law of zero-order reactions is a linear function.

Rate = k

The integrated rate law for a zero-order reaction when n=1,

ln[𝐴] = −𝑘𝑡 + ln[𝐴]0

In the above equation, [𝐴]0 is the initial concentration of the reactant 

and [𝐴] is the concentration attained after a time t.

So, y = mx + b

where, 𝑦 = [𝐴] ;𝑚 = −𝑘 ; 𝑥 = 𝑡 ; 𝑏 = [𝐴]0

First Order reactions

The first-order reaction is represented  as

[A]t = [A]0 e-kt

If we take 𝑡0 = 0, then

𝐥[𝑨] = −𝒌𝒕 + 𝐥𝐧[𝑨]0

which looks like y = mx + by

where,  𝑦 = ln[𝐴] ;𝑚 = −𝑘 ; 𝑥 = 𝑡 ; 𝑏 = ln[𝐴]0

Under a given set of reaction conditions, the half-life of a first-order reaction is constant and is independent of the concentration of the reactants.

Second-Order reactions

The second-order reaction can be represented as

𝟏/ [𝑨] = 𝒌𝒕 + 𝟏/ [𝑨]𝟎

Which, again, looks like: 

y = mx + by

where, 𝑦 = 1/[𝐴] ;𝑚 = 𝑘 ; 𝑥 = 𝑡 ; 𝑏 = 1/[𝐴]0

The rate constant for a second-order reaction depends on the initial concentration of a reactant.

Reaction order

The order of the reaction is the exponents in the rate law that describes the concentration effects of the reactants in the reaction rate. 

There are reaction orders of zero-order, first-order, second-order, and so on.

For example,

In this reaction 2NO(g) +2H2(g) → 2N2(g) +2H2O(g) , the rate law can be written as

Rate of the reaction = k  [NO]2[H2]

Here, [NO], [H2] is the molar concentration of the reactants NO, H2

k is the rate constant 

           x – order of NO 

           y – order of H2

The reaction order of NO is 2

The reaction order of H2 is 1

The overall order of the reaction is 3.

The reaction order provides an understanding of the change in the rate of the reaction that is expected with the increase in the concentration of the reactants.

 For example, we are considering the various orders of reaction with its reactant concentration like

• If the reaction is a zero-order reaction, we will have no effect on the reaction rate when we double the reactant concentration.
• If the reaction is of the first order,  the reaction rate will be doubled when the reactant concentration is doubled.
• In the second-order reactions, the reaction rate will be quadrupled when the reactant concentration is doubled
• In the third-order reactions, when the reactant concentration is doubled, the overall rate increases by eight times 

Rate Constant 

The rate constant can be expressed as

                 K = Rate / [A]x [B]y

The units of a rate constant for a particular rate law can be calculated by dividing the units of rate by the units of molarity in the concentration term of the rate law. 

The unit of k = mol.L-1  or M.

k can be calculated using the formula

k = (M.s-1)*(M-n) = M(1-n).s-1

You can find the units of the rate constants for zero, first, second, and nth-order reactions in the following tabular column:

Rate laws may exhibit fractional orders for some reactants, and when an increase in the concentration of one reactant, negative reaction orders are observed, causing a decrease in reaction rate.

Difference between the differential rate law and integrated rate law

Conclusion 

The rate law of a chemical reaction gives the relationship between the concentrations of reactants and their reaction rate. It is the mathematical description of how the rate of the reaction is affected by the changes in the amount of the substance. The rate law is determined by experiment only and is not reliably predicted by the stoichiometric reaction.