The decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) takes place according to I order as $$ 2 \mathrm{~N}_{2} \mathrm{O}_{5} \longrightarrow 4 \mathrm{NO}_{2}+\mathrm{O}_{2} $$ Calculate : (a) The rate constant, if instantaneous rate is \(1.4 \times 10^{-6}\) mol litre \(^{-1}\) \(\mathrm{sec}^{-1}\) when concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is \(0.04 \mathrm{M}\) (b) The rate of reaction when concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is \(1.20 \mathrm{M}\) (c) The concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) when the rate of reaction will be \(2.45 \times 10^{-5} \mathrm{~mol}\) litre \(^{-1} \mathrm{sec}^{-\mathrm{i}}\).

Understanding the rate constant in chemical reactions is crucial for grasping reaction kinetics. The rate constant, symbolized as 'k', is a proportionality factor that links the reaction rate to the concentrations of reactants for a reaction following a given order, in this case, first order.

For first-order reactions, the rate law is expressed as \[\text{rate} = k[\text{Concentration}]\], where '[Concentration]' refers to the molar concentration of the reactant. The units of k vary depending on the reaction order; for first-order reactions, it's typically s\(^{-1}\).

Calculation of the rate constant involves rearranging the rate law to solve for 'k' as follows: \[k = \frac{\text{rate}}{[\text{Concentration}]}\] Applying this to our given problem, given an instantaneous rate of \(1.4 \times 10^{-6}\) mol litre\(^{-1}\) sec\(^{-1}\) and a concentration of \(0.04\) M, the rate constant 'k' can be computed, allowing for further analyses and predictions about the reaction under different conditions.

Reaction kinetics is the study of the rates of chemical processes and the factors that affect these rates. It provides an understanding of how different variables such as concentration, temperature, and presence of catalysts impact the speed of chemical reactions.

In first order reactions, where the rate of reaction is directly proportional to the concentration of one reactant, the kinetics are relatively straightforward. The time it takes for a reactant concentration to decrease by half, known as the half-life, is constant and independent of its initial concentration. This characteristic feature is a key concept in studying the behavior of first order reactions.

In our exercise, knowing the reaction kinetics allows us to calculate different variables, such as the rate constant as well as predicting reaction rates at different concentrations, which is an application of the principles of reaction kinetics crucial for scientists and engineers in various fields.

The concentration of reactants is a fundamental factor that influences the rate of chemical reactions. In a first order reaction, the rate is directly proportional to the concentration of a single reactant. This relationship implies that if we double the concentration of the reactant, the rate of the reaction will also double.

To illustrate this concept with our exercise, if the concentration of \(\mathrm{N}_2\mathrm{O}_5\) is increased from \(0.04\) M to \(1.20\) M, we can predict that the reaction rate will increase, given that the rate constant remains the same. Conversely, if we know the rate of reaction and the rate constant, we can determine the concentration of the reactant needed to achieve that rate.

This quantitative relationship is encapsulated by the equation \[\text{Rate} = k[\text{Concentration}]\], and allows us to solve various problems regarding how much reactant is required to reach a certain rate or, inversely, the rate that a given concentration of reactant will achieve. It is a fundamental aspect of predicting and controlling reaction outcomes in practical applications such as pharmaceuticals, environmental engineering, and material science.