Problem 13 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The condition at which average rate can be equal to instantaneous rate of the reaction is (a) \(\Delta n=0\) (b) \(\Delta t \rightarrow 0\) (c) reaction is elementary (d) reaction is complex

Step-by-Step Solution

Answer

(b) \(\Delta t \rightarrow 0\) is the condition at which the average rate can be equal to the instantaneous rate of the reaction.

Step 1: Understanding Average Rate

The average rate of a reaction over a time period is the change in concentration of a reactant or product divided by the change in time. It represents the overall rate of change over the given time interval while the reaction proceeds.

Step 2: Understanding Instantaneous Rate

The instantaneous rate of a reaction is the rate at a particular moment in time. It can be thought of as the derivative of concentration with respect to time, and therefore, it is the limit of the average rate as the change in time approaches zero.

Step 3: Analyzing the Options

Option (a) suggesting \(\Delta n=0\) implies no change in the amount of substance, which does not guarantee equal average and instantaneous rates. Option (c) and (d) refer to the nature of the reaction (elementary or complex) but do not directly relate to the condition for equal rates. Option (b) \(\Delta t \rightarrow 0\) is the correct answer, as it suggests taking an infinitesimally small interval of time, making the average rate equal to the instantaneous rate.

Step 4: Conclusion

The only condition where the average rate of the reaction can be equal to the instantaneous rate is when the time interval considered for the average rate calculation is infinitesimally small, which makes (b) \(\Delta t \rightarrow 0\) the correct answer.

Key Concepts

Chemical Kinetics

Chemical kinetics is a branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It involves the study of how reaction rates are affected by various factors such as the chemical nature of the reactants, the concentration of reactants, temperature, and the presence of catalysts.

For students tackling problems in chemical kinetics, one important aspect is to understand the distinction between the average rate of a reaction and the instantaneous rate of a reaction. The average rate gives you a broad overview of the reaction's speed over a time interval, while the instantaneous rate is much more specific, telling you the reaction speed at a particular moment.

Reaction Rate

The reaction rate is a measure of the speed at which reactants are converted into products in a chemical reaction. Typically, it is expressed in terms of the change in concentration of a reactant or product per unit time. A higher reaction rate means that the concentration of a substance is changing more quickly over time.

In the context of homework problems, it is useful to consider that the reaction rate can be affected by factors such as the nature of the reactants, the surface area of solid reactants, and the temperature at which the reaction is carried out. It is also affected by the presence of a catalyst, a substance that speeds up a reaction without being used up itself. When evaluating problems, think about how these factors might be influencing the reaction rate you are asked to calculate or predict.

Instantaneous Rate of Reaction

In contrast to the average rate, the instantaneous rate of reaction refers to the rate at a specific moment in time. It is the speed of the reaction at a precise instant, similar to how an instantaneous speed of a car tells you its speed at a particular moment, without regard for its earlier or later speed.

For mathematical clarity, the instantaneous rate of a reaction is defined as the derivative of the concentration of a reactant with respect to time. In layman's terms, this means that it is the rate of reaction at an infinitesimally small time interval. One helpful tip in understanding this concept is visualizing a graph of concentration versus time; the slope of the tangent line at any point on the curve gives the instantaneous rate at that moment. This measurement is essential for scientists and engineers designing reactors and for predicting how changes in conditions might affect the rate of a reaction.