Problem 20 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

After \(20 \%\) completion, the rate of reaction: \(\mathrm{A} \rightarrow\) products, is 10 unit and after \(80 \%\) completion, the rate is \(0.625\) unit. The order of the reaction is (a) zero (b) first (c) second (d) third

Step-by-Step Solution

Answer

The order of the reaction is second (option c).

Step 1: Understanding the Given Information

The reaction rate for a certain chemical reaction, \(\mathrm{A} \rightarrow\) products, is given for two different percentages of completion: 10 units at 20\(\%\) completion and 0.625 units at 80\(\%\) completion. We need to relate the rate of reaction to the concentration of reactant A to find the order of the reaction.

Step 2: Relating Rate to Concentration

The rate of a reaction can generally be related to the concentration of the reactants by the rate law expression: \(rate = k[\mathrm{A}]^n\), where \(k\) is the rate constant, \(\mathrm{A}\) is the concentration of reactant A, and \(n\) is the order of the reaction. We will use this equation to find the value of \(n\).

Step 3: Applying the Rate Law to Given Rates

Let's assume the initial concentration of A is \(\mathrm{A_0}\). At 20\(\%\) completion, the remaining concentration is \(0.8\mathrm{A_0}\) and the rate is 10 units. At 80\(\%\) completion, the remaining concentration is \(0.2\mathrm{A_0}\) and the rate is 0.625 units. Therefore, we have two rate equations based on the rate law: \begin{align*} rate_1 &= k(0.8\mathrm{A_0})^n = 10 \ rate_2 &= k(0.2\mathrm{A_0})^n = 0.625 \end{align*} Dividing rate_1 by rate_2 gives \begin{align*} \frac{10}{0.625} &= \left(\frac{0.8}{0.2}\right)^n \end{align*} Simplifying this, we have \( 16 = (4)^n \).

Step 4: Calculating the Order of Reaction

To find the value of \(n\), we need to solve the equation \(16 = 4^n\). Since \(4^2 = 16\), then \(n = 2\). Therefore, the order of the reaction is second.

Key Concepts

Chemical Kinetics

Chemical kinetics is a fascinating area of chemistry that focuses on understanding the speed or rate at which chemical reactions occur. The rate of a reaction is influenced by factors such as temperature, pressure, concentration of reactants, and the presence of a catalyst.

It's crucial to familiarize oneself with key terms, for example, the concept of the reaction rate, which tells us how quickly reactants are converted into products over time. By investigating the behavior of reactants under different conditions, chemists can develop strategies to control the speeds of reactions — something that's incredibly important in industries like pharmaceuticals and environmental engineering.

To analyze reaction rates, chemists often graph the concentration of reactants over time. A steeper slope indicates a faster reaction, highlighting that even the graphical interpretation of data is important in the study of chemical kinetics.

Rate Law Expression

The rate law expression is central to the study of reaction kinetics. It mathematically relates the rate of a reaction to the concentration of its reactants. In its simplest form, the rate law can be written as:
\[ rate = k[\mathrm{A}]^n \]
Here, \( k \) is the reaction rate constant, which is specific to each chemical reaction at a given temperature. The symbol \( [\mathrm{A}] \) signifies the concentration of reactant A, and \( n \) is the order of the reaction with respect to that reactant.

The order of a reaction can be zero, first, second, third, or even fractional, and it determines how the rate varies with concentration. Determining this order is like a detective’s task, piecing together clues from experiments to reveal how the concentration affects the reaction's pace.

Reaction Rates

Reaction rates can be somewhat deceptive; they seem straightforward but contain complex underlying principles. We measure these rates as the change in concentration of a reactant or product per unit time. In simpler terms, it's like clocking how fast your car goes but in chemical reactions, we’re tracking how quickly reactants disappear or products form.

Determining Reaction Rates

Calculating reaction rates often involves measurements taken at different times. In laboratories, these rates are sometimes observed through color changes, precipitate formation, or volume of gas released.

Understanding how reaction rates change with different concentrations leads to the manipulation of these rates in practical applications. For instance, pharmaceutical companies control reaction rates to synthesize drugs efficiently, while environmental agencies might use them to manage pollutant degradation.

Concentration of Reactants

The concentration of reactants is a pivotal factor in determining how fast a reaction will proceed. Imagine you are attending a crowded concert; the likelihood of you bumping into someone is high due to the high 'concentration' of people. Similarly, in a chemical reaction, a higher concentration of reactants means the reacting particles collide more often, leading to an increased rate of reaction.

In a quantitative manner, concentration is usually expressed in moles per liter (M) in a solution. Subtle changes in concentration can greatly affect the rate of reaction, reinforcing the need for accurate measurements and control when conducting experiments in the lab. This sensitivity also explains why reactions can suddenly speed up or slow down as reactants are consumed or products accumulate.