Question

For the reaction \(A \longrightarrow 2 B+C\), the following data are obtained for \([\mathrm{A}]\) as a function of time: \(t=0 \mathrm{min}\) \([\mathrm{A}]=0.80 \mathrm{M} ; 8 \mathrm{min}, 0.60 \mathrm{M} ; 24 \mathrm{min}, 0.35 \mathrm{M} ; 40 \mathrm{min}\) \(0.20 \mathrm{M}\) (a) By suitable means, establish the order of the reaction. (b) What is the value of the rate constant, \(k ?\) (c) Calculate the rate of formation of \(\mathrm{B}\) at \(t=30 \mathrm{min}\).

Short answer

By plotting the \(ln[A]\) against time, a straight line indicates first order reaction. The absolute value of the slope of the line provides the rate constant \(k\). Then use the rate equation for the formation of \(B\), \(Rate = 2k[A]\) to find the rate at \(t=30 \mathrm{min}\).

Step-by-step solution

Establish the order of the reaction

In the given data, the concentration of \(A\) is reducing over time, so the reaction order can be determined by plotting the log of concentration of \(A\) against time (a linear relationship suggests a first order reaction). A semi-log plot can be drawn, plotting the natural logarithm (ln) of concentration of \(A\) against time \(t\). If the plot yields a straight line, the reaction is first-order.

Determine the rate constant, \(k\)

The slope of the semi-log plot \(ln[A]\) vs \(t\) is equal to \(-k\) (negative because it is a reactant). Calculate the slope of the line and the rate constant \(k\) can be determined by taking absolute value of the slope.

Calculate rate of formation of \(B\)

For a first-order reaction like \(A \longrightarrow 2 B+C\), the rate of formation of \(B\) is twice the rate constant times the concentration of \(A\). So at \(t=30 \mathrm{min}\), interpolate the concentration of \(A\) from the given data, multiply it by twice the rate constant to get the rate of formation of \(B\) at \(t=30 \mathrm{min}\).

Key concepts

Reaction Order

The reaction order is an essential concept in reaction kinetics that helps us understand how the concentration of reactants influences the rate of a chemical reaction. To determine the order of a reaction, we need to observe how the concentration of the reactant changes over time. If the concentration of the reactant and time data produce a straight line when plotted using a logarithmic scale, we can confirm that the reaction is of first-order.

In this exercise, we have a reaction involving a reactant \( A \) gradually converting to products over time. By taking the natural logarithm of the concentration of \( A \) and plotting these values against time, a linear relationship indicates a first-order reaction.

• A first-order reaction means that the rate of reaction is directly proportional to the concentration of the reactant.
• For first-order reactions, the half-life is independent of the initial concentration of the reactant.
• The slope of the plot on a semi-log scale provides further insight into the reaction kinetics.

Rate Constant

The rate constant, \( k \), is a crucial parameter that provides insights into the speed of a chemical reaction. It is a unique value for every reaction at a given temperature and can be influenced by factors like temperature and catalyst presence. For a first-order reaction, the rate constant is determined from the slope of the natural log of the concentration of the reactant plotted against time.

In the example given, \( k\) is calculated from the slope of the line obtained in the semi-logarithmic plot of \( \ln[A] \) versus time. Since the line is straight, indicating a first-order reaction, the slope of the line is equal to \( -k \). Taking the absolute value of the slope gives us the rate constant.

• The rate constant provides information on how fast or slow a reaction proceeds.
• For first-order reactions, the unit of \( k \) is \( \text{time}^{-1} \).
• The value of \( k \) remains constant as long as the conditions (such as temperature) remain unchanged.

Rate of Formation

The rate of formation is an important concept that tells us how quickly products are generated in a chemical reaction. Specifically, for the formation of a product like \( B \) in a reaction, it is crucial to understand how it relates to the reactant consumption.

In a reaction such as \( A \rightarrow 2B + C \), the appearance of \( B \) happens twice as fast as the disappearance of \( A \), since two moles of \( B \) are formed for every mole of \( A \) that reacts. To calculate the formation rate of \( B \), we need to know the concentration of \( A \) at the time in question and the rate constant. The rate of formation at any given time, such as 30 minutes in this case, can be calculated with the formula: Rate = \( 2k[A] \).

• The rate of formation tells us the speed at which a specific product is produced.
• For first-order reactions, the rate of formation can be easily estimated once \( k \) and concentration of the reactant are known.
• The factor of "2" in this context depends on the stoichiometry of the reaction, which shows how many units of \( B \) are formed per unit of \( A \) reacted.