Question

The decomposition of \(\mathrm{XY}\) is second order in \(\mathrm{XY}\) and has a rate constant of \(7.02 \times 10^{-3} \mathrm{M}^{-1} \cdot \mathrm{s}^{-1}\) at a certain temperature. a. What is the half-life for this reaction at an initial concentra- tion of \(0.100 \mathrm{M} ?\) b. How long will it take for the concentration of XY to decrease to \(12.5 \%\) of its initial concentration when the ini- tial concentration is \(0.100 \mathrm{M}\) ? When the initial concentra- tion is \(0.200 \mathrm{M} ?\) c. If the initial concentration of \(\mathrm{XY}\) is \(0.150 \mathrm{M}\), how long will it take for the concentration to decrease to \(0.062 \mathrm{M} ?\) d. If the initial concentration of \(\mathrm{XY}\) is \(0.050 \mathrm{M},\) what is the concentration of XY after \(5.0 \times 10^{1}\) s? After \(5.50 \times 10^{2}\) s?

Short answer

a. The half-life for the reaction at 0.100 M is approximately 142.7 seconds. b. The time to decrease to 12.5% for initial concentrations of 0.100 M and 0.200 M need to be calculated using the second-order integrated rate law. c. The time to decrease to 0.062 M from 0.150 M also requires use of the second-order integrated rate law with the respective concentrations. d. Concentrations after 50 s and 550 s must be found using the same law with initial concentration 0.050 M and respective times.

Step-by-step solution

Understanding second-order kinetics

For second-order reactions, where the rate of decay is proportional to the square of the concentration of a reactant, the rate law can be written as rate = k[XY]^2, and the half-life formula for a second-order reaction is given by t_(1/2) = 1 / (k[XY]_0), where k is the rate constant and [XY]_0 is the initial concentration of XY.

Calculating the half-life (Part a)

Substitute the given rate constant, k = 7.02 × 10^{-3} M^{-1}s^{-1}, and initial concentration, [XY]_0 = 0.100 M, into the half-life formula to find the half-life for the reaction. t_(1/2) = 1 / (k[XY]_0) = 1 / (7.02 × 10^{-3} M^{-1}s^{-1} × 0.100 M) = 1.427 M s / (7.02 × 10^{-3} M^{-1}s^{-1}).

Calculating the time for concentration to decrease to 12.5% for 0.100 M [XY]_0 (Part b)

For a second-order reaction, the relationship between concentration and time can be given by 1/[XY] = kt + 1/[XY]_0. When the concentration is at 12.5% of the initial, [XY] = 0.125 × [XY]_0. We can solve for time, t, using the initial concentration [XY]_0 = 0.100 M.

Calculating the time for concentration to decrease to 12.5% for 0.200 M [XY]_0 (Part b)

Repeat the process from Step 3 with [XY]_0 = 0.200 M to find the time it takes for the concentration to decrease to 12.5% of its initial value.

Calculating the time for the concentration to decrease to 0.062 M (Part c)

Using the same equation from Step 3, solve for time t when the initial concentration [XY]_0 is 0.150 M and the final concentration [XY] is 0.062 M.

Finding the concentration after given times for 0.050 M [XY]_0 (Part d)

For the given times, plug in the rate constant and the time into the second-order integrated rate law to find the concentrations after 50 s and 550 s when the initial concentration is 0.050 M.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of the rates at which chemical processes occur and the factors that affect these rates. It's fundamental in understanding how reactions happen and how to control them. In the context of the exercise, we deal with a second-order reaction, which implies that the reaction rate depends on the concentration of one reactant raised to the second power or two reactants, each raised to the first power.

For second-order reactions, the rate law can be expressed as rate = k[XY]^2, where k is the reaction rate constant and [XY] is the concentration of the reactant. Unlike first-order reactions, which have a constant half-life, the half-life of a second-order reaction depends on the initial concentration of the reactant as well as the rate constant. This relationship must be comprehended to effectively apply the half-life formula and understand how the reaction progresses over time.

In chemical kinetics, understanding the factors that influence reaction rates is also crucial. These can include the physical state of the reactants, the concentration of reactants, the temperature at which the reaction occurs, and the presence of catalysts. By manipulating these variables, chemists can control the speed of chemical reactions, which is essential for various applications in industry and research.

Half-Life Calculation

The half-life of a reaction is the time it takes for the concentration of a reactant to decrease to half its initial value. In our exercise, we focus on the half-life of a second-order reaction. While first-order reactions have a constant half-life, irrespective of the reactant's concentration, the half-life for second-order reactions varies with initial concentration.

The formula used to calculate the half-life of a second-order reaction is given by:

\[ t_{1/2} = \frac{1}{{k[XY]_0}} \]

In this formula, \( t_{1/2} \) represents the half-life, \( k \) is the reaction rate constant, and \( [XY]_0 \) is the initial concentration of the reactant. For a given reaction, if the initial concentration is known along with the rate constant, the half-life can be calculated. This measurement is instrumental for predicting how long it will take for a reactant to reach a certain concentration and is essential for processes that rely on precise timing, such as drug delivery and industrial synthesis.

Reaction Rate Constant

The reaction rate constant, denoted by \( k \), is a numerical value that represents the speed of a chemical reaction. For a second-order reaction, the units of \( k \) are typically \( M^{-1}s^{-1} \), which indicates that the reaction rate constantfactor is based on the concentration of the reactants in molarity (M) and the time in seconds (s).

The rate constant is determined experimentally and is a vital component in the rate laws that relate the reaction rate to the concentration of reactants. In our exercise, the given rate constant is \( 7.02 \times 10^{-3} M^{-1}s^{-1} \). This constant allows us to calculate not only the half-life but also how the concentration of a reactant will change over time in a second-order reaction.

It's important to note that the rate constant is dependent on temperature and can change with the presence of a catalyst. The Arrhenius equation is often used to describe how the rate constant varies with temperature. Understanding the rate constant not only helps in predicting reaction behavior but also provides insights into the mechanism of the reaction and its energy profile, which are important for designing new chemical processes and understanding existing ones.