Problem 98
Question
For a reaction involving the decomposition of a hypothetical substance \(\mathrm{Y}\), these data are obtained: Rate \([\mathrm{Y}]\) \(\begin{array}{lllll}\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{~min}^{-1}\right) & 0.288 & 0.245 & 0.202 & 0.158 \\ & 0.200 & 0.170 & 0.140 & 0.110\end{array}\) (a) Determine the order of the reaction. (b) Write the rate law for the decomposition of \(Y\). (c) Calculate \(k\) for the experiment above.
Step-by-Step Solution
Verified Answer
First-order reaction with rate law: \\( ext{Rate} = k[Y] \\). Rate constant \\( k = 0.694 \: ext{min}^{-1} \\).
1Step 1: Understand the Data Relationships
We are given rate data dependent on concentrations \([Y]\). The task is to determine the relationship between these rates and the concentrations. To determine the order of the reaction, observe how the rate changes as \([Y]\) changes.
2Step 2: Determine the Reaction Order
The reaction order can be determined by examining how changes in concentration affect the rate. Assume \text{Rate} = k[Y]^n\. Calculate \(n\) using different pairs: let the first pair represent \( [Y]_1 = 0.288 \: ext{mol L}^{-1} \, ext{Rate}_1 = 0.200 \: ext{mol L}^{-1}\text{min}^{-1}\) and the second pair \( [Y]_2 = 0.245 \: ext{mol L}^{-1} \, ext{Rate}_2 = 0.170 \: ext{mol L}^{-1}\text{min}^{-1}\).
Key Concepts
Reaction OrderRate LawRate Constant
Reaction Order
In reaction kinetics, the reaction order tells us how the rate of a chemical reaction depends on the concentration of reactants. Knowing the reaction order is crucial to understanding how different concentration levels will affect the speed of the reaction. We define the order of reaction as an exponent value in the rate law equation, which indicates the power to which the concentration of a reactant is raised.
To determine the reaction order, observe how changes in the concentration of reactants influence the rate. For example, if doubling the concentration leads to doubling the reaction rate, the reaction order with respect to that reactant is likely first order (n = 1). If doubling the concentration leads to a fourfold increase, it's second order (n = 2). The reaction can be zero, fractional, or even negative order, depending on how the concentration change affects the rate. Analyzing experimental data is critical to determining these values.
To determine the reaction order, observe how changes in the concentration of reactants influence the rate. For example, if doubling the concentration leads to doubling the reaction rate, the reaction order with respect to that reactant is likely first order (n = 1). If doubling the concentration leads to a fourfold increase, it's second order (n = 2). The reaction can be zero, fractional, or even negative order, depending on how the concentration change affects the rate. Analyzing experimental data is critical to determining these values.
Rate Law
The rate law provides a mathematical relationship that shows the role of reactant concentrations in influencing the rate of reaction. It is usually expressed as:
\[ ext{Rate} = k[Y]^n\]
Where \( ext{Rate} \) is the rate of reaction, \( [Y] \) is the concentration of reactant Y, \( n \) is the reaction order, and \( k \) is the rate constant. The rate law is specific to each chemical reaction and must be determined experimentally. It tells us how sensitive the reaction rate is to changes in the concentration of the reactants. The form of the rate law suggests mechanisms through which reactants are turned into products.
By analyzing concentration and rate data, one can derive the order of the reaction with respect to each participating species and construct the rate law to model how the reaction proceeds under various conditions.
\[ ext{Rate} = k[Y]^n\]
Where \( ext{Rate} \) is the rate of reaction, \( [Y] \) is the concentration of reactant Y, \( n \) is the reaction order, and \( k \) is the rate constant. The rate law is specific to each chemical reaction and must be determined experimentally. It tells us how sensitive the reaction rate is to changes in the concentration of the reactants. The form of the rate law suggests mechanisms through which reactants are turned into products.
By analyzing concentration and rate data, one can derive the order of the reaction with respect to each participating species and construct the rate law to model how the reaction proceeds under various conditions.
Rate Constant
The rate constant, denoted as \( k \), is a crucial part of the rate law that relates the rate of reaction to the concentrations of reactants and the reaction order. This constant is determined experimentally and is unique to each reaction. It varies with temperature and can also be affected by the presence of catalysts.
The units of the rate constant depend on the overall order of the reaction and are derived to satisfy the rate equation. For a first-order reaction, for example, the units of \( k \) are \( ext{min}^{-1} \), showing that it's reliant on time. Being able to calculate \( k \) gives an opportunity to predict reaction rates at different concentrations, which aids in designing chemical processes and understanding reaction mechanisms.
Ultimately, the rate constant is an essential factor that allows chemists to predict and control the progress of reactions in both laboratory and industrial settings.
The units of the rate constant depend on the overall order of the reaction and are derived to satisfy the rate equation. For a first-order reaction, for example, the units of \( k \) are \( ext{min}^{-1} \), showing that it's reliant on time. Being able to calculate \( k \) gives an opportunity to predict reaction rates at different concentrations, which aids in designing chemical processes and understanding reaction mechanisms.
Ultimately, the rate constant is an essential factor that allows chemists to predict and control the progress of reactions in both laboratory and industrial settings.
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