The differential rate law, also known as the rate equation, describes the instantaneous rate of a chemical reaction as a function of the concentrations of the reactants. It is a fundamental concept in chemical kinetics that helps determine the order of a reaction and the rate constant.
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The differential rate law is expressed as: $\text{Rate} = k[A]^m[B]^n$, where $k$ is the rate constant, $[A]$ and $[B]$ are the concentrations of the reactants, and $m$ and $n$ are the reaction orders with respect to $A$ and $B$, respectively.
The reaction order can be determined experimentally by measuring the reaction rate at different initial concentrations of the reactants and analyzing the relationship between the rate and the concentrations.
The rate constant $k$ is specific to a given chemical reaction and depends on factors such as temperature, pressure, and the presence of catalysts.
The differential rate law is the starting point for deriving the integrated rate law, which describes the change in reactant concentration over time.
Understanding the differential rate law is crucial for predicting the progress of a chemical reaction, designing reaction conditions, and analyzing experimental kinetic data.
Review Questions
Explain the relationship between the differential rate law and the reaction order.
The differential rate law, $\text{Rate} = k[A]^m[B]^n$, describes the instantaneous rate of a chemical reaction as a function of the concentrations of the reactants. The exponents $m$ and $n$ in the rate law represent the reaction orders with respect to reactants $A$ and $B$, respectively. The reaction order indicates the dependence of the reaction rate on the concentration of each reactant. For example, if the reaction order with respect to $A$ is 2 (i.e., $m = 2$), then the reaction rate is proportional to the square of the concentration of $A$. Determining the reaction orders is an important step in understanding the mechanism of a chemical reaction.
Describe how the differential rate law can be used to derive the integrated rate law.
The differential rate law, $\text{Rate} = k[A]^m[B]^n$, is the starting point for deriving the integrated rate law, which describes the change in reactant concentration over time. To obtain the integrated rate law, the differential rate law is integrated with respect to time, assuming the reaction is carried out under certain conditions (e.g., constant temperature, constant volume). The integration process involves separating the variables and solving the resulting differential equation. The integrated rate law provides a mathematical expression that relates the concentration of a reactant to the reaction time, which is useful for predicting the progress of a chemical reaction and analyzing experimental kinetic data.
Explain the significance of the rate constant $k$ in the differential rate law and discuss the factors that can influence its value.
The rate constant $k$ in the differential rate law, $\text{Rate} = k[A]^m[B]^n$, is a proportionality constant that relates the reaction rate to the concentrations of the reactants. The value of the rate constant is specific to a given chemical reaction and depends on various factors, such as temperature, pressure, and the presence of catalysts. An increase in temperature, for example, typically leads to an increase in the rate constant, as it provides more energy to the reacting molecules and facilitates the formation of the activated complex. The presence of a catalyst, on the other hand, can alter the reaction pathway and lower the activation energy, thereby increasing the rate constant. Understanding the factors that influence the rate constant is crucial for optimizing reaction conditions and predicting the progress of chemical reactions.
The exponent to which the concentration of a reactant is raised in the rate equation, indicating the dependence of the reaction rate on the concentration of that reactant.
A proportionality constant that relates the reaction rate to the concentrations of the reactants, and is specific to a given chemical reaction under a set of conditions.