Question

How would you express the rate of the chemical reaction \(\mathrm{A} \rightarrow \mathrm{B}\) based on the concentration of Reactant \(\mathrm{A}\) ? How would that rate compare with the reaction rate based on the Product \(\mathrm{B}\) ?

Short answer

The rate of the chemical reaction A → B can be expressed based on the concentration of Reactant A using the rate law equation: \(Rate = k[A]^m\), where k is the rate constant and m is the reaction order with respect to reactant A. To express the rate based on the concentration of Product B, we can use the equation \(Rate = k_B[B]^n\), where k_B is the rate constant for product B, and n is the reaction order with respect to product B. Comparing these rates requires more information about the reaction orders and initial concentrations.

Step-by-step solution

The general form of the rate law equation for a reaction is given as: $$ Rate = k[A]^m $$ Where: \\ Rate = rate of the reaction \\ k = rate constant \\ [A] = concentration of reactant A \\ m = order of the reaction with respect to reactant A (it can either be a positive integer or a zero) \\ #Step 2: Determine the rate based on the concentration of Reactant A#

As our aim is to express the rate based on the concentration of reactant A, we rewrite the rate law equation as follows: $$ Rate = k[A]^m $$ Here, the rate of the reaction is expressed based on the concentration of reactant A. #Step 3: Express the rate based on the concentration of Product B#

To express the rate based on the concentration of product B, we must first consider the stoichiometry of the reaction. As the stoichiometry of the reaction A → B is 1:1, the concentration of B will be equal to the initial concentration of A minus the remaining concentration of A at a given time. Let the initial concentration of A be [A]₀ and the concentration of A at time t be [A]. Then the concentration of B at time t can be given as: $$ [B] = [A]_0 - [A] $$ Now, we rewrite the rate law expression for Product B: $$ Rate = k_B [B]^n $$ Where: \\ k_B = rate constant of the reaction based on the product B \\ n = order of the reaction with respect to product B \\ #Step 4: Compare the reaction rate based on Reactant A and Product B#

We have two different rate expressions: 1. Based on Reactant A: \(Rate = k[A]^m\) 2. Based on Product B: \(Rate = k_B [B]^n\) These two expressions represent the same reaction, but they depend on the concentration of either Reactant A or Product B. By comparing these two equations, we can see that the exponents m and n represent the reaction order for each species and that the rate constants k and k_B may have different values. In order to compare the rates, we have to know the reaction orders m and n and the initial concentration [A]₀. Otherwise, we cannot directly compare the rates just based on these expressions without more information. In summary, we have expressed the rate of the chemical reaction A → B based on the concentration of Reactant A and Product B through the rate law equations. To compare the rates, we need more information about the reaction orders and initial concentrations.

Key concepts

Rate Law Equation

Understanding the rate law equation is fundamental in studying the kinetics of a chemical reaction. The equation illustrates how the rate of a chemical reaction depends on the concentration of the reactants. In simple terms, the rate law represents a mathematical equation that relates the reaction rate with the concentrations of reactants and a proportionality constant called the rate constant.

For a given reaction, say \(\mathrm{A} \rightarrow \mathrm{B}\), the rate law can be written as: \[Rate = k[A]^m\] Here, \(k\) is the rate constant, which is a measure of how quickly the reaction occurs. It's specific to each reaction and can be affected by factors like temperature and catalysts. The term \(\mathrm{A}\) represents the concentration of reactant A, and \(m\) signifies the reaction order with respect to A. The reaction order tells us how the rate is affected by the concentration of the reactant; it is normally determined experimentally.

Understanding the rate law equation is crucial because it gives insights into the dynamics of the reaction. It can also be used to predict how the reaction rate will change if the conditions are altered, such as increasing the reactant concentration or adding a catalyst.

Reaction Order

The reaction order is a critical concept in chemical kinetics as it specifies the dependency of the reaction rate on the concentration of each reactant. The reaction order is an exponent in the rate law equation that provides this information. If we have a rate law like \(Rate = k[A]^m\), the value of \(m\) indicates the reaction's order with respect to reactant A.

For instance, if \(m\) is 1, the reaction is first-order with respect to A, meaning the rate doubles if the concentration of A doubles. A zero-order reaction (\(m = 0\)) implies that changing A's concentration has no effect on the rate. For a second-order reaction (\(m = 2\)), the rate will quadruple if A's concentration doubles.

These orders are not limited to whole numbers; they can also be fractions, indicating complex relationships between reaction rate and concentrations. Determining the reaction order is essential for understanding the mechanism of the reaction and for designing chemical processes because it affects how reactants are used and products are formed over time. It is also especially useful when considering how changes in conditions can affect the yield and rate of a chemical process.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. It is described by the balanced chemical equation, which gives the mole ratio in which reactants combine and products form. For example, the simple reaction \(\mathrm{A} \rightarrow \mathrm{B}\) suggests a one-to-one stoichiometric ratio between A and B.

Understanding stoichiometry is essential because it allows chemists to predict the amounts of products that can be formed from a given amount of reactants. This is particularly important in industrial applications where yield and efficiency need to be maximized. Stoichiometry also plays a vital role when relating the concentration of reactants to the rate of the reaction. As we have seen in the rate law equation for product B: \[Rate = k_B [B]^n\], the rate can depend on the concentration of product B, which can be derived from stoichiometry by knowing that \[B] = [A]_0 - [A]\], considering \(\mathrm{A}\) as the initial reactant and the equation's stoichiometric coefficients.

In the context of reaction rates, the stoichiometric coefficients become particularly important when dealing with reactions that do not have a 1:1 stoichiometry. In such cases, the rates at which reactants are consumed and products are produced can differ markedly, and understanding the stoichiometry is required to correctly apply the rate law.