Problem 81 - Introductory Chemistry Atoms First

Question

Given the general form of the rate law, Rate \(=k[\operatorname{Reactant} 1]^{x}[\text { Reactant } 2]^{y}\) answer the following questions: (a) Which part of the rate law reflects the inherent factors of the reaction? (b) What is the general name for the exponents \(x\) and \(y ?\) (c) How do we calculate the overall order of a chemical reaction? (d) Suppose the reaction is second-order with respect to reactant 1 and first- order with respect to reactant \(2 .\) What are the values of \(x\) and \(y\), and what is the overall order of a reaction with only these two reactants? (e) Suppose reactant 1 does not appear in the rate law. What does this say about the value of its order? What is the meaning of the value of its order?

Step-by-Step Solution

Answer

(a) The inherent factors of the reaction are reflected in the rate constant (k). (b) The exponents \(x\) and \(y\) are called "reaction orders." (c) The overall order of a chemical reaction is calculated by adding the orders of each individual reactant: \(x + y\). (d) If the reaction is second-order with respect to reactant 1 and first-order with respect to reactant 2, then \(x = 2\) and \(y = 1\). The overall order of the reaction is \(2 + 1 = 3\). (e) If reactant 1 does not appear in the rate law, its order \(x\) is equal to 0. This means the reaction rate is independent of the concentration of reactant 1.

Step 1: (a) Inherent factors of the reaction

The inherent factors of the reaction are reflected in the rate constant (k). The rate constant is specific to a particular reaction and takes into account intrinsic properties like temperature and the nature of the reactants.

Step 2: (b) General name for the exponents x and y

The general name for the exponents x and y in the rate law are called "reaction orders" or more specifically, the "orders" with respect to reactant 1 and reactant 2, respectively.

Step 3: (c) Calculation of overall order of a chemical reaction

To calculate the overall order of a chemical reaction, simply add the orders of each individual reactant present in the rate law. In this case, the overall order of the reaction would be given by \(x + y\).

Step 4: (d) Values of x, y, and the overall order of the reaction

According to the problem, the reaction is second-order with respect to reactant 1 and first-order with respect to reactant 2. This means that \(x = 2\) and \(y = 1\). To find the overall order of the reaction, we add the individual orders: \(Overall order = x + y = 2 + 1 = 3\). So, the overall order of the reaction with these two reactants is 3.

Step 5: (e) Value and meaning of reactant 1 order

If reactant 1 does not appear in the rate law, it means that its order, in this case x, is equal to 0. This signifies that the reaction rate is independent of the concentration of reactant 1; changes in its concentration have no effect on the rate of the reaction.

Key Concepts

Rate Constant

The rate constant, often represented by the symbol 'k', is a crucial aspect of chemical kinetics as it quantifies the speed of a chemical reaction. Essentially, it is a measure of how rapidly a reaction occurs under certain conditions. The rate constant is unique for each reaction and varies with temperature, pressure, and catalysts if present. It does not change with the concentrations of the reactants but is profoundly influenced by environmental factors such as temperature. For instance, a higher temperature usually increases the rate constant, thereby accelerating the reaction.

Understanding the rate constant is essential, as it helps in predicting how a reaction’s rate will change under different circumstances. When we talk about its units, they can vary depending on the overall order of the reaction, ranging from liters per mole per second to reciprocal seconds for zero-order reactions.

Reaction Orders

The reaction orders, designated as 'x' and 'y' in a rate law, indicate the dependence of the reaction rate on the concentration of each reactant. Each order represents the power to which the concentration of a reactant is raised in the rate law equation. They provide insight into the mechanism of the reaction by telling us how altering the concentration of a reactant affects the rate. For instance, a reaction order of 2 for a reactant means that if the concentration of that reactant is doubled, the reaction rate will increase by a factor of four.

These orders of reaction are determined experimentally and can be whole numbers, fractions, or even zero. In cases where the order is zero, this indicates that changes in the reactant's concentration do not influence the rate of the reaction.

Overall Order of Reaction

The overall order of a reaction is simply the sum of the individual reaction orders with respect to each reactant present in the rate law. It represents the reaction’s overall dependence on the concentration of all reactants. For example, if a reaction has a rate law with a second-order dependence on one reactant and first-order on another, the overall order would be three. This particular value provides important information about how the rate will change with varying concentrations of all reactants involved.

Calculation of the overall order is crucial for understanding reaction kinetics and in formulating rate laws based on experimental data. It also influences the units of the rate constant 'k', which must be consistent with the rate equation’s overall units.

Concentration Dependence

The concentration dependence of a reaction rate expresses how the speed of a reaction varies with the concentration of reactants. This concept is embodied in the rate law, where the rate is proportional to the product of the concentrations raised to their respective orders. In practical terms, if a reactant has an order greater than zero, an increase in its concentration will lead to an increase in the reaction rate.

It is essential to grasp concentration dependence because it enables us to predict and control reaction rates. Moreover, it sheds light on the reaction mechanism: if a reactant appears with a nonzero order in the rate law, it plays a direct role in the rate-determining step of the mechanism. By contrast, a zero-order indicates no direct influence on rate by changes in that reactant's concentration.