Problem 48 - Chemistry The Science in Context

Question

Determine the overall order of the following rate laws and the order with respect to each reactant. a. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}]^{1 / 2}\) b. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}][\mathrm{C}]\) c. Rate \(=k[\mathrm{A}][\mathrm{B}]^{3}[\mathrm{C}]^{1 / 2}\)

Step-by-Step Solution

Answer

Question: Determine the overall order of the reactions and the order with respect to each reactant for the given rate laws: a. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}]^{1 / 2}\) b. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}][\mathrm{C}]\) c. Rate \(=k[\mathrm{A}][\mathrm{B}]^{3}[\mathrm{C}]^{1 / 2}\) Answer: a. Order with respect to A: 2, Order with respect to B: \(\frac{1}{2}\), Overall order: 2.5 b. Order with respect to A: 2, Order with respect to B: 1, Order with respect to C: 1, Overall order: 4 c. Order with respect to A: 1, Order with respect to B: 3, Order with respect to C: \(\frac{1}{2}\), Overall order: 4.5

Step 1: Order with respect to A

In this rate law, the exponent of A is 2. Therefore, the order with respect to A is 2.

Step 2: Order with respect to B

The exponent of B is \(\frac{1}{2}\). Therefore, the order with respect to B is \(\frac{1}{2}\).

Step 3: Overall order

Add up the orders with respect to A and B: \(2+\frac{1}{2}=2.5\). Hence, the overall order of this rate law is 2.5. b. Rate \(=k[\mathrm{A}]^{2}[\mathrm{B}][\mathrm{C}]\)

Step 4: Order with respect to A

The exponent of A is 2. Therefore, the order with respect to A is 2.

Step 5: Order with respect to B

The exponent of B is 1. Therefore, the order with respect to B is 1.

Step 6: Order with respect to C

The exponent of C is 1. Therefore, the order with respect to C is 1.

Step 7: Overall order

Add up the orders with respect to A, B, and C: \(2+1+1=4\). Hence, the overall order of this rate law is 4. c. Rate \(=k[\mathrm{A}][\mathrm{B}]^{3}[\mathrm{C}]^{1 / 2}\)

Step 8: Order with respect to A

The exponent of A is 1. Therefore, the order with respect to A is 1.

Step 9: Order with respect to B

The exponent of B is 3. Therefore, the order with respect to B is 3.

Step 10: Order with respect to C

The exponent of C is \(\frac{1}{2}\). Therefore, the order with respect to C is \(\frac{1}{2}\).

Step 11: Overall order

Add up the orders with respect to A, B, and C: \(1+3+\frac{1}{2}=4.5\). Hence, the overall order of this rate law is 4.5.

Key Concepts

reaction_order

In chemistry, the reaction order is a crucial concept for understanding how the concentration of reactants affects the rate of reaction. The reaction order with respect to a particular reactant is the exponent to which its concentration term is raised in the rate law. This value indicates how sensitive the rate is to changes in that reactant's concentration.
There are two main types of reaction orders:
- **Individual Reaction Order**: Relates to a single reactant and is given by its exponent in the rate law.
- **Overall Reaction Order**: The sum of all the exponents in the rate law. It gives an overall picture of how the combination of reactant concentrations affects the rate.Knowing the reaction order allows chemists to determine the rate law equation and predict how changes in concentration will influence the reaction speed. For instance, in the rate law \(Rate = k[\mathrm{A}]^2[\mathrm{B}]^{1/2}\), the overall order is 2.5 because it is the sum of the individual orders: 2 for \([\mathrm{A}]\) and 0.5 for \([\mathrm{B}]\). This provides insight into the mechanism of the reaction.

rate law

The rate law is an equation that expresses the rate of a chemical reaction in terms of the concentrations of the reactants and the rate constant. It is typically written as:
\[ \text{Rate} = k [\text{A}]^m [\text{B}]^n \ldots \]
Where \(k\) is the rate constant, and \(m\) and \(n\) are orders of reaction with respect to reactants \([\text{A}]\) and \([\text{B}]\). The rate law provides a detailed, mathematical description of the reaction rate's dependence on the concentrations of various substances. This is particularly useful in experimental settings to calculate the rate or predict the effect of changes in concentration.

Understanding the rate law allows for:

• Predicting how changes in concentration affect the reaction rate.
• Determining which reactant has the greatest influence on the reaction rate.
• Providing clues to the reaction mechanism by indicating which reactants are involved in the rate-determining step.

The rate law does not depend on the balanced chemical equation but must be determined experimentally. This makes it a powerful tool for chemists in research and development.

reactant order

Reactant order refers to the order of a reaction specifically concerning a certain reactant. It is essentially the power to which the concentration of a reactant is raised in the rate law equation. This reflects how changes in that reactant's concentration affect the overall rate of reaction.
For any reactant \([\mathrm{A}]\), the order \(m\) is defined as:

• **Zero Order**: Rate does not change with concentration of the reactant. Often denoted as \([\mathrm{A}]^0\).
• **First Order**: Rate is directly proportional to the concentration. Represented as \([\mathrm{A}]^1\).
• **Second Order**: Rate changes with the square of the concentration. Written as \([\mathrm{A}]^2\).

Different reaction orders for different reactants can indicate various influences on the rate. For instance, in the rate law \(Rate = k[\mathrm{A}]^2[\mathrm{B}]^1\), \([\mathrm{A}]\) has a second order, while \([\mathrm{B}]\) is first order, suggesting \([\mathrm{A}]\) plays a more significant role in determining the rate as compared to \([\mathrm{B}]\). Explaining reactant orders is essential for understanding each reactant's impact in complex reactions.

rate constant

The rate constant, represented as \(k\), is a significant factor in the rate law equation relating the concentration of reactants and the measured reaction rate. It is a proportionality constant that is specific to a given reaction at a particular temperature and is usually determined experimentally.
The value of \(k\) is influenced by:

• **Temperature**: Generally, the rate constant increases with temperature due to the Arrhenius equation.
• **Catalysts**: The presence of a catalyst can alter \(k\) by lowering the activation energy.
• **Nature of Reactants**: Different reactants and their states (solid, liquid, gas) can affect \(k\).

Understanding the rate constant is crucial because it helps calculate the reaction rate under various conditions, and for comparing the rate of similar reactions. In the rate law \(Rate = k[\mathrm{A}]^2[\mathrm{B}]\), \(k\) determines how fast the reaction proceeds when \([\mathrm{A}]\) and \([\mathrm{B}]\) are at standard concentrations. Therefore, specifying \(k\) helps chemists optimize industrial processes or experimental conditions in the lab.