Problem 31 - Chemistry The Central Science In Si Units

Question

Consider the following reaction: $$ \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}(a q)+\mathrm{OH}^{-}(a q) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)+\mathrm{Cl}^{-}(a q) $$ The rate law for this reaction is first order in \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}\) and first order in \(\mathrm{OH}^{-}\). When \(\left[\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{Cl}\right]=4.0 \times 10^{-3} \mathrm{M}\) and \(\left[\mathrm{OH}^{-}\right]=2.5 \times 10^{-2} \mathrm{M},\) the reaction rate at 310 \(\mathrm{K}\) is \(5.20 \times 10^{-2} \mathrm{M} / \mathrm{s}\). (a) What is the value of the rate constant? (b) What are the units of the rate constant?

Step-by-Step Solution

Answer

The rate constant \( k \) is 520 \( \text{M}^{-1} \text{s}^{-1} \).

Step 1: Write the Rate Law Expression

The rate law expression for the given reaction can be written as \( \text{Rate} = k [\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{Cl}] [\mathrm{OH}^-] \) where \( k \) is the rate constant.

Step 2: Substitute Known Values

Substitute the given concentrations and rate into the rate law expression. The expression becomes \( 5.20 \times 10^{-2} = k (4.0 \times 10^{-3}) (2.5 \times 10^{-2}) \).

Step 3: Solve for the Rate Constant

Rearrange the equation to solve for \( k \): \[ k = \frac{5.20 \times 10^{-2}}{(4.0 \times 10^{-3})(2.5 \times 10^{-2})} \].

Step 4: Calculate the Rate Constant

Using a calculator, compute \( k = \frac{5.20 \times 10^{-2}}{1.0 \times 10^{-4}} = 520 \).

Step 5: Determine Rate Constant Units

Since the reaction is second order (first order in each reactant: \([\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{Cl}]\) and \([\mathrm{OH}^-]\)), the units of the rate constant \( k \) are \( \text{M}^{-1}\text{s}^{-1} \).

Key Concepts

Rate Law

The rate law is an equation that tells us how the concentration of reactants affects the rate of a chemical reaction. Essentially, it helps us understand how fast a reaction takes place under certain conditions. Typically, the rate law expression for a reaction can be written in the form:
\[ \text{Rate} = k [A]^x [B]^y \]where:

• \( k \) is the rate constant, a unique value for each reaction at a given temperature.
• \( [A] \) and \( [B] \) are the concentrations of the reactants.
• \( x \) and \( y \) define the order of the reaction with respect to each reactant.

The overall rate law gives us a mathematical model to predict how changes in concentrations impact the reaction rate.

Reaction Order

Reaction order is a key term in the field of chemical kinetics, representing the power to which the concentration of a reactant is raised in the rate law. In simpler terms, it tells us how sensitive the rate of a reaction is to the concentration of each reactant.

For a given reaction, if the rate is first-order in relation to a reactant like \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl} \), it implies that doubling this concentration will double the reaction rate, assuming other conditions are constant. Similarly, if it is first-order in \( \mathrm{OH}^{-} \), changing its concentration will also proportionally affect the rate.

The total reaction order is the sum of the orders with respect to each reactant. In cases where both reactants are first order, the reaction's total order is second order, as explained in the example above. Understanding reaction orders helps in designing experiments and controlling reaction conditions effectively.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that focuses on understanding the rates of chemical reactions and the factors that affect these rates. It is essential for predicting how quickly a reaction will occur, which is crucial in industrial applications and research.

• It involves studying rate laws and determining reaction orders.
• Helps in understanding the mechanism by which a reaction proceeds.
• Analyzes the influence of various factors like temperature, pressure, and catalysts.

A firm grasp of chemical kinetics allows chemists to manipulate reactions for desired outcomes, optimize reaction conditions, and improve safety and efficiency in chemical processes.

Units of Rate Constant

The units of the rate constant \( k \) depend on the total order of the reaction. These units are crucial for ensuring the rate law maintains consistent units across its variables and result. In the rate law, the reaction rate has units of concentration per unit time (such as \( \text{M/s} \)), and the concentrations \([A]\) and \([B]\) have units of molarity (\( \text{M} \)). To understand the units of \( k \), consider the reaction order:

Second-Order Reaction Units:

For a second-order reaction like the one given, where both reactants contribute to the reaction order:

• Rate units: \([\text{M/s}]\)
• Concentration units for each reactant: \(\text{M}^1\)

Therefore, to balance the dimensions in the rate law, the rate constant \( k \) ends up with units of \( \text{M}^{-1}\text{s}^{-1} \).
Knowing these units helps in ensuring the calculations in kinetics are done correctly, allowing scientists to find \( k \) accurately from experimental data.