Problem 86 - Introductory Chemistry Atoms First

Question

The reaction \(\mathrm{A}+2 \mathrm{~B}+\mathrm{C} \rightarrow \mathrm{AB}_{2} \mathrm{C}\) has a rate that does not change when more \(C\) is added to the reaction flask. Suppose the overall reaction order is 2, and the reaction is first-order with respect to \(\mathrm{A}\). (a) Write the rate law for this reaction, (b) What is the order for reactant \(C ?\)

Step-by-Step Solution

Answer

(a) The rate law for this reaction is \(r = k[\mathrm{A}][\mathrm{B}]\). (b) The order for reactant C is 0.

Step 1: Analyze the given information

We are given a reaction: \(\mathrm{A} + 2 \mathrm{B} + \mathrm{C} \rightarrow \mathrm{AB}_{2}\mathrm{C}\) We know that the overall reaction order is 2, and it is first-order with respect to A. Also, the rate doesn't change when more C is added to the reaction flask.

Step 2: Write the general form of the rate law

The general rate law for a reaction is given by: \(r = k[\mathrm{A}]^m[\mathrm{B}]^n[\mathrm{C}]^p\) where r is the rate of the reaction, k is the rate constant, and m, n and p are the reaction orders with respect to reactants A, B and C, respectively.

Step 3: Substituting the given information

We are given that the reaction is first-order with respect to reactant A, so m = 1. Also, as the rate doesn't change when more C is added, it means the reaction is zero-order with respect to reactant C, so p = 0. So, our rate law becomes: \(r = k[\mathrm{A}]^1[\mathrm{B}]^n[\mathrm{C}]^0\)

Step 4: Determine the reaction order with respect to reactant B

We know that the overall reaction order is 2. The sum of the reaction orders with respect to each reactant should be equal to the overall reaction order. So, we can write: 1 + n + 0 = 2 Solving for n, we get n = 1.

Step 5: (a) Write the rate law for this reaction

Now that we have the reaction orders for all reactants, we can write the rate law for this reaction: \(r = k[\mathrm{A}][\mathrm{B}]\)

Step 6: (b) What is the order for reactant C?

The order for reactant C is 0, as we determined earlier when discussing that the rate doesn't change when more C is added to the reaction flask.

Key Concepts

Reaction Order

In chemical kinetics, the reaction order provides valuable insight into how the concentration of reactants affects the rate of a chemical reaction. The reaction order can be

• zero-order,
• first-order,
• second-order, or higher,

depending on the particular reactants involved.

It's essential to note that the reaction order is determined by empirical experimentation rather than stoichiometry. This concept applies to both specific reactants and the overall reaction.

For example, in the given exercise, the overall reaction order is 2, which means the combination of changes in the concentrations of the reactants influences the reaction's rate significantly. The reaction is also explicitly noted as first-order with respect to reactant A, meaning any change in its concentration will proportionally affect the rate. Understanding these orders helps in predicting how the reaction will progress over time when concentrations change.

Rate Constant

The rate constant, denoted by the symbol 'k', is a fundamental aspect of the rate law in chemical reactions. This constant provides a quantitative measure of the speed of the reaction.

The rate constant in a chemical reaction will differ based on

• temperature,
• the presence of a catalyst, and
• specific reaction conditions.

The importance of 'k' lies in its role within the rate law equation, such as:\[ r = k[\mathrm{A}]^1[\mathrm{B}]^n[\mathrm{C}]^0 \]Where 'k' influences the actual numerical rate of reaction. 'k' does not have a single fixed value; its value can change significantly with alterations in the reaction environment.

In the exercise, understanding 'k' helps us see how, alongside reactant concentrations, it determines the rate of the reaction mathematically and scientifically.

Zero-Order Reaction

A zero-order reaction is unique because the rate of the reaction is independent of the concentration of the reactants. This means that adding more reactant doesn't change the speed at which the reaction occurs.

For a component of a reaction to be deemed zero order, such as reactant C in the problem given, its concentration does not play a role in increasing or decreasing the rate. This is shown in the rate law where the exponent associated with reactant C is zero:\[ r = k[\mathrm{A}][\mathrm{B}][\mathrm{C}]^0 \]The zero exponent mathematically means any number raised to the power of zero equals one, confirming the concentration's irrelevance.

This type of reaction is crucial for understanding scenarios where certain reactants do not participate in determining the speed of the reaction, thereby simplifying the mechanics of evaluating reaction progress.

First-Order Reaction

A first-order reaction is where the rate directly depends on the concentration of a single reactant. In this case, the reactant's concentration directly influences the rate of reaction linearly.

In the original problem, reactant A displays first-order behavior, meaning any doubling or halving of concentration results in a proportional change in the reaction rate.

The representation of a first-order reaction in rate law format would be:\[ r = k[\mathrm{A}]^1[\mathrm{B}][\mathrm{C}]^0 \] This linear dependence makes first-order reactions simple to model and predict within calculations and experiments. It is a common reaction order encountered in various fields, from chemistry to biology.

Because of this straightforward relationship between concentration and rate, understanding first-order reactions aids in comprehending the kinetics and dynamics of many systems and reactions.