Problem 47 - Chemistry The Science in Context

Question

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

Step-by-Step Solution

Answer

a. Rate = k[A][B] b. Rate = k[A]^2[B] c. Rate = k[A][B]^3 Answer: a. The order with respect to reactant A is 1, the order with respect to reactant B is 1, and the overall order of the reaction is 2. b. The order with respect to reactant A is 2, the order with respect to reactant B is 1, and the overall order of the reaction is 3. c. The order with respect to reactant A is 1, the order with respect to reactant B is 3, and the overall order of the reaction is 4.

Step 1: Identify the exponents of the reactants

In the given rate law: Rate = k[A][B], we see the exponents are both 1 for reactants A and B, as it is not explicitly shown.

Step 2: Determine order with respect to each reactant

With the exponents as 1 for each reactant A and B, the order with respect to reactant A is 1, and the order with respect to reactant B is also 1.

Step 3: Determine the overall reaction order

Now, we will add the exponents together: Overall order = 1 (A) + 1 (B) = 2. b.

Step 1: Identify the exponents of the reactants

In the given rate law: Rate = k[A]^2[B], we see the exponent for reactant A is 2 and for reactant B is 1.

Step 2: Determine order with respect to each reactant

With the exponents as 2 for reactant A and 1 for reactant B, the order with respect to reactant A is 2, and the order with respect to reactant B is 1.

Step 3: Determine the overall reaction order

Now, we will add the exponents together: Overall order = 2 (A) + 1 (B) = 3. c.

Step 1: Identify the exponents of the reactants

In the given rate law: Rate = k[A][B]^3, we see the exponent for reactant A is 1 and for reactant B is 3.

Step 2: Determine order with respect to each reactant

With the exponents as 1 for reactant A and 3 for reactant B, the order with respect to reactant A is 1, and the order with respect to reactant B is 3.

Step 3: Determine the overall reaction order

Now, we will add the exponents together: Overall order = 1 (A) + 3 (B) = 4.

Key Concepts

Rate Law

When studying chemical reactions, understanding the rate law is important because it tells us about the speed of a reaction. The rate law expresses the rate of a reaction in terms of the concentration of its reactants. It has the general form:
Rate = k[A]^m[B]^n
where:

• k is the rate constant; a specific value that is constant for a given reaction at a specific temperature.
• [A] and [B] are the molar concentrations of the reactants.
• m and n indicate the reactant orders, showing how much the rate is affected by each reactant's concentration.

The exponents m and n are determined experimentally and show how the change in concentration of each reactant affects the rate. In essence, the rate law reveals both the speed of the reaction and how sensitive it is to the concentration of each reactant.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that analyzes the rates of chemical reactions and the factors that influence these rates. It's like figuring out the rules of a game, where instead of players and goals, we have molecules and energy. There are several factors to consider:

• Concentration: Higher concentrations of reactants can lead to faster reactions because there are more molecules to collide and react.
• Temperature: Increasing temperature often increases the rate because molecules have more energy to overcome activation barriers.
• Catalysts: Substances that speed up reactions without being consumed themselves. They provide an alternate pathway with a lower activation energy.
• Surface Area: For reactions involving solids, a greater surface area allows more collisions between reactants.

By studying these factors, chemists can predict how a reaction will proceed under different conditions, which is invaluable in industrial and laboratory settings.

Reactant Order

The term "reactant order" describes how the concentration of a particular reactant affects the rate of the reaction. Each reactant in a reaction has its own order, which is indicated by the exponent in the rate law equation.
For example, in the rate equation
Rate = k[A]^2[B]^1,
the reactant order for A is 2, meaning the rate increases with the square of A's concentration. The reactant order for B is 1, indicating a linear relationship where the rate doubles with the doubling of B's concentration.

• A reactant order of 0 implies that changes in concentration do not affect the reaction rate.
• A reactant becomes first-order when its concentration directly affects the rate, doubling the rate if doubled.
• A second-order reactant has a squared effect on the rate, meaning the rate increases by four times if its concentration is doubled.
• Higher integers or even fractional orders are possible, showing complex interactions.

Understanding reactant order helps chemists infer reaction mechanisms and predict how changes in concentration will alter the speed of a chemical reaction.