Question

Consider a hypothetical reaction between \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) that is zero order in A, second order in B, and first order in C. (a) Write the rate law for the reaction. (b) How does the rate change when [A] is tripled and the other reactant concentrations are held constant? (c) How does the rate change when [B] is doubled and the other reactant concentrations are held constant? (d) How does the rate change when [C] is tripled and the other reactant concentrations are held constant? (e) By what factor does the rate change when the concentrations of all three reactants are doubled? (f) By what factor does the rate change when the concentrations of all three reactants are cut in half?

Short answer

Solution: (a) The rate law is given by Rate = k * [B]^2 * [C]. (b) The rate remains unchanged when [A] is tripled. (c) The new rate is 4 times the initial rate when [B] is doubled. (d) The new rate is 3 times the initial rate when [C] is tripled. (e) The rate changes by a factor of 8 when all three reactants' concentrations are doubled. (f) The rate changes by a factor of \(\frac{1}{8}\) when all three reactants' concentrations are cut in half.

Step-by-step solution

(a) Write the rate law for the reaction

The rate law for a reaction can be expressed as: Rate = k * [A]^m * [B]^n * [C]^p where m, n, and p are the orders of the reactants A, B, and C respectively, and k is the rate constant. In this case, the reaction is zero order in A, second order in B, and first order in C. Therefore, the rate law is: Rate = k * [A]^0 * [B]^2 * [C]^1 This simplifies to: Rate = k * [B]^2 * [C]

(b) How does the rate change when [A] is tripled and the other reactant concentrations are held constant?

Since the reaction is zero order in A, tripling the concentration of A will not have any impact on the rate of the reaction. Thus, the rate remains unchanged.

(c) How does the rate change when [B] is doubled and the other reactant concentrations are held constant?

The reaction is second order in B, so if the concentration of B doubles, we can represent this change in the rate law: New Rate = k * [2B]^2 * [C] New Rate = k * (2^2) * [B]^2 * [C] New Rate = 4 * (k * [B]^2 * [C]) The new rate is 4 times the initial rate.

(d) How does the rate change when [C] is tripled and the other reactant concentrations are held constant?

The reaction is first order in C, so if the concentration of C triples, we can represent this change in the rate law: New Rate = k * [B]^2 * [3C] New Rate = 3 * (k * [B]^2 * [C]) The new rate is 3 times the initial rate.

(e) By what factor does the rate change when the concentrations of all three reactants are doubled?

We will have the concentrations of A, B, and C doubled: New Rate = k * [2A]^0 * [2B]^2 * [2C]^1 New Rate = k * [B]^2 * [C] * (2^2) * 2 The factor by which the rate changes is (2^2) * 2 = 8.

(f) By what factor does the rate change when the concentrations of all three reactants are cut in half?

We will have the concentrations of A, B, and C halved: New Rate = k * [\(\frac{1}{2}\)A]^0 * [\(\frac{1}{2}\)B]^2 * [\(\frac{1}{2}\)C]^1 New Rate = k * [B]^2 * [C] * (\(\frac{1}{2}\))^2 * \(\frac{1}{2}\) The factor by which the rate changes is (\(\frac{1}{2}\))^2 * \(\frac{1}{2}\) = \(\frac{1}{8}\).

Key concepts

Order of Reaction

The order of a reaction is a key concept used in chemistry to describe how the rate of a chemical reaction depends on the concentration of the reactants. Essentially, it determines the reaction's sensitivity to changes in concentration.
For example, consider a reaction involving substances A, B, and C. If this reaction is zero order in A, it means that changes in the concentration of A do not affect the reaction rate whatsoever. That's a superpower of sorts for the reaction involving A - the rate is independent of how much A you have around.
However, the reaction is second order in B and first order in C. This means, respectively, that the rate of the reaction will change if the concentration of B or C is changed. Specifically, doubling the concentration of B will lead to a fourfold increase in the rate, while tripling the concentration of C will result in a threefold increase in the rate of reaction.
This understanding of the order of reaction is crucial in predicting how altering reactant concentrations will affect the speed of reactions. It's like knowing the unique recipe that determines how the reaction takes place at different speeds.

Chemical Kinetics

Chemical kinetics is the branch of chemistry focused on the speed, or rate, at which reactions occur. It allows chemists to understand the factors that accelerate or decelerate a reaction, which is vital in both laboratory and industrial settings.
Kinetics involves studying the concentration of reactants and how they influence the rate of reactions. By using different orders of reactions — such as zero order for A, second order for B, and first order for C as in our exercise — kinetics helps us identify and predict the behavior of reactions over time.
Importantly, chemical kinetics also considers other factors beyond concentrations, such as temperature and catalysts, which can significantly affect reaction rates. By integrating these factors, chemists can control a reaction much like a maestro conducting an orchestra, ensuring events unfold with perfect timing.

• Zero Order: Reaction rate is constant, regardless of concentration.
• First Order: Reaction rate changes linearly with concentration.
• Second Order: Rate increases exponentially as concentration rises.

Understanding these principles helps create more efficient industrial processes, optimize reactions for desired outcomes, and innovate in fields ranging from pharmaceuticals to materials science.

Rate Constant

The rate constant, often denoted as \(k\), is a crucial parameter in the rate law equation of a chemical reaction. It essentially acts like a dial that adjusts the speed at which a reaction proceeds.
In mathematical terms, the rate law for a reaction is expressed as:\[\text{Rate} = k \times [\text{A}]^m \times [\text{B}]^n \times [\text{C}]^p\]where \(m, n,\) and \(p\) represent the orders of reactants A, B, and C, respectively. This formula shows how the concentration of reactants and the rate constant together determine the reaction rate.
The beauty of the rate constant is that it is unique to every reaction at a given temperature. It doesn't just hover there; it's a key player influenced by environmental factors such as temperature, pressure, and presence of catalysts.
For example, in our exercise, the rate constant \(k\) stays the same regardless of changes in the reactant concentrations — unless external conditions vary. Think of \(k\) as the heart of the reaction's rate law equation; it beats at a rhythm set by the chemical and environmental surroundings, affecting how vigorously the reaction proceeds.