Problem 2

Question

For Reaction Mixture 1 , the student found that 92 seconds were required. On dividing Equation 5 for Reaction Mixture 1 by Equation 5 for Reaction Mixture $4,$ and after canceling out the common terms $\left(k^{\prime} \text { and terms in }\left[\mathrm{I}^{-}\right] \text {and }\left[\mathrm{BrO}_{3}^{-}\right]\right)$, they got the following equation: $$\frac{10.9}{45}=\left(\frac{0.020}{0.040}\right)^{p}=\left(\frac{1}{2}\right)$$ Recognizing that $10.9 / 45$ is about equal to one-fourth, they obtained an approximate value for $p .$ What was that value?$p=$_______. By taking logarithms of both sides of the equation, the student got an exact value for $p$. What was that value? Since orders of reactions are often integers, the student rounded their value of $p$ to the nearest integer and reported that value as the order of the reaction with respect to $\mathbf{H}^{+}$. $p=$_______.

Step-by-Step Solution

Verified

Answer

The value of $p$ is 2.

1Step 1: Understand the Problem

We are given the equation $\frac{10.9}{45} = \left(\frac{0.020}{0.040}\right)^p$ which simplifies to $\frac{1}{4} = \left(\frac{1}{2}\right)^p$. We need to solve for $p$ by using the property of logarithms that says if $a^b = c$, then $b = \log_a(c)$.

2Step 2: Set Up the Equation

We recognize $\frac{1}{4}$ as $\left(\frac{1}{2}\right)^2$. Hence the equation simplifies to $(\frac{1}{2})^2 = (\frac{1}{2})^p$.

3Step 3: Applying the Concept of Equality

Since $(\frac{1}{2})^2 = (\frac{1}{2})^p$, by the property of exponents, we equate powers: $p = 2$.

4Step 4: Confirm Using Logarithms

Taking log base $\frac{1}{2}$ on both sides of the equation gives us $\log_{\frac{1}{2}}\left(\frac{1}{4}\right) = p$. Since $\frac{1}{4} = \left(\frac{1}{2}\right)^2$, $p$ is indeed $2$ as previously determined.

5Step 5: Conclusion and Reporting

The student found that the exact value for $p$ using logarithms is $2$. Since reaction orders are often integers, the order is rounded to an integer (though in this case, no rounding is needed).

Key Concepts

KineticsChemical reactionsLogarithmsRate laws

Kinetics

Kinetics is a branch of chemistry that studies the rates at which chemical reactions occur. It helps us understand how different variables affect these rates, such as concentration, temperature, and catalysts. By studying kinetics, we can predict how quickly a reaction will proceed and the factors that can speed up or slow it down. Understanding kinetics is crucial for designing industrial processes, developing new products, and even in biological systems where reaction rates influence functions like metabolism.
Key concepts in kinetics include:

Reaction rate: It's the speed at which reactants are converted into products. Expressed as change in concentration over time.
Activation energy: The minimum energy needed for a reaction to occur.
Rate constant (k): A proportionality constant in rate laws that is specific to each reaction at a given temperature.

By manipulating these elements, chemists can control the path and speed of reactions to achieve desired outcomes.

Chemical reactions

Chemical reactions are processes where substances, known as reactants, transform into different substances, called products. Understanding chemical reactions involves examining factors that affect them, such as energy changes, reaction conditions, and molecular interactions.
Some characteristics of chemical reactions include:

Reactants and Products: Reactants are the starting substances, while products are the substances formed.
Balancing Equations: Chemical equations must be balanced to obey the Law of Conservation of Mass, meaning the number of each type of atom must be the same on both sides.
Types of Reactions: These include synthesis, decomposition, single replacement, double replacement, and combustion, each with its own characteristics and conditions.

Understanding these elements helps predict how a chemical reaction will proceed and how different factors such as concentration and energy will influence it.

Logarithms

Logarithms are mathematical tools that help us solve equations involving exponents. In the context of kinetics and rate laws, logarithms can aid in finding reaction orders, as seen in the equation $\left(\frac{1}{2}\right)^p = \frac{1}{4}$. By applying logarithms, we simplify the solving process:

Logarithmic Equality: If $a^b = c$, then $b = \log_a(c)$.
Simplifying Equations: Logarithms help break down complex equations, making it easier to isolate variables.
Transcendence of Logarithms: They are used in various scientific fields to handle exponential growth or decay, sound intensity (in decibels), and even in economics.

Using logarithms efficiently allows for precise calculation of reaction orders, as it becomes especially useful when handling large or small numbers in chemical kinetics.

Rate laws

Rate laws are mathematical expressions that describe how the rate of a chemical reaction depends on the concentration of its reactants. They are pivotal in the field of kinetics as they help predict reaction behavior under various conditions. Key aspects of rate laws include:

Form of Rate Law: For a general reaction $aA + bB \rightarrow cC + dD$, the rate law might be expressed as $\text{Rate} = k[A]^m[B]^n$, where $k$ is the rate constant, and $m$ and $n$ are the reaction orders with respect to $A$ and $B$, respectively.
Order of Reaction: The sum of the powers in the rate law gives the overall reaction order, influencing how changes in reactant concentrations affect the reaction rate.
Determination: Empirical data is used to determine the rate laws, often requiring experiments to measure how rates depend on concentrations.

Understanding rate laws is essential for anyone studying chemical kinetics because it allows chemists to define exactly how a reaction will progress and provides insights into the mechanisms of the reactions.