Question

The following data relate to the reaction \(\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}\) . Find the order with respect to each reactant. \(\begin{array}{lll} {[\mathrm{Al}(\mathrm{M})} & {[\mathrm{B}](\mathrm{M})} & {\text { Rate }(\mathrm{M} / \mathrm{s})} \\ {0.08} & {0.06} & {0.012} \\\ {0.08} & {0.03} & {0.006} \\ {0.04} & {0.06} & {0.003}\end{array}\)

Short answer

The reaction is second order with respect to A and first order with respect to B.

Step-by-step solution

Identify the Rate Law Expression

The rate law for the reaction \(\text{A} + \text{B} \rightarrow \text{C}\) can be written as \(\text{Rate} = k[\text{A}]^m[\text{B}]^n\), where \(m\) and \(n\) are the orders of the reaction with respect to \(\text{A}\) and \(\text{B}\), respectively.

Compare Experiments to Determine Order with Respect to B

When comparing two experiments where \([\text{A}] \) is kept constant to find \(n\):\( \frac{0.012}{0.006} = \frac{0.06^n}{0.03^n} \) simplifies to \(\frac{0.012}{0.006} = \frac{1}{2}^n \rightarrow 2 = 2^m\). Therefore, \(n = 1\).

Compare Experiments to Determine Order with Respect to A

When comparing two experiments where \([\text{B}] \) is kept constant to find order with respect to \(\text{A}\):\( \frac{0.012}{0.003} = \frac{0.08^m}{0.04^m} \) simplifies to \(\frac{0.012}{0.003} = \frac{2}{1} = 2^m \rightarrow 4 = 2^m\). Therefore, \(m = 2\).

Key concepts

rate law

The rate law is an equation that links the rate of a reaction to the concentration of the reactants. It is particularly important in chemical kinetics to understand how different factors influence the speed of chemical reactions. For our given reaction, \(A + B \rightarrow C\), the rate law can generally be expressed as:

\[ \text{Rate} = k[A]^m[B]^n \]

The symbol \(k\) is the rate constant, while the exponents \(m\) and \(n\) denote the order of the reaction with respect to reactants A and B, respectively. By determining these orders, we can predict how changes in the concentration of reactants will affect the reaction rate.

reaction order

Reaction order illustrates how the concentration of each reactant influences the rate of the reaction. It can be a whole number, a fraction, or even zero. Let’s break down how to determine the reaction order from experimental data.

In our example, we used data from a table to isolate the effects of each reactant:

• To find the order with respect to B, we compared experiments where \([A]\) was constant (0.08 M) and solved the ratio of rates to find \(n = 1\).
  
  \[ \frac{0.012}{0.006} = \left( \frac{0.06}{0.03} \right)^n \rightarrow 2 = 2^n \rightarrow n = 1 \]
  
• To find the order with respect to A, we compared experiments where \([B]\) was constant (0.06 M) and solved for \(m = 2\).
  
  \[ \frac{0.012}{0.003} = \left( \frac{0.08}{0.04} \right)^m \rightarrow 4 = 2^m \rightarrow m = 2 \]
  

Ultimately, we found that the reaction is first order with respect to B and second order with respect to A.

chemical kinetics

Chemical kinetics is the study of the speed or rate at which chemical reactions occur and the factors that affect this rate. This field helps us understand and predict the behavior of reactions over time.

Key concepts include:

• **Rate of Reaction:** How quickly reactants are converted into products.
  
• **Activation Energy:** The energy barrier that must be overcome for a reaction to proceed.
  
• **Catalysts:** Substances that increase the reaction rate without being consumed in the reaction.
  
• **Temperature:** Higher temperatures generally increase the reaction rate.
  

In our exercise, understanding the rate law and reaction order are fundamental aspects of chemical kinetics that provide insights into how the reaction progresses.

rate constant

The rate constant \(k\) is a crucial component of the rate law and is unique for every reaction at a given temperature. It incorporates factors such as the probability of particles colliding with the correct orientation and energy.

Some important points about the rate constant include:

• **Units Depend on Reaction Order:** For instance, in our reaction with rate law \(\text{k[A]^2[B]}\), the units of \(k\) would be \(M^{-2}s^{-1}\).
  
• **Temperature Dependent:** As temperature increases, the rate constant typically increases due to more particles having the required activation energy.
  
• **Catalysts:** Adding a catalyst can change the rate constant by providing an alternative pathway with lower activation energy.
  
  Knowing the rate constant helps chemists design experiments and predict the behavior of chemical reactions in various conditions, making it a foundational concept in reaction kinetics.