Problem 32 - Differential Equations and Linear Algebra

Question

Chemicals \(A\) and \(B\) combine in the ratio \(2: 3 .\) Initially there are \(10 \mathrm{g}\) of \(\mathrm{A}\) and \(15 \mathrm{g}\) of \(\mathrm{B}\) present, and after \(5 \min , 10 \mathrm{g}\) of \(\mathrm{C}\) has been produced. Determine the amount of C that has been produced in 30 min. How long will it take for the reaction to be \(50 \%\) complete?

Step-by-Step Solution

Answer

The amount of C produced after 30 minutes is 300g, and the reaction will be 50% complete after 1 minute.

Step 1: Find the rate of the reaction for each chemical

Since the chemicals A and B have different initial amounts, we need to find the rate at which both chemicals react. We will use the amount of C produced in 5 minutes to find the rate of reaction for each chemical. Let the rate of reaction for A be a, and the rate of reaction for B be b. Then we can write the equations: \(2a = C_{produced}\) \(3b = C_{produced}\) Here, \(C_{produced}\) is the amount of C produced in 5 minutes, which is 10g. We can then solve these equations to find the values of a and b.

Step 2: Solve for the rate of reaction for A (a)

From our first equation, we can find the rate of reaction for A as follows: \(2a = C_{produced}\) \(2a = 10\) \(a = 5 \ g/min\)

Step 3: Solve for the rate of reaction for B (b)

Similarly, we can solve for the rate of reaction for B from our second equation: \(3b = C_{produced}\) \(3b = 10\) \(b = \frac{10}{3} \ g/min\)

Step 4: Determine the amount of C produced in 30 minutes

Now, we can use our rate of reaction values to calculate the amount of C produced after 30 minutes \(C_{30min} = 2a \times 30 \) \(C_{30min} = 2 \times 5 \times 30\) \(C_{30min} = 300 \ g\) The amount of C produced after 30 minutes is 300g.

Step 5: Calculate when the reaction is 50% complete

To find when the reaction is 50% complete, we need to determine when half of the initial amount of A or B has reacted. Let t be the time taken for the reaction to be 50% complete, then: \(2a \times t = 0.5 \times (initial \ amount \ of \ A)\) \(2 \times 5 \times t = 0.5 \times 10\) \(t = 1 \ min\) The reaction will be 50% complete after 1 minute.

Key Concepts

Reaction Rates

Reaction rates are a fundamental aspect of chemical kinetics, providing insight into how quickly a chemical reaction occurs. In the case of chemicals A and B reacting to form C, the rate at which C is produced can be determined using the amounts of A and B that react over a specific period. For our example, we learned from the given solution that in 5 minutes, 10 grams of C was produced.
This indicates the rate at which the reaction occurs under those specific conditions. Reaction rates can be affected by various factors, including temperature, concentration, and the presence of a catalyst. In this example, the rate of reaction was calculated separately for chemicals A and B, allowing us to determine how quickly each reacts to form the product C.

• The rate of reaction for A, denoted as a, is calculated as 5 grams per minute.
• The rate for B, denoted as b, is approximately 3.33 grams per minute.

Understanding reaction rates is crucial for predicting how long it takes to produce a desired amount of product or to reach a certain extent of reaction, like 50% completion in our example.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It helps determine how much of each substance is needed and produced during the reaction. In our exercise, the stoichiometry is presented as a ratio of 2:3 between chemicals A and B. This means that for every 2 units of A, 3 units of B are required to react completely.
This ratio is crucial in calculating the precise amounts needed to produce a certain amount of product. It also allows us to understand the relationship between reactants and predict the formation of products. Using stoichiometry, the step-by-step solution calculated the amount of C produced after 30 minutes based on the initial quantities of A and B.

• If 10 grams of C are produced in 5 minutes, the stoichiometric ratios ensure that the calculation of 300 grams of C in 30 minutes holds consistency with the initial problem statement.
• Stoichiometry helps determine when a reactant is completely consumed, which is pivotal for calculating the completion time of a reaction.

Chemical Reactions

Chemical reactions involve the breaking and forming of bonds between atoms, leading to a change in the substance's chemical structure. In our example, both chemicals A and B react to form a new substance, C. This transformation is based on the stoichiometric ratios provided and follows the conservation of mass principle, which states that the mass of reactants must equal the mass of the products.
Chemical reactions can be influenced by various factors, such as temperature and pressure; however, they will proceed according to their inherent kinetic pathways dictated by the reaction mechanism. The reaction from A and B to C involves specific steps where the reactants intermingle and progressively transform into products.

• The calculation of when the reaction is 50% complete involves determining when half of the initial reactants have transformed into products.
• This specific reaction reaches 50% completion in just 1 minute, showcasing the rapid nature of this chemical transformation.

By understanding these basic principles, one can predict the outcomes of chemical reactions and manipulate conditions to optimize yields or minimize undesirable byproducts.