Chapter 8 Conceptual Project:

Creating a New Element

Recall from Section 3.7 our discussion of a chemical reaction where reactants A and B produce a new product substance C, a process represented by

$A+B\to C$.

In this project, we will derive and use a differential equation that describes such a process.

1. Suppose that in the above reaction for each gram of reactant A, b grams of B are used to form C. If we start with initial amounts $A_0$ and $B_0$, respectively, and $X\left(t\right)$ denotes in grams the amount of substance C already formed at time t, find the remaining amounts of reactants A and B at any time during the process.

2. Given that the rate of formation of substance C at any time is proportional to the product of the remaining amounts of reactants A and B, respectively, find a differential equation in terms of $X\left(t\right)$ that describes the process.

   (As in Question 1, let $A_0$ and $B_0$ stand for the initial amounts.)

3. Suppose a product substance C is being formed from reactant substances A and B and that for each gram of substance A, $3$ grams of B are used to form C. As in Question 1, let $X\left(t\right)$ denote the amount of C formed at time t, and assume that the initial amounts of reactants A and B are $A_0=60\text{grams}$ and $B_0=40\text{grams}$, respectively. Find the initial value problem describing this reaction. (Hint: Use your answer to Question 2.)

4. If $20$ grams of the product compound form during the first $5$ minutes, use the model you obtained in Question 3 to predict how much of the product compound C is present $10$ minutes into the process.

5. Use your model from Question 3 to predict what happens as $t\to \infty$. Interpret your answer.