Consider the general reaction $$ \mathrm{aA}+\mathrm{bB} \longrightarrow \mathrm{cC} $$ and the following average rate data over some time period \(\Delta t\) : $$ \begin{aligned} -\frac{\Delta \mathrm{A}}{\Delta t} &=0.0080 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \\ -\frac{\Delta \mathrm{B}}{\Delta t} &=0.0120 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \\ \frac{\Delta \mathrm{C}}{\Delta t} &=0.0160 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \end{aligned} $$ Determine a set of possible coefficients to balance this general reaction.

We are given the average rate data for changes in concentration of reactants A, B, and product C over a certain time period: \[ \begin{aligned} -\frac{\Delta \mathrm{A}}{\Delta t} &=0.0080 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \\ -\frac{\Delta \mathrm{B}}{\Delta t} &=0.0120 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \\ \frac{\Delta \mathrm{C}}{\Delta t} &=0.0160 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \end{aligned} \]

To find the relationship between the change in concentrations of A, B, and C, we can first divide the rate of change of B by the rate of change of A and then the rate of change of C by the rate of change of A: \[ \frac{-\frac{\Delta \mathrm{B}}{\Delta t}}{-\frac{\Delta \mathrm{A}}{\Delta t}} =\frac{0.0120}{0.0080} = 1.5 \] \[ \frac{\frac{\Delta \mathrm{C}}{\Delta t}}{-\frac{\Delta \mathrm{A}}{\Delta t}} =\frac{0.0160}{0.0080} = 2 \] This tells us that B is changing 1.5 times as fast as A, while C is changing 2 times as fast as A.

Now, we use the relationships we found in Step 2 to balance the reaction: \[ \mathrm{aA} + \mathrm{bB} \longrightarrow \mathrm{cC} \] Since B is changing 1.5 times as fast as A, the coefficient b can be expressed as 1.5a. Similarly, since C is changing 2 times as fast as A, the coefficient c can be expressed as 2a. To find a set of possible coefficients, we can use multiples of these ratios. For example, we can set a = 2, which results in: \[ \mathrm{aA} = 2\mathrm{A} \] \[ \mathrm{bB} = 1.5(2) = 3\mathrm{B} \] \[ \mathrm{cC} = 2(2) = 4\mathrm{C} \] So one set of possible coefficients is a = 2, b = 3, and c = 4. And the balanced reaction would be: \[ 2\mathrm{A} + 3\mathrm{B} \longrightarrow 4\mathrm{C} \] However, there might be other possible combinations fitting the rate data. The most important part is to maintain the ratios between a, b, and c.