Reaction rate is expressed in terms of changes in concentration of reactants and products. Write a balanced equation for the reaction with this rate expression: $$ \text { Rate }=-\frac{\Delta\left[\mathrm{CH}_{4}\right]}{\Delta t}=-\frac{1}{2} \frac{\Delta\left[\mathrm{O}_{2}\right]}{\Delta t}=\frac{1}{2} \frac{\Delta[\mathrm{H}, \mathrm{O}]}{\Delta t}=\frac{\Delta\left[\mathrm{CO}_{2}\right]}{\Delta t} $$

In chemistry, concentration changes tell us how the amount of a substance changes over time. For a reaction, we look at how the concentrations of reactants and products change as the reaction proceeds.

For example, consider the rate expression given in the exercise: \[-\frac{\Delta[\text{CH}_4]}{\Delta t} = -\frac{1}{2} \frac{\Delta[\text{O}_2]}{\Delta t} = \frac{1}{2} \frac{\Delta[\text{H}_2\text{O}]}{\Delta t} = \frac{\Delta[\text{CO}_2]}{\Delta t}\].

This equation tells us how the concentrations of CH4, O2, H2O, and CO2 change over time. \[-\frac{\Delta[\text{CH}_4]}{\Delta t}\] represents the rate at which methane (CH4) is consumed. Because of the negative sign, it indicates a decrease in concentration.

Similarly, \[-\frac{1}{2} \frac{\Delta[\text{O}_2]}{\Delta t}\] shows that oxygen (O2) is also being consumed, and it is being consumed twice as fast as CH4.

On the other hand, \[\frac{1}{2} \frac{\Delta[\text{H}_2\text{O}]}{\Delta t}\] and \[\frac{\Delta[\text{CO}_2]}{\Delta t}\] represent the formation rates of water (H2O) and carbon dioxide (CO2), respectively. Their positive values indicate they are being produced during the reaction.

Stoichiometric coefficients are numbers that represent the number of molecules or moles of each substance that participate in a chemical reaction. These coefficients are determined from the balanced chemical equation and help understand the proportions in which substances react and are produced.

In the rate expression: \[-\frac{\Delta[\text{CH}_4]}{\Delta t} = -\frac{1}{2} \frac{\Delta[\text{O}_2]}{\Delta t} = \frac{1}{2} \frac{\Delta[\text{H}_2\text{O}]}{\Delta t} = \frac{\Delta[\text{CO}_2]}{\Delta t}\],

we can deduce the stoichiometric coefficients:

• CH4 has a coefficient of 1 (directly from the expression for CH4).
• O2 has a coefficient of 2 (since the rate is halved, indicating two O2 molecules are used for every CH4 molecule).
• H2O has a coefficient of 2 (since its rate is halved, implying two H2O molecules are produced for each CO2 molecule).
• CO2 has a coefficient of 1 (directly from the expression for CO2).

These stoichiometric coefficients are crucial for writing the balanced chemical equation.

A balanced chemical equation accurately represents a chemical reaction, showing the correct proportions of all reactants and products. Balancing chemical equations ensures that the Law of Conservation of Mass is satisfied; the number of atoms for each element must be the same on both sides of the equation.

For the reaction in the exercise, we can construct the balanced chemical equation based on the stoichiometric coefficients deduced earlier:

\[\text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O}\].

This equation shows that one molecule of methane (CH4) reacts with two molecules of oxygen (O2) to produce one molecule of carbon dioxide (CO2) and two molecules of water (H2O).

The coefficients in front of each substance (1 for CH4 and CO2, 2 for O2 and H2O) ensure the equation is balanced. We have the same number of carbon (C), hydrogen (H), and oxygen (O) atoms on both sides:

• 1 C atom in CH4 and 1 C atom in CO2.
• 4 H atoms in CH4 and 4 H atoms in 2 H2O.
• 4 O atoms in 2 O2 and 4 O atoms in CO2 + 2 H2O.