## Explanation

### Understanding the Coefficient in a Balanced Chemical Equation

In the context of a balanced chemical equation, the **coefficient** is a numerical value placed before a chemical formula. It holds several critical roles, such as:

### Quantitative Representation

The coefficient indicates the **relative number of moles** of each substance involved in the reaction. This is essential for maintaining the **Law of Conservation of Mass**, which states that mass cannot be created or destroyed in a chemical reaction.

### Stoichiometric Proportions

The coefficients provide the **stoichiometric proportions**, which are necessary for accurately predicting the amounts of reactants and products. For example, in the balanced chemical equation for the combustion of methane:

Here, the coefficient "2" in front of $\text{O}_2$ and $\text{H}_2\text{O}$ indicates that **2 moles of oxygen gas react with 1 mole of methane to produce 1 mole of carbon dioxide and 2 moles of water**.

### Representation in Multi-step Reactions

In complex reactions that occur in multiple steps, the coefficients offer insight into **intermediate stages**. For instance, in a reaction sequence involving the formation and decomposition of intermediates, coefficients help identify the **intermediate compounds' quantities** at each step.

### Mole Ratios

One of the primary uses of coefficients is in deriving **mole ratios**. These ratios are calculated from the coefficients and are vital for **reaction stoichiometry calculations**, ensuring that the proportions of reactants and products are consistent in chemical analysis. For instance, for the balanced reaction:

The mole ratio can be expressed as:

$\frac{\text{a}}{\text{b}} = \frac{\text{c}}{\text{d}}$By understanding the coefficients and their relationships, chemists can predict how much of a reactant is needed to completely react with a given amount of another reactant or how much product will be formed from a certain quantity of reactants.

### Conservation of Atoms

Lastly, coefficients ensure the **conservation of atoms** for each element involved in the reaction, aligning with the fundamental principles of chemical reactions. This means the total number of atoms for each element on the reactants' side is equal to the total number on the products' side.

For example, balancing the equation for the reaction of hydrogen and oxygen to form water:

$2\text{H}_2 + \text{O}_2 \rightarrow 2\text{H}_2\text{O}$The coefficients "2" before $\text{H}_2$ and $\text{H}_2\text{O}$ ensure that there are **4 hydrogen atoms on both sides** and the coefficient "1" before $\text{O}_2$ ensures there are **2 oxygen atoms on both sides**.

To sum up, coefficients are crucial not only for balancing chemical equations but also for providing a clear quantitative framework that chemists use for various calculations and predictions in chemical reactions.