Question

Explain why the coefficients in a balanced chemical equation represent not only molar amounts but also relative volumes for gases.

Short answer

The coefficients in a balanced chemical equation represent not only molar amounts but also relative volumes for gases because stoichiometry gives us the molar ratios needed for the reaction, and the ideal gas law, under constant temperature and pressure, allows us to relate the moles and volumes of gases. Thus, the coefficients indicate the relationship between reactants and products in terms of both their amounts and their volumes. For example, in the reaction \(2H_2 + O_2 \rightarrow 2H_2O\), the coefficients suggest a volume ratio of 2:1 for hydrogen to oxygen and a 1:1 ratio for water to hydrogen.

Step-by-step solution

Understanding Stoichiometry

Stoichiometry is the study of the relationship between the amounts of reactants and products in a chemical reaction. In a balanced chemical equation, the coefficients of the reactants and products represent the molar ratios in which the substances react and are formed. For example, consider the following balanced chemical equation: \[2H_2 + O_2 \rightarrow 2H_2O\] The coefficients tell us that 2 moles of hydrogen gas (H2) react with 1 mole of oxygen gas (O2) to produce 2 moles of water (H2O). These coefficients give us the stoichiometric or molar ratios needed for the reaction to occur.

Introducing the Ideal Gas Law

The ideal gas law can be used to relate the volume, pressure, temperature, and amount (in moles) of a gas. The ideal gas law is given by the equation: \[PV = nRT\] Where: P = pressure of the gas V = volume of the gas n = moles of the gas R = ideal gas constant T = temperature (in Kelvin) of the gas

Relating Coefficients to Volumes of Gases

Now, let's look at the relationship between moles and volume in the ideal gas law. If we rearrange the equation so that volume is on one side of the equation and everything else is on the other side, we get: \[V = \frac{nRT}{P}\] Assuming that the pressure (P) and temperature (T) are constant for all gases involved in the reaction, the ratio between the volumes of the gases is equal to the ratio between their amounts (in moles): \[\frac{V_1}{V_2} = \frac{n_1}{n_2}\] Returning to our earlier example: \[2H_2 + O_2 \rightarrow 2H_2O\] The coefficients in the balanced equation tell us the molar ratios between the reactants and products. These molar ratios can also be expressed as volume ratios: \[\frac{V_{H_2}}{V_{O_2}} = \frac{2}{1} \] and \[\frac{V_{H_2O}}{V_{H_2}} = \frac{2}{2}\] So, the volume ratio of hydrogen to oxygen in this reaction is 2:1, and the volume ratio of water to hydrogen is 1:1, assuming constant pressure and temperature.

Conclusion

The coefficients in a balanced chemical equation represent not only the molar amounts but also the relative volumes for gases involved in the reaction because stoichiometry deals with the relationship between the amounts (in moles) of reactants and products. Under constant temperature and pressure, the ideal gas law allows us to relate the moles and volumes of gases in the reaction, leading to the conclusion that the coefficients represent both molar amounts and relative volumes for gases.

Key concepts

Stoichiometry

Stoichiometry is a section of chemistry that deals with the amounts and proportions of reactants and products in chemical reactions. Understanding stoichiometry is essential because it allows chemists to predict how much of each substance is needed to react completely without any waste and how much product will be generated.

At the heart of stoichiometry are balanced chemical equations, which illustrate the molar ratios of the substances involved in a chemical reaction. For example, the balanced equation for the combustion of methane is:
\[CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O\]
This equation shows that one mole of methane reacts with two moles of oxygen to produce one mole of carbon dioxide and two moles of water. Stoichiometry uses these molar ratios to calculate necessary amount of reactants or the expected quantity of a product, thus guiding the practical aspect of running a chemical reaction.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that connects four key properties of gas: pressure (P), volume (V), number of moles (n), and temperature (T). This relationship is captured in the simple formula:
\[PV = nRT\]
Where R is the ideal gas constant. This law assumes gas particles are small, hard spheres with insignificant volume, no intermolecular forces, and collisions that are perfectly elastic.

In practice, the ideal gas law allows us to predict how a gas will behave under different conditions. For instance, if you increase the pressure exerted on a gas, its volume will decrease if the temperature and the number of moles remain constant. This law is a key principle in understanding gas behavior and is widely applied in both chemical reactions involving gases and various industrial processes.

Molar Ratios

Molar ratios are the proportions of reactants and products involved in a chemical reaction and are directly derived from the coefficients of a balanced chemical equation. They are a critical component of stoichiometry as they determine how much of each substance will interact and how much product will form.

For example, consider the synthesis of ammonia from nitrogen and hydrogen:\[N_2 + 3H_2 \rightarrow 2NH_3\]
The molar ratio of hydrogen to nitrogen is 3:1, meaning three moles of hydrogen gas are required for every one mole of nitrogen gas to produce ammonia. Similarly, the reaction yields two moles of ammonia for every one mole of nitrogen used. These ratios are essential for calculating reactant quantities and expected yields, especially when planning industrial-scale synthesis of chemicals.

Chemical Reaction

A chemical reaction is a process where substances, known as reactants, transform into different substances, known as products. During these reactions, chemical bonds are broken and new ones are formed, which can lead to changes in energy, color, temperature and even the state of the materials.

Reactions can be categorized into several types such as synthesis, decomposition, single replacement, and double replacement. An essential factor in understanding and describing chemical reactions is the law of conservation of mass, which states that the total mass of reactants must equal the total mass of products. Notably, the coefficients in balanced chemical equations convey not just the molar ratios of the reactants and products but also, in the case of gases, their relative volumes - another consequence of the ideal gas law. In conclusion, a firm grasp on chemical reactions facilitates a deeper comprehension of how different substances interact and the stoichiometry that governs these interactions.