How many milliliters of oxygen are required to react completely with \(175 \mathrm{~mL}\) of \(\mathrm{C}_{4} \mathrm{H}_{10}\) if the volumes of both gases are measured at the same temperature and pressure? The reaction is $$ 2 \mathrm{C}_{4} \mathrm{H}_{10}(g)+13 \mathrm{O}_{2}(g) \longrightarrow 8 \mathrm{CO}_{2}(g)+10 \mathrm{H}_{2} \mathrm{O}(g) $$

Avogadro's Law is a fundamental principle in chemistry, particularly when it comes to understanding gases. Stated simply, Avogadro's Law posits that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules or moles.

This law forms the foundation for the concept of molar volume, which is the volume one mole of gas occupies at a specified temperature and pressure, usually at Standard Temperature and Pressure (STP), which is 273.15 K (0 °C) and 1 atmosphere. At STP, one mole of any gas occupies approximately 22.4 liters.

For students grappling with gas volume calculations in a chemical reaction, remembering that the volumes are directly proportional to the number of moles (as long as the temperature and pressure are unchanged) is key. This means that if you know the volume of one gas in a reaction, you can determine the volume of another gas, assuming the reaction conditions remain constant.

When applying Avogadro's Law, it is essential to ensure that the gases involved are ideal or close to ideal conditions. Real gases at high pressures and low temperatures may not follow Avogadro's Law perfectly due to intermolecular forces and the volume occupied by the gas particles themselves.

Chemical reaction equations serve as the blueprint for chemical reactions. They tell us which reactants combine and what products are formed, as well as the proportions in which these substances react and are produced. A balanced chemical equation is vital because it respects the law of conservation of mass, meaning the number of atoms for each element involved in the reaction must be the same on both the reactant side and the product side.

It's crucial to understand how to read and interpret these equations. For instance, coefficients in front of a chemical formula indicate the number of moles of each substance involved in the reaction. The equation from our exercise,  \(2 \text{C}_4\text{H}_{10}(g) + 13 \text{O}_2(g) \rightarrow 8 \text{CO}_2(g) + 10 \text{H}_2\text{O}(g)\), tells us that two moles of butane (\(\text{C}_4\text{H}_{10}\)) react with thirteen moles of oxygen (\(\text{O}_2\)) to produce eight moles of carbon dioxide (\(\text{CO}_2\)) and ten moles of water (\(\text{H}_2\text{O}\)).

By fully grasping the chemical equation, students can more easily embark on exercises involving stoichiometry, predicting the outcomes of reactions, and understanding the quantitative relationships between reactants and products.

When it comes to applying stoichiometry to gas volume calculations, it's essential to understand that the volumes of gases involved in a chemical reaction can be directly related to the moles of gases, provided conditions of temperature and pressure are unchanged. This follows from Avogadro's Law, as previously discussed.

In the context of the exercise, the calculation of the oxygen volume required to react with a known volume of butane (\(\text{C}_4\text{H}_{10}\)) is straightforward once you understand the mole-to-volume relationship. Here's the process simplified:

• Identify the volume ratio from the balanced reaction equation, which reflects the mole ratio.
• Set up a proportion, based on Avogadro's Law, that relates the known volume of butane to the unknown volume of oxygen.
• Calculate the unknown volume using simple algebra.

It's important to note that a common error students might make is to forget that this method only works when comparing volumes of gases under the same conditions of temperature and pressure. Variations in these conditions would require a different approach, such as using the Ideal Gas Law (\(PV=nRT\)), where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature in Kelvin.