What volume will each of the following occupy at STP? (a) \(1.80 \times 10^{24}\) molecules of \(\mathrm{SO}_{3}\) (b) \(7.5 \mathrm{~mol} \mathrm{} \mathrm{C}_{2} \mathrm{H}_{6}\) (c) \(25.2\) g chlorine

Understanding the concept of Avogadro's number is crucial in performing gas volume calculations at STP (Standard Temperature and Pressure). Avogadro's number, typically denoted as \( 6.022 \times 10^{23} \), represents the number of units, such as atoms or molecules, in one mole of any substance. When dealing with gases, Avogadro's hypothesis asserts that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules.

This constant allows us to relate the microscopic world of atoms and molecules to the macroscopic one that we can measure, which is extremely helpful when converting between the number of entities and the amount in moles. For instance, in the exercise solution, Avogadro's number was used to determine the number of moles of sulfur trioxide (\( \text{SO}_{3} \)) given its number of molecules.

The ideal gas law is an equation of state for a hypothetical ideal gas. It is expressed as \( PV = nRT \), where \( P \) stands for pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. In the context of the textbook exercise, STP conditions are implied, meaning a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere.

By knowing that one mole of an ideal gas occupies 22.4 liters at STP, we can simplify the equation for volume calculations without needing to directly solve for each variable. In the examples given, the ideal gas law helps to calculate the volume that certain amounts of gases would occupy when the number of moles or the mass and the molar mass of the gas are known.

Molar volume is defined as the volume occupied by one mole of any gas at a specific temperature and pressure. For gases, a common reference is the molar volume at STP, which is 22.4 liters per mole. This value assumes that the gas behaves perfectly, according to the kinetic molecular theory, which simplifies complex real-world behavior to an ideal model.

In the exercises, this concept of molar volume is directly used to find the volume occupied by a given amount of gas at STP. By simply multiplying the molar volume by the number of moles calculated, we can find the final volume for each gas in question. It's important for students to understand not only how to use the molar volume in calculations but also why it is a vital part of understanding gas properties under standard conditions.