40 milligram diatomic volatile substance \(\left(X_{2}\right)\) is converted to vapour that displaced \(4.92 \mathrm{~mL}\) of air at 1 atm and \(300 \mathrm{~K}\). Atomic weight of element \(X\) is nearly: (a) 400 (b) 240 (c) 200 (d) 100

Understanding the behavior of gases is crucial in chemistry, and the Ideal Gas Law provides a practical tool for relating the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. The law is summarized by the equation
\(PV = nRT\),
where R is the ideal gas constant. An 'ideal' gas is a hypothetical gas whose molecules have no intermolecular forces and occupy no volume. Although real gases do not perfectly adhere to these assumptions, the Ideal Gas Law offers a close approximation for many conditions.

In solving problems that involve the Ideal Gas Law, students must ensure that their units are consistent. For example, pressure should be in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K). When given other units like milliliters or Celsius, conversions are necessary:

• 1 mL = 0.001 L
• Degrees Celsius to Kelvin = Degrees Celsius + 273.15

These conversions are essential to avoid any discrepancies in calculations. The Ideal Gas Law not only helps to calculate the number of moles of a gas given PVT conditions but also lays the ground for further calculations, such as determining the molar mass of a substance.

Molar mass is a fundamental concept in stoichiometry that is defined as the mass of one mole of a substance. It is typically measured in grams per mole (g/mol) and is vital for converting between mass and moles in chemical equations. The determination of molar mass becomes particularly significant when dealing with reactions involving gases, as in the original exercise.

To determine the molar mass of a gas, analytical methods such as mass spectrometry may be used; however, the molar mass can also be calculated from experimental data using the Ideal Gas Law. In the given problem, after finding the number of moles (n) of a gas using the Ideal Gas Law, the molar mass (M) can be calculated by dividing the mass of the gas sample by the number of moles:
\(M = \frac{\text{mass of the gas sample (g)}}{n}\).
If the substance is a compound, such as the diatomic molecule \(X_2\) mentioned in the exercise, this gives us the molar mass of the entire molecule. To find the atomic weight of a single atom within the molecule, further simple arithmetic is required.

Stoichiometry is the study of quantitative relationships in chemical reactions, often described as the 'mathematics of chemistry.' It is based on the laws of conservation of mass and the fixed composition of compounds. Stoichiometric calculations enable chemists to predict the amounts of reactants and products involved in a reaction.

The basic principle of stoichiometry is that in a chemical reaction, atoms are neither created nor destroyed, they are merely rearranged. Consequently, the amount of each element must be the same in the reactants and products. Therefore, knowing the molar mass of the substances involved is crucial for stoichiometry since it allows us to relate mass to moles and vice versa. For instance, to find the stoichiometric ratios in a balanced chemical equation, it's often helpful to convert all quantities to moles.

In the context of the original exercise, stoichiometry comes into play when translating the mass of a diatomic volatile substance (given in milligrams) into moles through the Ideal Gas Law and then into the atomic weight of the element (X). This process illustrates the interconnectedness of stoichiometry with other core chemical concepts.

Tackling chemical problems like calculating an element's atomic weight from experimental data is a multi-step process that involves understanding and applying chemical principles and equations. The steps typically begin with converting units, applying laws such as the Ideal Gas Law for gases, and performing stoichiometric calculations.

Effective chemical problem-solving requires a systematic approach:

• Clearly define what is known from the problem and what needs to be determined.
• Identify the correct chemical principles and equations to use.
• Perform unit conversions where necessary.
• Carefully execute the calculations step by step.
• Check the answer for consistency with the physical meaning and units.

In the given exercise, such an approach helps to seamlessly navigate from the volume of a gas to its moles, then to its molar mass, and finally to the atomic weight of the component element. This methodology is not only useful in solving homework problems but also crucial in laboratory settings and real-world applications.