An evacuated bulb of unknown volume is filled with a sample of \(\mathrm{H}_{2}\) gas at a temperature \(T\). The pressure of the gas in the bulb is \(756 \mathrm{~mm} \mathrm{Hg}\). A portion of the \(\mathrm{H}_{2}\) gas is transferred to a different flask and found to occupy a volume of \(40.0 \mathrm{~mL}\) at \(1.00\) atm and the same temperature \(T\). The pressure of the \(\mathrm{H}_{2}\) gas remaining in the original bulb drops to 625 \(\mathrm{mm} \mathrm{Hg}\) at the same temperature \(T\). Assuming \(\mathrm{H}_{2}\) is an ideal gas, what is the volume of the bulb?

Understanding chemical stoichiometry is pivotal when solving problems related to the quantities of reactants and products in chemical reactions. For students tackling IIT-JEE Chemistry problems, grasping the principles of stoichiometry is crucial because it underpins the calculations of chemical equations.

Stoichiometry allows you to determine the relationships between volumes of gases in reactions, given that these gases behave ideally. When utilizing stoichiometry in the context of the Ideal Gas Law, one can predict how changes in conditions (like temperature and pressure) affect the amount of substance (moles) involved. For example, if you're given the volume and pressure of a gas at a certain temperature, stoichiometry enables you to calculate the number of moles of the gas using the Ideal Gas Law equation, which is \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for moles of gas, \(R\) for the gas constant, and \(T\) for temperature in Kelvin.

This concept is used in our original exercise to connect the volume of the original bulb with the portion of \(H_2\) gas transferred to the different flask, using the relationship between the pressure, volume, and number of moles.

The gaseous state of matter is defined by its ability to fill any container, taking its shape, and by its particles moving freely and rapidly. In this state, the distance between gas particles is large compared to the size of the particles themselves, which leads to negligible intermolecular forces. This property is why gases are well-described by the Ideal Gas Law.

For exercises related to the gaseous state, it is essential to recognize that temperature affects gas volume and pressure (Charles's law and Gay-Lussac's law, respectively), while the volume is inversely proportional to pressure at constant moles and temperature (Boyle's law). The assumptions of the Ideal Gas Law hinge upon these observations and allow us to simplify and solve real-life chemistry problems.

In the context of our problem, assuming hydrogen gas behaves ideally means that we can apply the Ideal Gas Law directly without considering intermolecular forces or the volume occupied by the gas particles themselves. Keeping the temperature constant simplifies the problem further, as we can relate the pressure and volume of \(H_2\) gas before and after the transfer directly.

The relationship between the pressure and volume of a gas is a foundational concept in chemistry, especially in solving gas-related exercises. Boyle's law states that for a given mass of gas at constant temperature, the volume of a gas is inversely proportional to its pressure. When the volume goes up, the pressure goes down, and vice versa, as long as the temperature and amount of gas remain unchanged.

In our exercise, knowing this relationship helps us understand why the pressure of the gas drops after a portion is transferred to another flask. While the amount of \(H_2\) gas decreases, the volume of the original bulb remains the same, hence the pressure must decrease. By applying Boyle's law and the Ideal Gas Law, we can mathematically relate these changes to determine the unknown volume of the bulb, further illustrating the utility of understanding pressure-volume relationships in chemical calculations.

Students preparing for the Indian Institutes of Technology Joint Entrance Examination (IIT-JEE) face complex problems in Chemistry that test their understanding and application of various concepts, including the behavior of gases and stoichiometry. Such exams require a deep understanding of underlying principles and the ability to apply them to novel situations.

Problems like the one in our textbook involve multiple concepts that students must intertwine, and success hinges on grasping the finer details of each concept. For instance, IIT-JEE problems often require students to perform unit conversions, apply laws consistently, and extract information from given data methodically.

By practicing problems that combine different aspects of chemistry, like the ideal gas law with stoichiometry, and understanding the relationships between pressure and volume, students can develop a robust problem-solving methodology. This is critical for success in exams like the IIT-JEE, where time management and accuracy are key to selecting the proper steps to find a solution, as we demonstrate in the step-by-step resolution of the textbook exercise.