A \(15.0\) - \(\mathrm{L}\) tank is filled with \(\mathrm{H}_{2}\) to a pressure of \(2.00 \times 10^{2}\) atm. How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of \(1.00\) atm from the tank? Assume that there is no temperature change and that the tank cannot be emptied below \(1.00 \mathrm{~atm}\) pressure.

Gas laws govern the behavior of gases under various conditions of temperature, pressure, and volume. These laws are crucial for predicting how gases will respond when these conditions change. We refer to Boyle's Law, Charles's Law, Gay-Lussac's Law, and the Combined Gas Law, all of which are derived from the ideal gas law equation.

Boyle's Law, in particular, focuses on the relationship between pressure and volume, indicating that for a given mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, Boyle's Law is expressed as:
\( P_1V_1 = P_2V_2 \),
where \(P1\) and \(V1\) represent the initial pressure and volume, and \(P2\) and \(V2\) are the final pressure and volume, respectively. Understanding this law is key to many scientific and industrial applications, including calculating how much gas can be stored or transferred under changing pressures.

Applying Boyle's Law is straightforward once we understand the relationship between pressure and volume. Let's say a diver descends deeper into the ocean. As the depth increases, the water pressure increases, and according to Boyle's Law, the volume of air in the diver's lungs would decrease proportionally. Similarly, when filling balloons with gas, Boyle's Law helps to determine the final volume once the pressure is changed.

When dealing with calculations, it's important to keep the units consistent and remember that the product of the pressure and volume before the change must equal the product after the change, provided that the temperature remains constant. The practical application of Boyle's Law enables us to perform accurate calculations for a variety of real-life situations, including the inflation of balloons, as in the textbook exercise we're discussing.

Calculating gas volume using Boyle's Law can be incredibly useful in many scenarios, such as predicting the behavior of gases under pressure or determining the capacity needed to store gases. In our exercise, the challenge was to find out how many balloons could be inflated from a tank of hydrogen at a specific pressure without a temperature change.

By rearranging Boyle's Law, we solve for the unknown volume after pressure changes:
\( V_2 = \frac{P_1V_1}{P_2} \).
Here, \( V_2 \) is the volume we want to find, \( P_1 \) and \( V_1 \) are the known initial pressure and volume, and \( P_2 \) is the final pressure. After calculating the available gas volume that can be used for inflation, we divided this by the volume needed for one balloon to find out the number of balloons that can be inflated. Such calculations are not only essential for homework problems but also for various applications in fields like engineering, environmental science, and healthcare.