1\. A container with a piston contains a sample of gas. Initially, the pressure in the container is exactly 1 atm, but the volume is unknown. The piston is adjusted so that the volume is \(0.155 \mathrm{~L}\) and the pressure is \(956 \mathrm{~mm}\) Hg; what was the initial volume? 2\. The pressure of \(12.5 \mathrm{~L}\) of a gas is \(0.82 \mathrm{~atm}\). If the pressure changes to \(1.32 \mathrm{~atm}\), what will the final volume be? A sample of helium gas has a pressure of \(860.0 \mathrm{~mm}\) Hg. This gas is transferred to a different container having a volume of \(25.0 \mathrm{~L}\); in this new container, the pressure is determined to be \(770.0 \mathrm{~mm}\) Hg. What was the initial volume of the gas?

To better understand the behavior of gases under various conditions, scientists have formulated principles known as gas laws. These include Boyle's Law, Charles's Law, Gay-Lussac's Law, and the Ideal Gas Law, among others.

Boyle's Law, in particular, focuses on the pressure-volume relationship of a gas held at a constant temperature. The law states that the pressure of a fixed amount of gas is inversely proportional to its volume when temperature is held constant. The mathematical representation of Boyle's Law is often written as \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume, respectively.

Understanding Boyle's Law is crucial because it applies to several real-world situations, such as breathing, where the lungs expand and contract, altering the volume and consequently the pressure of the air inside.

The pressure volume relationship highlighted in Boyle's Law is a core concept in understanding how gases behave under pressure changes. When the volume of a gas is decreased, its particles are squeezed closer together, causing more frequent collisions and thus increasing the pressure. Conversely, when the volume is increased, the particles are further apart, and the pressure decreases, as the collisions become less frequent.

Practical Application

Take a syringe for example: when you pull the plunger back, the volume inside increases, leading to a decrease in pressure which draws liquid into the syringe. Pushing the plunger in the opposite direction decreases the volume and the liquid is expelled due to increased pressure. These concepts are directly applicable to many scientific fields, including respiratory physiology, engineering, and more.

To solve problems involving this relationship, it's essential to understand both the direct and inverse proportionality, and ensure consistent units of measurement for accurate calculations. For instance, when converting from mmHg to atm or vice versa, using a conversion factor is necessary to maintain the integrity of the calculations.

Solving chemistry problems, especially those related to gas laws, requires a methodical approach. It's important to follow a series of steps to ensure that the problems are approached in a consistent and effective manner. Here's a simplified method often used by chemists and students alike:

• Identify which gas law applies to the problem at hand.
• Write down the known variables and what you are solving for.
• Convert units where necessary to maintain consistency throughout the problem.
• Rearrange the gas law formula to solve for the unknown variable.
• Perform the calculations carefully and check that the units cancel out as expected.

By breaking down the problem, converting units (such as atm to mmHg), rearranging equations to isolate the unknown variable, and methodically plugging in the given information, students can competently tackle and solve complex problems. Even the most challenging chemistry exercises become more approachable when each step is followed with care and precision, as illustrated in the given Boyle's Law problems.