A sample of methane gas, \(\mathrm{CH}_{4}\), occupies a volume (L) \(58.2 \mathrm{~L}\) at a pressure of \(7.25 \mathrm{~atm}\). What volume will the gas occupy if the pressure is lowered to \(2.03\) atm?

The behavior of gases under various conditions of pressure, temperature, and volume is described by the gas laws, a series of fundamental principles in Chemistry. These laws are pivotal to understanding not only laboratory procedures but also the natural behavior of the atmosphere and technological applications such as engine operation. Boyle's Law, which was applied in the given problem, is one of these gas laws. It is defined for a constant temperature scenario – a thermodynamic state known as isothermal conditions. Other important gas laws include Charles's Law, which examines temperature-volume relationships, and Gay-Lussac's Law that looks at pressure-temperature correlations.

Gases are composed of particles that are in constant, random motion, and these laws are derived from how gas particles interact with one another and with their container. Real-life applications of gas laws include breathing – where the lungs expand and contract changing pressure and volume, to syringes and even the behavior of weather balloons.

The pressure-volume relationship, also referred to as Boyle’s Law in Chemistry, states that for a given mass of gas at constant temperature, the pressure exerted by the gas is inversely proportional to the volume it occupies. In simpler terms, if you compress a gas, making its volume smaller without changing its temperature, the pressure will rise, and vice versa.

To visualize this, imagine a balloon being squeezed; as its volume decreases, the pressure inside increases, making it harder to squeeze. This relationship is vital for chemists as well as engineers, who apply this knowledge in designing equipment and processes involving gases. The formula derived from this law, \(P1 * V1 = P2 * V2\), is a direct way to solve for an unknown when the conditions of a gas change, as seen in the textbook exercise.

Chemistry problem solving is a skill that involves analytical thinking and application of scientific principles to find solutions to quantitative and qualitative problems. This process often includes several steps: identifying the given information and what is being asked, understanding the relevant concepts and laws, setting up equations, and performing calculations. In the context of our exercise, the problem-solving steps start by identifying known variables (\(P1, V1, P2\)) and what to find (\(V2\)).

Next, by understanding Boyle’s Law, we rearrange the equation to isolate the variable of interest (\(V2 = (P1 * V1) / P2\)). The final steps involve substituting the values into the equation and calculating the answer while considering significant figures. This systematic approach to solving chemistry problems is crucial for achieving accurate results and gaining a deeper understanding of chemical processes.