Problem 43 - Chemistry for Allied Health

Question

A sample of oxygen gas has an initial pressure and volume of \(1.0 \mathrm{~L}\) and \(1.0\) atm. What is the final pressure if the volume is compressed to \(0.50 \mathrm{~L}\) ? Assume constant moles and temperature.

Step-by-Step Solution

Answer

The final pressure of the gas is 2.0 atm.

Step 1: Identify the Law

This problem requires us to use Boyle's Law, which describes the relationship between pressure and volume of a gas at constant temperature. According to Boyle's Law, \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.

Step 2: Define Initial Conditions

The initial pressure \( P_1 \) is given as 1.0 atm, and the initial volume \( V_1 \) is 1.0 L. These values will be used to calculate the final pressure \( P_2 \) once the volume changes.

Step 3: Apply Boyle's Law

Using the relationship \( P_1V_1 = P_2V_2 \), we substitute the known values into the formula:\[1.0 \text{ atm} \times 1.0 \text{ L} = P_2 \times 0.50 \text{ L}\]This equation will help us solve for the unknown final pressure \( P_2 \).

Step 4: Solve for Final Pressure

Rearrange the equation from the previous step to solve for \( P_2 \):\[P_2 = \frac{1.0 \text{ atm} \times 1.0 \text{ L}}{0.50 \text{ L}}\]Upon simplification, this yields:\[ P_2 = 2.0 \text{ atm} \]

Key Concepts

Pressure and Volume Relationship

The relationship between pressure and volume is fundamental to understanding how gases behave under different conditions. Boyle's Law clearly demonstrates this relationship, stating that at a constant temperature, the pressure of a given quantity of gas is inversely proportional to its volume. This means:

• When the volume decreases, the pressure increases.
• Conversely, when the volume increases, the pressure decreases.

This inverse relationship can be mathematically represented by the equation: \[P \times V = ext{constant}\]In simple terms, if you compress a gas by decreasing its volume, its molecules will collide more frequently against the walls of the container, resulting in higher pressure. This principle is what allows balloons to pop when overinflated and is also applied in many real-world devices like pneumatic systems.

Gas Laws

Gas laws are a set of rules that describe how gases behave in various conditions. Boyle's Law is just one part of the broader category of gas laws, which also includes:

• Charles's Law: Describes how gases expand when the temperature increases if the pressure is constant.
• Avogadro's Law: States that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules.

Boyle's Law is particularly useful in scenarios where the temperature remains constant. It helps predict changes in pressure when gas volume changes. Collectively, these laws can be combined into the ideal gas law, which unifies the relationships between pressure, volume, temperature, and the number of gas molecules. Understanding these laws not only aids in solving textbook problems but also provides valuable insights into everyday phenomena such as how air pressure in car tires changes with temperature.

Constant Temperature Gas Behavior

The behavior of gases at constant temperature is especially important when applying Boyle's Law. Here, temperature stability ensures that the only factors affecting gas pressure and volume changes are themselves. At constant temperature:

• A decrease in volume results in an increase in pressure.
• An increase in volume results in a decrease in pressure.

This consistency is because temperature is directly linked to the kinetic energy of gas molecules. As long as the temperature remains unchanged, molecules move at a steady speed. Practically, this means you can rely on predictable results when applying Boyle's Law in scenarios such as inflating a bike tire, or in calculating the behavior of gases in laboratory experiments. Stable temperature conditions isolate pressure and volume as the primary variables, making it easier to predict and understand gas behavior in real-world applications.