Problem 91 - Chemistry: The Central Science

Question

A gas bubble with a volume of \(1.0 \mathrm{~mm}^{3}\) originates at the bottom of a lake where the pressure is \(3.0 \mathrm{~atm}\). Calculate its volume when the bubble reaches the surface of the lake where the pressure is 695 torr, assuming that the temperature doesn't change.

Step-by-Step Solution

Answer

The volume of the gas bubble when it reaches the surface of the lake is approximately \(3.27 \mathrm{~mm}^3\), assuming that the temperature doesn't change.

Step 1: Convert initial pressure to torr

To convert the initial pressure from atm to torr, use the conversion factor: 1 atm = 760 torr Initial Pressure in torr = (Initial Pressure in atm) * (760 torr/atm)

Step 2: Apply Boyle's Law

According to Boyle's Law: \(P_1V_1 = P_2V_2\) We can rearrange this equation to find the final volume (\(V_2\)): \(V_2 = \frac{P_1V_1}{P_2}\) Now, plug in the values: - Initial Pressure (\(P_1\)): 3.0 atm (converted to torr in Step 1) - Initial Volume (\(V_1\)): 1.0 mm³ - Final Pressure (\(P_2\)): 695 torr

Step 3: Calculate the final volume

Using the values from Step 2, plug them into the equation from Step 2 to calculate the final volume, \(V_2\): \(V_2 = \frac{(3.0 \times 760 \mathrm{~torr})(1.0 \mathrm{~mm}^3)}{695 \mathrm{~torr}}\) Solve for \(V_2\): \(V_2 \approx 3.27 \mathrm{~mm}^3\) The volume of the gas bubble when it reaches the surface of the lake is approximately 3.27 mm³, assuming that the temperature doesn't change.

Key Concepts

Gas Laws

Understanding the behavior of gases is a fundamental aspect of chemistry and physical science. Gas laws are simple mathematical relationships that describe how various properties of gases—pressure, volume, temperature, and number of moles—interact with each other. One of the most basic gas laws is Boyle's Law, which focuses on the relationship between pressure and volume at a constant temperature for a fixed amount of gas.

Other essential gas laws include Charles's Law, which describes the volume-temperature relationship at constant pressure, and the Ideal Gas Law, a more comprehensive formula combining Boyle's, Charles's, and Avogadro's laws. These laws are critical for understanding how gases react under different physical conditions and have practical applications ranging from breathing to the behavior of weather balloons.

Pressure Volume Relationship

Boyle's Law is an example of a gas law that illustrates the pressure-volume relationship. It states that for a given amount of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. Mathematically, this is expressed as: \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( P_2 \) are the initial and final pressures, and \( V_1 \) and \( V_2 \) are the initial and final volumes respectively.

This principle means that if the pressure increases, the volume decreases, and vice versa, as long as the amount of gas and the temperature do not change. This can be visually represented by the compression of a spring or the expansion of a balloon. The relationship is crucial not only in laboratory settings but also in everyday applications, such as in the function of the human lungs during breathing or in the automotive airbags deployment systems.

Chemistry Problem Solving

Chemistry problem solving often involves applying the scientific method and using the pertinent gas laws to find a solution. To solve problems effectively, one must first understand the concepts and then apply them methodically, as demonstrated in the exercise involving Boyle's Law. Here's a distilled approach to chemical problem solving:

• Clearly identify the given information and what you are solving for.
• Choose the appropriate gas law - in this case, Boyle's Law due to the constant temperature and the manipulation of pressure and volume.
• Make sure all units are consistent before applying the gas law formula.
• Carefully perform the arithmetic operation, maintaining the proper significant figures.

By following these steps, students can learn to tackle various chemistry problems methodically, building a solid foundation in scientific reasoning and quantitative analysis.