Question

A small bubble rises from the bottom of a lake, where the temperature and pressure are \(8^{\circ} \mathrm{C}\) and \(6.0 \mathrm{~atm}\), to the water's surface, where the temperature is \(25^{\circ} \mathrm{C}\) and pressure is \(1.0 \mathrm{~atm}\). Calculate the final volume of the bubble if its initial volume was \(2 \mathrm{~mL}\). (a) \(14 \mathrm{~mL}\) (b) \(12.72 \mathrm{~mL}\) (c) \(11.31 \mathrm{~mL}\) (d) \(15 \mathrm{~mL}\)

Short answer

The final volume of the bubble when it reaches the surface is approximately \( 11.31 \text{ mL} \).

Step-by-step solution

Understand the Process and Identify the Variables

We need to calculate the final volume of the bubble when it rises to the water's surface. The change in conditions (temperature and pressure) indicates we should use the combined gas law, which is \( P_1 V_1 / T_1 = P_2 V_2 / T_2 \). Here, \( P_1 \) and \( P_2 \) are the initial and final pressures, \( V_1 \) and \( V_2 \) are the initial and final volumes, and \( T_1 \) and \( T_2 \) are the initial and final temperatures in kelvins. The given values are \( P_1 = 6.0 \text{ atm} \), \( V_1 = 2 \text{ mL} \), \( T_1 = 8^\circ \text{C} \) (convert to Kelvin by adding 273), \( P_2 = 1.0 \text{ atm} \) and \( T_2 = 25^\circ \text{C} \) (also convert to Kelvin by adding 273).

Convert Temperatures to Kelvin

Temperature must be in Kelvin for the gas law equations. Convert both temperatures from Celsius to Kelvin by adding 273.15. Thus, \( T_1 = 8 + 273.15 = 281.15 \text{ K} \) and \( T_2 = 25 + 273.15 = 298.15 \text{ K} \) .

Rearrange the Combined Gas Law to Solve for Final Volume

With the combined gas law \( P_1 V_1 / T_1 = P_2 V_2 / T_2 \), we can rearrange to solve for \( V_2 \) by multiplying both sides by \( T_2 / P_2 \) and get \( V_2 = (P_1 V_1 T_2) / (T_1 P_2) \) .

Insert the Known Values

Insert the known values into the rearranged combined gas law to find the final volume: \( V_2 = (6.0 \text{ atm} \times 2 \text{ mL} \times 298.15 \text{ K}) / (281.15 \text{ K} \times 1.0 \text{ atm}) \) .

Calculate the Final Volume

Perform the calculation from the previous step: \( V_2 = (6.0 \times 2 \times 298.15) / (281.15) \). This gives \( V_2 = 36.0 \times 298.15 / 281.15 \) which simplifies to approximately \( V_2 = 11.31 \text{ mL} \) .

Key concepts

Physical Chemistry

Physical chemistry lies at the heart of understanding the behavioral principles of matter, particularly when dealing with gases. It is a fascinating field that blends the laws of physics with chemical processes to describe the phenomena we observe, like how a bubble behaves as it ascends through water in a lake. In our bubble example, principles from physical chemistry allow us to predict the final volume of the bubble at the surface by considering how gases expand when they're heated or decompressed. These principles anchor in molecular interactions and the energy change accompanying temperature and pressure variations.
Physical chemistry isn't just theoretical; it has real-world applications. For instance, divers must understand these principles to avoid decompression sickness, and engineers must account for them when designing pressure vessels or handling gases in industrial processes. For students to fully grasp these concepts, they should visualize the scenarios, perhaps with diagrams or animations, to see how molecules behave under different pressures and temperatures.

Gas Law Calculations

Gas law calculations are crucial in determining a gas's behavior under varying conditions. The combined gas law, which we used in our bubble example, mathematically relates temperature, pressure, and volume. It's a powerful tool when we don't have to keep one of these variables constant, as we would with Charles's or Boyle's law.When solving problems with the combined gas law, it's key to keep units consistent—pressures in atmospheres, volumes in liters or milliliters, and temperatures in Kelvin. Also, remember the importance of converting Celsius to Kelvin, because Kelvin provides an absolute temperature scale crucial for gas calculations.
For students, practicing these calculations with different scenarios can solidify understanding. Building from simpler to more complex problems, they could start with direct proportionality under Boyle's law (holding temperature constant) and then advance to scenarios where both temperature and pressure change, thus employing the combined gas law.

Temperature-Pressure-Volume Relationships

Understanding the temperature-pressure-volume relationships in gases is about grasping how temperature or pressure changes affect a gas's volume. When a gas heats up, its particles move faster and tend to take up more space, thus increasing the volume if the pressure remains constant. Similarly, if you decrease the pressure on a gas while maintaining the temperature, the gas will expand to fill a greater volume.In the context of our bubble exercise, as the bubble rises, pressure decreases due to less water column above, and temperature increases because water near the surface is generally warmer. According to the combined gas law, the bubble's volume should increase as it ascends to the surface, which is articulated by the inverse relationship between pressure and volume, and the direct relationship between temperature and volume.
To better internalize these concepts, visual aids like graphs demonstrating these relationships can be helpful for students. Also, engaging in laboratory experiments, where they can manipulate the temperature and pressure of gases and observe the volume changes, can provide invaluable practical insight.