Question

A submersible balloon is sent to the bottom of the ocean. On shore, the balloon had a capacity of \(162 \mathrm{~L}\) when it was filled at \(21.0^{\circ} \mathrm{C}\) and standard pressure. When it reaches the ocean floor, which is at \(5.92^{\circ} \mathrm{C},\) the balloon occupies \(18.8 \mathrm{~L}\) of space. What is the pressure on the ocean floor?

Short answer

The pressure on the ocean floor is approximately 8.68 atm.

Step-by-step solution

Understand Problem Setup

We need to find the pressure on the ocean floor, knowing the initial and final volumes and temperatures of the balloon. The initial volume is 162 L at 21.0°C, while the final volume is 18.8 L at 5.92°C.

Convert Temperatures to Kelvin

To use the ideal gas law, convert temperatures from Celsius to Kelvin by adding 273.15. Initial temperature: \(21.0 + 273.15 = 294.15 \; K\). Final temperature: \(5.92 + 273.15 = 279.07 \; K\).

Apply Combined Gas Law

The combined gas law is given by \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\). Assume standard pressure \(P_1 = 1 \; atm\) for initial conditions. Plug in known values: \(\frac{1 \times 162}{294.15} = \frac{P_2 \times 18.8}{279.07}\).

Solve for Final Pressure

Rearrange the combined gas law for \(P_2\): \[ P_2 = \frac{1 \times 162}{294.15} \times \frac{279.07}{18.8} \]Calculate \(P_2\): \[ P_2 \approx 8.676 \; atm \].

Verify Calculation

Check the calculations by reassessing each step, ensuring unit consistency and proper conversion factors. The calculated final pressure should be logically consistent with the conditions given the reduced temperature and volume.

Key concepts

Combined Gas Law

The Combined Gas Law is a crucial concept in chemistry that helps us understand how changes in the condition of a gas—like its pressure, temperature, or volume—can affect one another. The formula for the Combined Gas Law is:

• \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \)

This formula combines three other gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law, making it a powerful tool for solving problems involving gases in changing conditions. Here, \(P\) stands for pressure, \(V\) for volume, and \(T\) for temperature in Kelvin. This law assumes the amount of gas remains constant.
When using this equation, it's very important to ensure that temperatures are always in Kelvin. This ensures that our calculations are consistent and that we get accurate results.
Overall, the Combined Gas Law is essential for solving gas problems when you know the initial and final states of the system.

Temperature Conversion

Temperature conversion is vital because many physics and chemistry equations require temperature to be in Kelvin rather than Celsius. To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature.
For instance, when the balloon is initially filled, its temperature is given as 21°C. Converting this to Kelvin gives:

• 21 + 273.15 = 294.15 K

Similarly, the temperature when the balloon reaches the ocean floor is given as 5.92°C. In Kelvin, this is:

• 5.92 + 273.15 = 279.07 K

These conversions are crucial. Using temperatures in Kelvin not only follows the proper convention when applying the Gas Laws but also ensures that calculations make sense and are applicable across a wider range of scientific principles.

Volume Change

Understanding volume change is essential in studying gases. Volume, expressed in liters or cubic meters, is directly related to the pressure and temperature changes experienced by a gas.
In this exercise, the initial volume of the balloon is 162 L, but once it reaches the ocean floor, its volume reduces significantly to 18.8 L.
Several factors can cause changes in volume:

• Pressure changes: As pressure increases, volume decreases if temperature is constant (Boyle's Law).
• Temperature changes: As temperature decreases, volume decreases if pressure is constant (Charles's Law).

Considering that the balloon's volume decreased as it was submerged, it is clear that the external pressure from the water increased, overwhelming the effect of the smaller drop in temperature. These volume changes are predictable using the Combined Gas Law and are pivotal in calculating changes in pressure.

Pressure Calculation

Pressure calculations allow us to determine how much force a gas exerts on the walls of its container. In the given problem, we find the pressure at the ocean floor using the Combined Gas Law. The calculation setup is as follows:

• Use initial pressure \(P_1 = 1 \, atm\) (standard atmospheric pressure).
• Initial volume \(V_1 = 162 \, L\).
• Final volume \(V_2 = 18.8 \, L\).
• Initial temperature \(T_1 = 294.15 \, K\).
• Final temperature \(T_2 = 279.07 \, K\).

Plug these values into the combined gas law:

• \( \frac{1 \times 162}{294.15} = \frac{P_2 \times 18.8}{279.07} \)

Rearranging and solving for \(P_2\), the final pressure gives:

• \(P_2 \approx 8.676 \, atm\).

This result reflects the significant increase in pressure as the balloon is submerged, showcasing the powerful influence of depth on pressure.