Problem 51 - Chemistry The Science in Context

Question

Underwater Archeology \(\mathrm{A}\) scuba diver releases a balloon containing \(153 \mathrm{L}\) of helium attached to a tray of artifacts at an underwater archaeological site (Figure \(\mathrm{P} 6.51\) ). When the balloon reaches the surface, it has expanded to a volume of 352 L. The pressure at the surface is 1.00 atm; what is the pressure at the underwater site? Pressure increases by 1.0 atm for every \(10 \mathrm{m}\) of depth; at what depth was the diver working? Assume the temperature remains constant.

Step-by-Step Solution

Answer

Answer: The initial pressure at the underwater archaeological site is approximately 2.30 atm, and the diver was working at a depth of 13 meters.

Step 1: Set up Boyle's Law equation

\(P_1V_1 = P_2V_2\) where \(P_1\) is the initial pressure, \(V_1\) is the initial volume, \(P_2\) is the final pressure, and \(V_2\) is the final volume of the gas. We are given the following values: - Initial volume (\(V_1\)): \(153 \mathrm{L}\) - Final volume (\(V_2\)): \(352 \mathrm{L}\) - Final pressure (\(P_2\)): \(1.00 \mathrm{atm}\)

Step 2: Solve for the initial pressure

Plug the given values into the Boyle's Law equation: \(P_1 \times 153 = 1.00 \mathrm{atm} \times 352\) Now, solve for \(P_1\): \(P_1 = \frac{1.00 \times 352}{153} \approx 2.30 \mathrm{atm}\) The initial pressure at the underwater site is approximately \(2.30 \mathrm{atm}\).

Step 3: Determine the depth using the pressure information

We are given that pressure increases by 1.0 atm for every 10 m of depth. Since the pressure at the surface is 1.00 atm, we can find the depth by subtracting the surface pressure from the initial pressure: \(P_{\mathrm{depth}} = P_1 - P_2 = 2.30 \mathrm{atm} - 1.00 \mathrm{atm} = 1.30 \mathrm{atm}\) Now, we can use the pressure-depth relationship to find the depth: \(\mathrm{Depth} = \frac{P_{\mathrm{depth}}}{\mathrm{Pressure \, change \, per \, meter}} \times \mathrm{Meters \, per \, pressure \, change}\) \(= \frac{1.30}{1.0 \mathrm{atm/10\,m}} \times 10 \mathrm{m}\) \(= 1.3 \times 10 \mathrm{m} = 13 \mathrm{m}\) The diver was working at a depth of 13 meters.

Key Concepts

Scuba Diving Physics

Scuba diving involves the fascinating application of physics, particularly when managing the complexities of buoyancy and pressure changes underwater. One crucial principle is Boyle's Law, which explains how gas volumes change with pressure. When a diver descends, the pressure increases, compressing air spaces in their gear such as air tanks or balloons. This changes the volume of any encased gas, directly impacting buoyancy. Understanding these interactions is critical to managing equipment and ensuring diver safety.

In our exercise scenario, a diver releases a helium-filled balloon underwater. As it ascends, the external pressure decreases, allowing the balloon to expand. By calculating these changes using Boyle's Law, divers can predict how underwater operations will unfold, ensuring precise control over objects' buoyancy to avoid accidents.

Underwater Pressure

The concept of underwater pressure is vital in scuba diving. As a diver descends into the depths of the ocean, the water pressure increases with the depth. For every 10 meters descended, pressure increases by approximately 1 atmosphere (atm).

This exercise highlights how underwater pressure affects objects. With a balloon initially at a depth where pressure is greater than at the surface, we calculate the pressure it experiences. Knowing that the surface pressure is 1 atm, we deduced that the initial underwater pressure was 2.30 atm, specifically caused by the depth affecting the helium balloon's volume.

• Pressure increases with depth: For every 10 meters, add 1 atm.
• Pressure affects volume: Higher pressure compresses the gas, lowering its volume.

Understanding how pressure varies with depth helps divers make informed decisions about buoyancy control and depth management.

Gas Volume Expansion

Gas volume expansion is a fundamental concept that allows divers to predict how gases behave under different pressures. According to Boyle's Law, the volume of a gas is inversely proportional to the pressure it experiences, assuming the temperature stays the same. In other words, as pressure decreases, gas volume increases, and vice-versa.

In this problem, we observe that a helium balloon released underwater expands from 153 L to 352 L as it reaches the surface, demonstrating how the reduction in pressure facilitates volume expansion.

• Boyle's Law Equation: \( P_1V_1 = P_2V_2 \)
• Initial condition: smaller volume, higher pressure.
• Final condition: larger volume, lower pressure.

Being able to calculate these changes is crucial for divers to maintain control over buoyancy during ascent or descent. Through comprehensive understanding and application of gas laws, divers manage their ascents to avoid risks like decompression sickness.