Problem 22 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Dry air with the following composition is used to fill a SCUBA cylinder for a dive. (Section 8.6) $$\begin{array}{lc} \hline \text { Gas } & \text { Composition of dry air by volume } \\ \hline \mathrm{N}_{2} & 78 \% \\ \mathrm{O}_{2} & 21 \% \\ \mathrm{Ar} & 1 \% \\ \hline \end{array}$$ (a) At \(10 \mathrm{m}\) depth, a diver experiences an external pressure of 2 atm. Write an expression for the total pressure of the air in terms of the partial pressures of \(\mathrm{N}_{2}, \mathrm{O}_{2},\) and \(\mathrm{Ar}\). (b) What is the molar percentage of oxygen in the air inhaled at 2 atm? (c) What is the partial pressure of \(\mathrm{O}_{2}\) in air inhaled at: (1) \(1 \mathrm{atm}\); (ii) 2 atm? How does the number of molecules of \(\mathrm{O}_{2}\) inhaled per breath at \(10 \mathrm{m}\) depth compare with the number inhaled per breath at sea level? Suggest why some deep-sea divers dive with a gas mixture containing \(10 \%\) oxygen.

Step-by-Step Solution

Answer

(a) Total pressure is the sum of partial pressures: 2 atm = 1.56 atm (\(N_2\)) + 0.42 atm (\(O_2\)) + 0.02 atm (\(Ar\)). (b) Molar percentage of oxygen is 21%. (c) \(O_2\) partial pressure is 0.21 atm at 1 atm; 0.42 atm at 2 atm. More \(O_2\) molecules are inhaled at 2 atm. Divers use lower \(O_2\) mixtures to avoid oxygen toxicity.

Step 1: Understanding Partial Pressures

Partial pressure is calculated by multiplying the total pressure by the fraction of the particular gas in the mixture. For a gas component with percentage composition \(x\), the partial pressure \(P_x\) is given by \(P_x = P_{total} \times \frac{x}{100}\). The total pressure \(P_{total}\) at 10m depth is 2 atm.

Step 2: Write Expression for Total Pressure

The total pressure of the dry air at 10m depth can be expressed as the sum of the partial pressures of nitrogen, oxygen, and argon. Therefore, \(P_{total} = P_{N_2} + P_{O_2} + P_{Ar}\). Each partial pressure can be calculated using their respective fraction compositions (78% for \(N_2\), 21% for \(O_2\), and 1% for \(Ar\)).

Step 3: Calculate Partial Pressures at 2 atm

Using the expression from Step 1, calculate \(P_{N_2} = 2 \times 0.78 = 1.56\) atm, \(P_{O_2} = 2 \times 0.21 = 0.42\) atm, and \(P_{Ar} = 2 \times 0.01 = 0.02\) atm. Confirm that \(1.56 + 0.42 + 0.02 = 2\) atm, matching the total pressure.

Step 4: Molar Percentage of Oxygen at 2 atm

Since the composition does not change with pressure, the molar percentage of oxygen remains the same as the volume percentage, which is 21%.

Step 5: Calculate Partial Pressure of Oxygen at 1 atm

At 1 atm, the partial pressure of oxygen \(P_{O_2}\) is calculated as \(1 \times 0.21 = 0.21\) atm.

Step 6: Calculate Partial Pressure of Oxygen at 2 atm

Using the partial pressure formula, at 2 atm the partial pressure of oxygen is \(0.42\) atm, as calculated in Step 3.

Step 7: Compare Molecules of \(O_2\) Inhaled at Different Depths

Since pressure and the concentration of the gas determine the number of molecules inhaled, the increased pressure at 10m depth (2 atm) results in a higher number of \(O_2\) molecules inhaled compared to sea level (1 atm) due to the higher partial pressure of oxygen.

Step 8: Reason for Using a Lower Oxygen Mixture for Deep-Sea Diving

Deep-sea divers may use a gas mixture with 10% oxygen to avoid oxygen toxicity, which can occur at high pressures due to increased amounts of oxygen inhaled, leading to potential harm.

Key Concepts

Gas Composition

Gas composition is crucial when filling SCUBA tanks for divers. Dry air is predominantly composed of three gases:

• Nitrogen (\(N_2\)), which makes up 78% of the air by volume,
• Oxygen (\(O_2\)), which accounts for 21% by volume,
• Argon (\(Ar\)), which is 1% by volume.

These percentages tell us about the volume ratio of each gas in a mixture of dry air. When the air is compressed into a SCUBA tank, the ratio remains constant, though the volume of the air is reduced. This consistency is vital because divers rely on specific oxygen levels for safe breathing underwater.

Understanding these compositions allows divers to calculate partial pressures, which tell us how much of each gas is exerted within any given volume. For divers, knowing the partial pressures of oxygen and nitrogen is especially important because it helps determine safe diving times and depths.
Each gas behaves as if it alone occupies the entire volume, which helps us understand the total pressure within a tank or at a certain depth.

SCUBA Diving

SCUBA diving involves breathing compressed air through a cylinder to explore underwater environments. The acronym "SCUBA" stands for "Self-Contained Underwater Breathing Apparatus." During a dive, understanding gas behaviors in terms of physics and chemistry is critical for a diver’s health and safety.

At sea level, the atmospheric pressure is 1 atm, but when diving, this pressure increases with depth. For example, at 10 meters underwater, the pressure is roughly 2 atm due to the weight of the water above.

• This means divers are experiencing twice the pressure as on land.
• The increased pressure affects how the body absorbs gases, particularly nitrogen and oxygen.

It's also important to note that leverage changes the body’s physiologic responses. With increased pressure comes increased solubility of gases in body tissues. Divers must be cautious about how much nitrogen is absorbed to avoid decompression sickness, commonly known as "the bends." This condition arises when dissolved gases come out of solution too quickly as a diver ascends.
SCUBA divers learn about buoyancy, which involves controlling ascent and descent to manage the body’s exposure to pressure changes safely.

Diving Physics and Chemistry

Diving physics and chemistry are fundamental in ensuring that divers have safe underwater experiences. When diving, the concepts of pressure, volume, and gas laws intersect critically.

One primary principle is Boyle’s Law, which describes how pressure and volume of a gas are inversely proportional at constant temperature. During descent, as pressure increases, gas volume decreases, which is why divers need to understand and adjust the air in their buoyancy control devices.

• For every 10 meters of seawater, the pressure increases by approximately 1 atm.
• This increase affects how gases are compressed in a diver’s body.

Another essential aspect is Dalton’s Law of Partial Pressures. This law states that the total pressure exerted by an air mixture is the sum of the pressures of each gas in the mixture. Therefore, at a depth where the total pressure is 2 atm, the partial pressure of each gas component is crucial for life support.
For safety, divers might use gas mixtures with reduced oxygen levels to prevent oxygen toxicity. At higher pressures, like those encountered in deeper dives, there's a risk that too much oxygen can lead to toxic effects on the central nervous system.
These scientific principles help divers make informed decisions about diving depth, duration, and necessary decompression stops, ultimately leading to safer diving adventures.