Problem 9 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Divers' "bends' are caused by the formation of bubbles of nitrogen in blood as the solubility reduces when the diver returns to the surface. The solubility of nitrogen in water at 1.00 atm pressure is \(13.0 \mathrm{mg} \mathrm{kg}^{-1}\) at body temperature of \(37^{\circ} \mathrm{C}\) and increases linearly with pressure. In water, the pressure increases at the rate of 1.00 atm per \(10 \mathrm{m}\) depth. Estimate the volume of gas that comes out of solution when a diver who has \(4.5 \mathrm{kg}\) of blood rapidly ascends from a depth of \(50 \mathrm{m}\) of water to the surface. Assume the solubility of nitrogen in blood is the same as in water. (Section 8.2)

Step-by-Step Solution

Answer

Approximately 0.234 L of nitrogen gas is released.

Step 1: Determine Increased Pressure at Depth

The given problem states that the pressure increases by 1.00 atm for every 10 meters of depth. Therefore, at a depth of 50 meters, the pressure is 1.00 atm (surface pressure) + 5.00 atm (pressure increase due to depth) = 6.00 atm.

Step 2: Calculate Nitrogen Solubility at Depth

The solubility of nitrogen in water at 37°C and 1.00 atm is given as 13.0 mg/kg. Since the solubility increases linearly with pressure, at a pressure of 6.00 atm, the solubility is 6 times 13.0 mg/kg, which equals 78.0 mg/kg.

Step 3: Calculate Total Nitrogen Dissolved at Depth

Given the diver has 4.5 kg of blood, the total amount of nitrogen dissolved at 6.00 atm (50 m depth) is 78.0 mg/kg × 4.5 kg = 351 mg.

Step 4: Calculate Reduced Solubility at Surface

At the surface, the pressure is 1.00 atm and the solubility of nitrogen is 13.0 mg/kg. The total nitrogen that should be dissolved in 4.5 kg of blood at this condition is 13.0 mg/kg × 4.5 kg = 58.5 mg.

Step 5: Calculate Excess Nitrogen Gas Released

The excess nitrogen gas that comes out of the blood when the diver ascends is the difference between the nitrogen dissolved at depth and at the surface: 351 mg - 58.5 mg = 292.5 mg of nitrogen gas.

Step 6: Convert Mass of Gas to Volume

To convert the mass of the nitrogen gas to volume, use the ideal gas law, assuming standard conditions of temperature and pressure at the surface. Nitrogen has a molar mass of 28 g/mol, so 292.5 mg is 0.2925 g or 0.01045 mol. At standard conditions (1.00 atm, 273.15 K), 1 mole of gas occupies 22.4 L. Therefore, 0.01045 mol × 22.4 L/mol = 0.2341 L.

Key Concepts

Diver's Decompression Sickness

Decompression sickness, often known as "the bends," is a serious condition that divers must be aware of. It occurs when nitrogen, absorbed into the bloodstream under high pressure while diving, forms bubbles as pressure decreases during ascent. These bubbles can cause joint pain, dizziness, and even severe complications like paralysis or stroke.

When a diver descends, the increased pressure underwater leads to more nitrogen dissolving in their blood than what is typically present at surface level. As the diver ascends and pressure decreases, the solubility of nitrogen in the blood also decreases. If the diver rises too quickly, the nitrogen doesn't have enough time to return to the lungs for safe exhalation. Instead, it forms bubbles in tissues or direct into the bloodstream, causing the symptoms of decompression sickness.

To prevent this, divers practice controlled ascents and often include decompression stops while surfacing, allowing nitrogen to safely exit the blood.

• High pressure under water = more nitrogen absorbed.
• Rapid ascent leads to nitrogen bubbles forming.
• Controlled ascents prevent decompression sickness.

Pressure Effects on Solubility

In general, the solubility of gases in liquids increases with an increase in pressure. This principle is crucial in understanding what happens to divers as they go deeper underwater. As pressure increases with depth, more nitrogen dissolves into the liquid, which in this case is the diver's blood.

At the surface, atmospheric pressure is 1 atm. However, every 10 meters depth in water adds an additional 1 atm of pressure. This is why a diver at 50 meters experiences a total pressure of 6 atm (1 atm of surface pressure + 5 atm from the depth).

This increase in pressure enables more nitrogen to dissolve in the blood. Thanks to the linear relationship between pressure and gas solubility, as pressure doubles, so does the solubility of nitrogen. This means the deeper the diver goes, the more nitrogen the blood holds, and hence the greater the potential for "the bends" if ascend is too fast.

• Pressure increase with depth leads to more dissolved gas.
• Solubility of nitrogen is linearly related to pressure.
• Important for divers to manage ascent carefully.

Nitrogen Solubility in Water

Nitrogen is an inert gas that naturally dissolves in water, including the human body's internal fluids such as blood. At 1 atm pressure and a temperature of 37°C (body temperature), nitrogen's solubility in water is 13 mg per kg. This solubility changes predictably with changes in pressure, a principle that's exploited in diving and provides critical information for planning safe dives.

As the pressure increases, the solubility of nitrogen in water increases proportionally. For example, at a depth of 50 meters, where pressure is 6 atm, the solubility becomes six times the solubility at the surface, resulting in nitrogen solubility of 78 mg per kg.

This property aids divers in calculating how much nitrogen is absorbed into their bloodstream during a dive, which informs their decompression schedules.

• Nitrogen's solubility is a constant factor at constant temperature.
• Directly proportional to pressure.
• Used by divers for planning safe decompression.

Ideal Gas Law Applications

The ideal gas law relates the pressure, volume, and temperature of a gas to its amount in moles. This law is incredibly useful in diving physics, especially when determining the volume of gas released from a diver's blood during ascent.

The ideal gas law is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is temperature. In the situation of a diver ascending, the difference in nitrogen solubility from depth to surface allows us to find how much nitrogen gas comes out of solution.

For example, after calculating the difference in nitrogen content between blood at depth vs. the surface, you can determine the gas volume using standard temperature and pressure where 1 mole of gas occupies 22.4 liters. So, if you know the number of moles of nitrogen released, multiply by 22.4 liters/mol to find the volume of nitrogen gas produced.

• Relates gas properties: pressure, volume, temperature.
• Essential for calculating gas volume changes in divers.
• Utilizes standard conditions for practical dive calculations.