Question

At ordinary body temperature \(\left(37^{\circ} \mathrm{C}\right)\) the solubility of \(\mathrm{N}_{2}\) in water in contact with air at ordinary atmospheric pressure \((1.0 \mathrm{~atm})\) is \(0.015 \mathrm{~g} / \mathrm{L}\). Air is approximately \(78 \mathrm{~mol} \% \mathrm{~N}_{2}\). Calculate the number of moles of \(\mathrm{N}_{2}\) dissolved per liter of blood, which is essentially an aqueous solution. At a depth of \(100 \mathrm{ft}\) in water, the pressure is \(4.0 \mathrm{~atm}\). What is the solubility of \(\mathrm{N}_{2}\) from air in blood at this pressure? If a scuba diver suddenly surfaces from this depth, how many milliliters of \(\mathrm{N}_{2}\) gas, in the form of tiny bubbles, are released into the bloodstream from each liter of blood?

Short answer

The number of moles of N₂ dissolved per liter at ordinary body temperature and atmospheric pressure is \( \frac{0.015 \thinspace g}{28.02 \thinspace g/mol} \approx 5.36 \times 10^{-4} \thinspace mol/L \). The solubility of N₂ in blood at a pressure of 4.0 atm is \( 0.015 \thinspace g/L \times \frac{4.0 \thinspace atm}{1.0 \thinspace atm} = 0.060 \thinspace g/L \). When the scuba diver suddenly surfaces, the mass of N₂ released into the bloodstream from each liter of blood is approximately \( (0.060 - 0.015) \thinspace g = 0.045 \thinspace g \). This corresponds to a volume of N₂ gas released of approximately \(20.4 \thinspace mL\).

Step-by-step solution

Calculate the number of moles of N₂ dissolved per liter at ordinary body temperature and atmospheric pressure

First, we have to determine the amount of N₂ in air. Since air is about 78 mol% N₂, we can calculate the actual moles of N₂ based on that percentage. Given that solubility of N₂ in water is 0.015 g/L, we need to find the number of moles of N₂ that corresponds to this amount. To do that, we have to convert the mass to moles using the molar mass of N₂: Moles of N₂ = (0.015 g) / (28.02 g/mol) (molar mass of N₂ is 28.02 g/mol)

Calculate the solubility of N₂ in blood at a pressure of 4.0 atm

Now, we need to determine the solubility of N₂ in blood at 100 ft depth in water with a pressure of 4.0 atm. According to Henry's law, the solubility of a gas is directly proportional to the partial pressure of the gas. Therefore, we can calculate the new solubility of N₂ in blood at 4.0 atm as follows: New solubility = (old solubility) * (new pressure / old pressure) New solubility = (0.015 g/L) * (4.0 atm / 1.0 atm)

Calculate the volume of gas released when a scuba diver surfaces

When a scuba diver suddenly surfaces, the pressure on the dissolved N₂ rapidly decreases from 4.0 atm to 1.0 atm, causing the excess N₂ to be released in the form of bubbles. We can calculate the mass of N₂ released by finding the difference in solubility at the two pressures: Mass of N₂ released = (Solubility at 4.0 atm - Solubility at 1.0 atm) * volume of blood Then, we can convert the mass of N₂ released to moles of N₂: Moles of N₂ released = Mass of N₂ released / (28.02 g/mol) Finally, we have to find the volume of N₂ released in milliliters (mL), using the ideal gas law (PV=nRT) at body temperature and atmospheric pressure: Volume of N₂ released = (Moles of N₂ released * R * Temperature) / Pressure Where R is the ideal gas constant: 0.0821 L atm / (mol K). After calculating the volume of N₂ released, convert the result from L to mL.

Key concepts

Henry's Law

Henry's Law is crucial for understanding how gases dissolve in liquids, a topic intimately related to our overall well-being especially in situations involving drastic changes in pressure, such as scuba diving. In its essence, Henry's Law states that the amount of dissolved gas in a liquid is directly proportional to the pressure of that gas above the liquid. This can be mathematically represented as:
\[ C = kP \]
where \( C \) is the concentration of the gas in the liquid, \( k \) is the Henry's law constant for the solute-solvent pair and \( P \) is the partial pressure of the gas above the liquid. When we know the solubility of a gas such as \( \mathrm{N}_2 \) at a certain pressure, Henry's law allows us to predict how the solubility will change if the pressure changes.
For example, if a scuba diver is at a pressure of 1.0 atm and the solubility of nitrogen in the blood is known, one can use Henry's law to predict how this solubility increases as the pressure increases, which is directly related to the depths of the dive. Divers must pay close attention to these changes to prevent decompression sickness, commonly known as 'the bends'.

Molar Mass

The molar mass of a substance is the mass in grams of one mole of that substance. It's an indispensable property for converting between the mass of a substance and the number of moles, as it essentially serves as a bridge between the macroscopic and microscopic worlds. For instance, the molar mass of nitrogen \( \mathrm{N}_2 \) is 28.02 g/mol.
In the context of the exercise, knowing the molar mass of nitrogen allows you to calculate the number of moles of dissolved \( \mathrm{N}_2 \) based on the given solubility in grams per liter. It is crucial for solving problems related to gas solubility and is used whenever we measure the quantity of a substance in the laboratory or in our environment. Students can apply this practice to determine the moles of any compound necessary to react or to be produced in a chemical reaction.

Partial Pressure

Partial pressure is a fundamental concept when it comes to gas mixtures. Each gas in a mixture exerts pressure as if it were the only gas present in the volume, and the pressure it exerts is its partial pressure. For air, which is a mixture of gases, the partial pressure of each gas is proportional to its mole fraction in the mixture.
For example, nitrogen makes up approximately 78% of air's composition by moles, so the partial pressure of nitrogen in the atmosphere is about 78% of the total atmospheric pressure. This is an important factor when using Henry's Law to determine the solubility of gases since the pressure in the formula specifically refers to the partial pressure of the gas in question, which for nitrogen under standard conditions would be approximately \( 0.78 \times 1.0 \) atm.

Scuba Diving and The Bends

For divers, understanding the principles of gas solubility in liquids, like Henry's law, partial pressures, and molar mass, is more than academic—it's a matter of safety. 'The bends', more formally known as decompression sickness, occurs when a diver ascends too quickly after spending time at high pressure underwater. At depth, the increased pressure allows more nitrogen to dissolve into the diver's blood. If the pressure decreases too quickly during ascension, nitrogen doesn't have enough time to exit the bloodstream slowly; instead, it forms bubbles.
These bubbles can cause joint pain, block blood vessels, or even be life-threatening. The exercise presented illustrates precisely why the bends occur, by showing the increased solubility of nitrogen at higher pressures and the potential consequences of rapid pressure change. Divers use decompression stops to slowly adjust to surface pressure, giving nitrogen time to safely leave their bloodstream.