Question

At ordinary body temperature \(\left(37^{\circ} \mathrm{C}\right),\) the solubility of \(\mathrm{N}_{2}\) in water at ordinary atmospheric pressure is \(0.015 \mathrm{~g} / \mathrm{L}\) Air is approximately \(78 \mathrm{~mol} \% \mathrm{~N}_{2}\). (a) Calculate the number of moles of \(\mathrm{N}_{2}\) dissolved per liter of blood, assuming blood is a simple aqueous solution. (b) At a depth of \(30.5 \mathrm{~m}\) in water, the external pressure is \(405 \mathrm{kPa}\). What is the solubility of \(\mathrm{N}_{2}\) from air in blood at this pressure? (c) 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

(a) The number of moles of \(\mathrm{N}_{2}\) dissolved per liter of blood is \(5.36 \times 10^{-4} \thinspace mol/L\). (b) The solubility of \(\mathrm{N}_{2}\) from air in blood at a depth of 30.5 meters and a pressure of 405 kPa is given by \(C_{new} = k_H \times 405\thinspace kPa\), where \(k_H\) is the Henry's law constant calculated from the initial solubility. (c) To find the volume of \(\mathrm{N}_{2}\) gas bubbles released into the bloodstream when a scuba diver surfaces suddenly, first calculate the difference in solubility (ΔC) and number of moles released (Δn). Finally, calculate the volume (V) using the ideal gas law and convert it to milliliters.

Step-by-step solution

Calculate the number of moles of nitrogen dissolved per liter of blood

We are given the solubility of nitrogen gas in water at atmospheric pressure and body temperature, which is 0.015 grams per liter. We also know that air is approximately 78 mol% nitrogen. We will use this information to calculate the number of moles of nitrogen dissolved per liter of blood. First, we need to know the molar mass of nitrogen: - Molar mass of nitrogen (\(N_2\)) = 28 g/mol Now, we can calculate the number of moles of nitrogen dissolved in 1 liter of blood: No of moles = \(\frac{Solubility (grams/liter)}{Molar \thinspace mass (grams/mol)}\) No of moles = \(\frac{0.015 \thinspace g/L}{28 \thinspace g/mol}\) = \(5.36 \times 10^{-4} \thinspace mol/L\)

Calculate the solubility of nitrogen gas at 30.5 meters depth

At a depth of 30.5 meters, the external pressure is 405 kPa. Using Henry's law, we can calculate the solubility of nitrogen gas in blood at this pressure. Henry's law states that the concentration of a gas in a liquid is directly proportional to the partial pressure of the gas: \(C = k_H \times Po\) Where: - C is the concentration of the gas in the liquid (mol/L) - k_H is the Henry's law constant for the specific gas and liquid at the given temperature - Po is the partial pressure of the gas (kPa) First, we will calculate the partial pressure of nitrogen in the air: Partial pressure of \(N_2\) = fraction of nitrogen × atmospheric pressure. The atmospheric pressure at sea level is approximately 101.3 kPa. Therefore, the partial pressure of nitrogen in air is: Partial pressure of \(N_2\) = 0.78 × 101.3 kPa = 79 kPa Next, we will determine the value of Henry's law constant from the given solubility of nitrogen in water at body temperature: \(k_H = \frac{C}{Po}\) We use the initial nitrogen mole concentration calculated in step 1: \(k_H = \frac{5.36 \times 10^{-4} \thinspace mol/L}{79\thinspace kPa}\) Now we can calculate the solubility of nitrogen gas in blood at a depth of 30.5 meters and a pressure of 405 kPa: \(C_{new} = k_H \times P_{new}\) Where: - \(C_{new}\) is the new solubility of nitrogen in blood at 30.5 meters depth - \(P_{new}\) is the new pressure of nitrogen gas at 30.5 meters depth \(C_{new} = k_H \times 405\thinspace kPa\)

Calculate the volume of nitrogen gas bubbles released when a diver surfaces

When a scuba diver suddenly surfaces, the pressure returns to atmospheric pressure, and the solubility of nitrogen gas in the blood decreases. The excess nitrogen gas forms tiny bubbles in the bloodstream. We will calculate the volume of these nitrogen gas bubbles in milliliters, released per liter of blood. First, we need to calculate the difference in solubility at the surface and at a depth of 30.5 meters. ΔC = \(C_{new} - C_{old}\) Where: - ΔC is the difference in solubility between the surface and the depth Now, we can find the number of moles of nitrogen released per liter of blood by multiplying the ΔC value with the initial volume in liters: Δn = ΔC × V Finally, we can calculate the volume of nitrogen gas released using the ideal gas law: \(PV = nRT\) Where: - P is the pressure (in atm) - V is the volume (in L) - n is the number of moles of gas - R is the ideal gas constant (0.0821 L atm/mol K) - T is the temperature (in K) Rearrange the equation to find the volume: \(V = \frac{nRT}{P}\) The volume can be converted to milliliters by multiplying by 1000.

Putting the values together

Now we have all the equations and values to solve the problem. 1. Calculate the value of \(k_H\): \(k_H = \frac{5.36 \times 10^{-4} \thinspace mol/L}{79\thinspace kPa}\) 2. Calculate the solubility of \(N_2\) gas in blood at 30.5 meters depth: \(C_{new} = k_H \times 405\thinspace kPa\) 3. Calculate the difference in solubility between the surface and the depth: ΔC = \(C_{new} - C_{old}\) 4. Calculate the number of moles of nitrogen released per liter of blood: Δn = ΔC × V 5. Calculate the volume of nitrogen gas released: \(V = \frac{nRT}{P}\) 6. Convert the volume to milliliters: V (mL) = V (L) × 1000 Now, we have a step-by-step solution to find the solubility of nitrogen gas and the volume of nitrogen gas released when a scuba diver surfaces suddenly.

Key concepts

Solubility of Gases

The solubility of gases refers to how well a gas can dissolve in a liquid, such as nitrogen in blood. This solubility depends on several factors like temperature, pressure, and the nature of the gas and solvent. When gases dissolve in liquids, they do so in a way that balances the rate of the gas escaping from the liquid with the rate of the gas entering it.
This means that at a specific temperature and pressure, there is a precise amount of gas that will dissolve in a given volume of liquid. In the exercise, the solubility of nitrogen gas in water at body temperature and atmospheric pressure is given as 0.015 g/L.
Knowing the solubility helps in calculating how gases, such as oxygen and nitrogen, interact with the human body, which is crucial for understanding phenomena such as the bends in scuba diving. To determine how much nitrogen dissolves in blood, we use solutions and derive the number of moles using its molar mass.

Ideal Gas Law

The ideal gas law is a fundamental principle relating pressure, volume, and temperature of a gas with its number of moles. It is expressed as \(PV = nRT\), where \(P\) is the pressure, \(V\) the volume, \(n\) the number of moles, \(R\) the ideal gas constant, and \(T\) the temperature in Kelvin.
This law helps us understand gas behavior under different conditions and is essential to calculate changes in gas volumes when a diver surfaces. When solving the problem, the ideal gas law is used to estimate how much nitrogen gas will be released from a diver's blood when surfacing.

• The ideal gas constant \(R\) is 0.0821 L atm/mol K, which provides a means to relate the temperatures and pressures in diving scenarios.
• Understanding these dynamics aids in grasping how sudden pressure changes affect solubility and the formation of bubbles in the bloodstream.

Partial Pressure

Partial pressure is the pressure that a gas in a mixture would exert if it occupied the entire volume on its own. It is an essential concept in gases' solubility and behavior.
In a gas mixture, each component gas's partial pressure is proportional to its fraction in the mixture. For nitrogen, which makes up about 78% of air, its partial pressure can be calculated by multiplying this fraction by the total pressure of the air.

• At sea level, with atmospheric pressure around 101.3 kPa, nitrogen's partial pressure is approximately 79 kPa.
• At deeper depths, like 30.5 meters underwater where pressure is 405 kPa, the partial pressure increases proportionally.

This increase in partial pressure allows more gas to dissolve in liquids, a critical factor in calculating the diver's blood nitrogen level.

Molar Mass of Nitrogen

The molar mass of nitrogen is crucial in determining how many moles of nitrogen are present when calculating its solubility in blood. Nitrogen is a diatomic molecule, meaning it exists naturally as \(N_2\) molecules.
The molar mass of a nitrogen molecule, \(N_2\), is 28 g/mol. This value is derived from the atomic mass of nitrogen, which is approximately 14 g/mol. By using the molar mass, we can convert between the mass of nitrogen dissolved in blood and the number of moles.

• Given that the solubility of nitrogen gas in water is 0.015 g/L, we calculate moles using \(\text{No of moles} = \frac{\text{Solubility (grams/liter)}}{\text{Molar mass (grams/mol)}}\).
• This conversion is essential for further calculations involving Henry's law and the ideal gas law to solve problems related to gas behavior under pressure changes.

Understanding the molar mass is not only about solving this exercise but also about grasping how chemical quantities are interrelated in real-world applications.