Question

The solubility of \(\mathrm{N}_{2}\) in blood is a serious problem for divers breathing compressed air \(\left(78 \% \mathrm{~N}_{2}\right.\) by volume \()\) at depths greater than \(50 \mathrm{ft}\). (a) What is the molarity of \(\mathrm{N}_{2}\) in blood at \(1.00 \mathrm{~atm} ?\) (b) What is the molarity of \(\mathrm{N}_{2}\) in blood at a depth of \(50 . \mathrm{ft} ?\) (c) Find the volume (in \(\mathrm{mL}\) ) of \(\mathrm{N}_{2}\), measured at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\), released per liter of blood when a diver at a depth of \(50 . \mathrm{ft}\) rises to the surface \(\left(k_{\mathrm{H}}\right.\) for \(\mathrm{N}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is \(7.0 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{atm}\) and at \(37^{\circ} \mathrm{C}\) is \(6.2 \times 10^{-4} \mathrm{~mol} / \mathrm{L}\) -atm; assume \(d\) of water is \(1.00 \mathrm{~g} / \mathrm{mL}\) ).

Short answer

At 1 atm, the molarity of N₂ in blood is approximately 4.84 × 10⁻⁴ mol/L. At 50 ft depth, it is approximately 1.22 × 10⁻³ mol/L. The volume of N₂ released is about 18 mL per liter of blood when rising to the surface.

Step-by-step solution

Understanding and Applying Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas in contact with the liquid. The formula for Henry's Law is: \[ C = k_H \times P \] where \( C \) is the concentration or molarity of the gas, \( k_H \) is Henry's constant, and \( P \) is the partial pressure of the gas.

Calculating the Partial Pressure of N₂ at 1 atm

Given that nitrogen makes up 78% of the air by volume at the surface, the partial pressure of \( \mathrm{N}_2 \) at the surface (1 atm) is: \[ P_{N_2} = 0.78 \times 1.00 \text{ atm} = 0.78 \text{ atm} \]

Finding the Molarity of N₂ at 1 atm

Using Henry's Law and the given Henry's constant for \( \mathrm{N}_2 \) at 37°C, we can find the molarity of \( \mathrm{N}_2 \) in blood at 1 atm: \[ C = 6.2 \times 10^{-4} \text{ mol/L} \cdot \text{atm} \times 0.78 \text{ atm} = 4.836 \times 10^{-4} \text{ mol/L} \]

Calculating the Partial Pressure of N₂ at 50 ft Depth

At a depth of 50 ft underwater, the pressure increases by approximately 1 atm for every 33 ft. Therefore, the total pressure at 50 ft is: \[ P_{total} = 1 + \frac{50}{33} \text{ atm} \approx 2.515 \text{ atm} \] The partial pressure of \( \mathrm{N}_2 \) at this depth is: \[ P_{N_2} = 0.78 \times 2.515 \text{ atm} \approx 1.9617 \text{ atm} \]

Finding the Molarity of N₂ at 50 ft Depth

Using Henry's Law again with the new partial pressure: \[ C = 6.2 \times 10^{-4} \text{ mol/L} \cdot \text{atm} \times 1.9617 \text{ atm} = 1.2152 \times 10^{-3} \text{ mol/L} \]

Determining the Volume of N₂ Released when Surfacing

When the diver surfaces, the change in molarity of \( \mathrm{N}_2 \) is: \[ 1.2152 \times 10^{-3} \text{ mol/L} - 4.836 \times 10^{-4} \text{ mol/L} = 7.316 \times 10^{-4} \text{ mol/L} \] Using the Ideal Gas Law \(PV = nRT\) to find the volume of gas released: \[ V = \frac{nRT}{P} \] For \( n = 7.316 \times 10^{-4} mol \), \( R = 0.0821 \text{ L·atm/(K·mol)} \), \( T = 298 K \), and \( P = 1 atm \): \[ V = \frac{(7.316 \times 10^{-4} \text{ mol})(0.0821 \text{ L·atm/(K·mol)})(298 \text{ K})}{1 \text{ atm}} \approx 0.0180 \text{ L} \] This converts to 18 mL of \( \mathrm{N}_2 \) per liter of blood.

Key concepts

gas solubility

Gas solubility refers to how well a gas dissolves in a liquid. When we talk about gas solubility in blood, we need to understand how gases, like nitrogen, mix in the blood. Henry's Law helps us here. Henry's Law states that the concentration of a gas in a liquid is directly proportional to the pressure of that gas above the liquid.
For instance, in our body, gases such as oxygen and nitrogen mix with our blood. When a diver goes deep underwater, the pressure increases, causing more nitrogen to dissolve in the blood. Conversely, when they come back up, the pressure decreases, and nitrogen can escape from the blood.
This concept is crucial for divers because if they rise too quickly, the nitrogen can form bubbles in their blood, leading to decompression sickness, also known as 'the bends'. Let's see more details on how this works.

partial pressure

Partial pressure is the pressure a gas would exert if it alone occupied the entire volume. For gases in a mixture, like air, each gas has its own partial pressure.
For example, at sea level, the air pressure is about 1 atm (atmosphere). Since nitrogen makes up 78% of the air, its partial pressure is 0.78 atm. This applies to all gases in the air.
When diving, the total pressure increases as the diver goes deeper. This affects the partial pressures of all gases. If a diver is at 50 feet deep, the pressure increases by roughly 1 atm for every 33 feet. This means at 50 feet, the pressure is 2.515 atm, considering the water pressure.
So, the partial pressure of nitrogen at this depth would now be 0.78 times 2.515 atm. With this increased partial pressure, more nitrogen will dissolve in the diver's blood.

ideal gas law

The Ideal Gas Law is a fundamental equation in chemistry, represented as \( PV = nRT \). It relates the pressure (P), volume (V), amount (n, in moles), gas constant (R), and temperature (T) of a gas.
In our scenario with nitrogen in blood, we use this law to determine how much gas is released when a diver surfaces. When the pressure drops, some nitrogen comes out of the blood. Using \( nRT \) over the pressure, we can figure out the volume of gas released.
For instance, with the change in nitrogen concentration from 50 feet depth to the surface, we find the volume of gas that would be released in blood upon decompression. This helps us predict and manage the risk of decompression sickness, keeping divers safe.