Question

Divers get the bends. Divers returning from deep dives can get the bends from nitrogen gas bubbles in their blood. Assume that blood is largely water. The Henry's law constant for \(\mathrm{N}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is \(86,000 \mathrm{~atm}\). The hydrostatic pressure is \(1 \mathrm{~atm}\) at the surface of a body of water and increases by approximately \(1 \mathrm{~atm}\) for every 33 feet in depth. Calculate the \(\mathrm{N}_{2}\) solubility in the blood as a function of depth in the water, and explain why the bends occur.

Short answer

The solubility of \( \text{N}_2 \) in blood at depth \( d \) is \( 86,000 \times \frac{d}{33} \) atm. The bends occur due to rapid ascent and formation of nitrogen gas bubbles.

Step-by-step solution

Understand Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to the partial pressure of that gas above the liquid. The formula is given by \[ C = k_H P \] where \( C \) is the solubility, \( k_H \) is Henry's law constant, and \( P \) is the partial pressure of the gas.

Calculate the Hydrostatic Pressure at Depth

The hydrostatic pressure increases by approximately 1 atm for every 33 feet of depth. Therefore, at a depth of \( d \) feet, the total pressure \( P_{\text{total}} \) is \[ P_{\text{total}} = 1 + \frac{d}{33} \text{ atm} \]

Apply Henry's Law

Using Henry's law \[ C = k_H P \], substitute the values of \( k_H \) (86,000 atm) and \( P_{\text{total}} \): \[ C = 86,000 \times P_{\text{total}} = 86,000 \times \left(1 + \frac{d}{33}\right) \text{ atm} \] Simplifying, we get \[ C = 86,000 \left(1 + \frac{d}{33}\right) \]}, {

Understand the Bends

When a diver ascends too quickly, the rapid decrease in pressure causes nitrogen gas to come out of solution too quickly, forming bubbles in the blood. This is because the solubility of nitrogen decreases rapidly with pressure, leading to bubble formation.

Key concepts

Henry's Law Constant

Henry's Law is a fundamental principle that helps us understand how gases dissolve in liquids. It states that the solubility of a gas in a liquid is directly proportional to the partial pressure of that gas above the liquid. This relationship is represented by the formula: \[ C = k_H P \] where:
C = Solubility of the gas
k_H = Henry's Law constant
P = Partial pressure of the gas
The Henry's Law constant for \(N_2\) in water at 25°C is given as 86,000 atm. This high constant indicates that \(N_2\) dissolves readily in water under increased pressure conditions.
Understanding this concept is crucial for divers, as the solubility of gases like nitrogen in their blood changes with depth.

Gas Solubility

Gas solubility is a measure of how much of a gas can dissolve in a liquid at a given temperature and pressure. For example, nitrogen gas (N2) dissolves in water based on Henry's Law. The solubility is calculated using: \[ C = k_H P \] where C is the solubility.
When diving, the pressure increases as a diver descends. This increased pressure results in a greater solubility of nitrogen in the diver's blood. At surface level (1 atm), the solubility is relatively low. However, deeper waters exert increased pressure, thus raising nitrogen solubility in the blood. Knowing how solubility changes with pressure helps manage safe diving practices.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid due to its weight. When underwater, this pressure increases with depth. For divers, the hydrostatic pressure increases by 1 atm for every 33 feet. Thus, at a depth of d feet, the pressure P_total is: \[ P_{\text{total}} = 1 + \frac{d}{33} \text{ atm} \]
For example, at 66 feet, the pressure would be: \[ P_{\text{total}} = 1 + \frac{66}{33} = 3 \text{ atm} \]
This additional pressure significantly affects gas solubility. Understanding hydrostatic pressure helps divers anticipate how deeply they can dive and recognize the need to manage ascent rates to avoid rapid changes in pressure.

Decompression Sickness

Decompression sickness, or 'the bends,' occurs when dissolved gases, like nitrogen, form bubbles in the blood. This can happen during a rapid ascent as the pressure decreases quickly. At high pressures underwater, more nitrogen dissolves. When a diver ascends too quickly, the decreased pressure results in a rapid drop in gas solubility: \[ C = 86,000 \times \frac{P_{\text{total}}}{33} \]
Bubbles form because gas is forced out of solution, leading to severe pain and potential harm.
To prevent this, divers must ascend slowly, allowing the body to safely expel excess nitrogen. Understanding how pressure changes affect gas solubility helps divers avoid decompression sickness and ensures a safer diving experience.