Question

The solubility of nitrogen gas is \(1.90 \mathrm{~mL} / \mathrm{dL}\) of blood at 1.00 atm. What is the solubility of nitrogen gas in a scuba diver's blood at a depth of 225 feet and a pressure of 8.00 atm?

Short answer

The solubility of nitrogen gas is 15.2 mL/dL at 8.00 atm.

Step-by-step solution

Understanding Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to the pressure of that gas above the liquid. It can be expressed as \( S_1 / P_1 = S_2 / P_2 \), where \( S \) refers to solubility and \( P \) is the pressure.

Identifying Initial Conditions

We are given that the initial solubility \( S_1 \) of nitrogen gas is \( 1.90 \text{ mL/dL} \) at an initial pressure \( P_1 \) of \( 1.00 \text{ atm} \).

Identifying Final Conditions

The final pressure \( P_2 \) is \( 8.00 \text{ atm} \) at a depth of 225 feet. We need to find the final solubility \( S_2 \).

Applying Henry's Law

Using the formula \( \frac{S_1}{P_1} = \frac{S_2}{P_2} \), plug in the known values: \( \frac{1.90 \text{ mL/dL}}{1.00 \text{ atm}} = \frac{S_2}{8.00 \text{ atm}} \).

Solving for \( S_2 \)

Rearrange the equation to solve for \( S_2 \): \( S_2 = \frac{1.90 \text{ mL/dL} \times 8.00 \text{ atm}}{1.00 \text{ atm}} \). Calculate \( S_2 \): \( S_2 = 15.2 \text{ mL/dL} \).

Key concepts

Solubility

Solubility refers to the ability of a substance, like a gas, to dissolve in a liquid. In context with Henry's Law, solubility represents how much of a gas can be dissolved in a liquid such as blood, under specific pressure conditions. The principle here is that as pressure increases, so does the solubility of a gas, meaning more gas can be dissolved. Understanding solubility is crucial for scenarios involving gases under pressure, such as deep-sea diving or in carbonated drinks.
In deep-sea diving, the increased atmospheric pressures encountered at depth cause more nitrogen to dissolve in a diver's blood. Conversely, as pressure decreases, less nitrogen remains dissolved, which is a key aspect to manage in order to avoid conditions like decompression sickness. These solubility principles help in calculating how pervasive a gas can be within a liquid, based on external pressure conditions.

Gas Laws

Gas laws are the scientific laws that describe how gases behave and interact under varying conditions of temperature, volume, and pressure. Henry's Law is one such gas law, specifically focusing on the relationship between pressure and solubility of gases.
For example, when students study gas laws, they encounter equations that relate different variables like pressure, volume, temperature, and the moles of gas. Each law provides foundational insights into how gas properties change in a variety of circumstances. In this exercise, Henry's Law demonstrates how gas solubility is dependent directly on pressure. It uses the relationship \( S_1 / P_1 = S_2 / P_2 \), which showcases direct proportionality, making it easier to understand how increased external pressure raises gas solubility.
When applying this law, it's important not only to be careful with the units but also to double-check the conditions of temperature and type of gas involved, as these could affect how accurately the simple applications hold.

Pressure and Depth

The relationship between pressure and depth is an essential factor in understanding how gas solubility changes in different environments, such as underwater. As any diver descends into the ocean, they experience an increase in pressure due to the weight of the water above them.
To put it simply, every 33 feet (or roughly 10 meters) of water depth adds about 1 additional atmosphere of pressure. So, at a depth of 225 feet, a diver experiences approximately 8 atmospheres of pressure, not including the atmospheric pressure at the surface. This increased pressure affects the solubility of gases like nitrogen in a diver's blood, demonstrated by the exercise example where it increased from 1.90 mL/dL at 1 atm to 15.2 mL/dL at 8 atm.
Understanding this concept is crucial in ensuring divers safely control their ascent to let the gasses dissolve and escape properly. This helps in preventing illnesses like decompression sickness or "the bends," which occurs when gases are not regulated properly upon returning to the surface.