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 125 feet and a pressure of 4.50 atm?

Short answer

At 4.50 atm, the solubility of nitrogen gas is 8.55 mL/dL.

Step-by-step solution

Understanding Initial Conditions

The initial solubility of nitrogen gas is given as \(1.90 \, \text{mL/dL}\) (milliliters per deciliter) at a pressure of \(1.00 \, \text{atm}\). This is our reference condition for using Henry's Law, which relates pressure and solubility.

Using 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. The mathematical formula is: \[ S_1 / P_1 = S_2 / P_2 \]where \(S_1\) and \(P_1\) are the initial solubility and pressure, and \(S_2\) and \(P_2\) are the final solubility and pressure we aim to find.

Substituting Given Values into Henry's Law

We know \(S_1 = 1.90 \, \text{mL/dL}\), \(P_1 = 1.00 \, \text{atm}\), and \(P_2 = 4.50 \, \text{atm}\). We need to solve for \(S_2\) using the equation:\[ \frac{1.90}{1.00} = \frac{S_2}{4.50} \]

Solving for Final Solubility

Rearrange the equation to find \(S_2\):\[ S_2 = 1.90 \, \text{mL/dL} \times 4.50 \]\[ S_2 = 8.55 \, \text{mL/dL} \]

Verification

Double-check the calculations to ensure that multiplying the initial solubility by the pressure ratio gives the expected final solubility. The proportional relationship in Henry's Law confirms this computation. Everything checks out.

Key concepts

Solubility

Solubility, in simple terms, is how much of a substance can dissolve in a liquid. Imagine trying to mix sugar into a cup of hot tea. Some point, you can't dissolve any more sugar no matter how much you stir. This is where the solution becomes saturated, and the sugar is at its maximum solubility.

In the context of gases, solubility refers to how much of a gas can dissolve into a liquid, such as blood. This is crucial for situations like human physiology and scuba diving, where gases, like nitrogen, play significant roles. Normally, gases are not very soluble. However, under high pressure conditions, more gas can dissolve into the liquid, increasing its solubility. This concept is central in understanding how gases behave under different pressures, as with Henry's Law.

Understanding solubility helps us comprehend scenarios such as why sodas go flat when opened. It's because the gas dissolved in the liquid under pressure escapes when the pressure is released.

Gas Pressure

Gas pressure is the force that gases exert when they collide with the walls of a container. Under ambient conditions, gases have a certain pressure, but this changes based on factors like volume, temperature, and the presence of other gases.

Henry's Law tells us that the solubility of a gas in a liquid is directly proportional to the pressure of that gas above the liquid. This means if the pressure increases, more gas molecules are pushed into the liquid, increasing their solubility. It's like squeezing more people into a crowded room by adding more pressure at the door.

In the exercise you saw, higher pressure at greater depths forces more nitrogen into the diver's blood. This exemplifies how pressure changes deeply impact gas solubility. Remember, divers need to ascend slowly to safely release this pressure without harmful issues like decompression sickness.

Scuba Diving

Scuba diving offers a fascinating application of gas pressure and solubility. Divers must understand the impact of increased pressure under water. At sea level, atmospheric pressure is around 1 atm. As divers descend, the pressure increases significantly, approximately by 1 atm every 10 meters (33 feet) of depth.

This rise in pressure affects the gases dissolved in the blood. At greater depths, more nitrogen dissolves into the bloodstream because of increased pressure. Diving too fast or staying deep for extended periods can lead to excess nitrogen dissolving. When ascending, a sudden decrease in pressure causes nitrogen to form bubbles, which can be extremely dangerous.

Thus, understanding Henry's Law and the concept of gas solubility is critical for divers to avoid "the bends" (decompression sickness). Safe diving practices involve controlling ascent rates and spending time at certain depths to allow nitrogen to safely escape the blood.