Question

The solubility of ${\mathrm{N}}_2$ in blood at ${37}^{\circ }\mathrm{C}$ and at a partial pressure of 0.80 atm is $5.6\times {10}^{-4}\mathrm{mol}/\mathrm{L}$. A deep-sea diver breathes compressed air with the partial pressure of ${\mathrm{N}}_2$ equal to $4.0\mathrm{atm}$. Assume that the total volume of blood in the body is $5.0\mathrm{L}$. Calculate the amount of ${\mathrm{N}}_2$ gas released (in liters at ${37}^{\circ }\mathrm{C}$ and $1\mathrm{atm})$ when the diver returns to the surface of the water, where the partial pressure of ${\mathrm{N}}_2$ is $0.80\mathrm{atm}$.

Short answer

The volume of $N_2$ gas released is approximately 0.285 liters.

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. Mathematically, it is expressed as: $S_1/P_1=S_2/P_2$, where $S$ is the solubility and $P$ is the partial pressure. Here, $S_1=5.6\times {10}^{-4}$ mol/L, $P_1=0.80$ atm, and $P_2=4.0$ atm.

Calculate Solubility at 4.0 atm

Using Henry's Law, solve for $S_2$, the solubility at $P_2=4.0$ atm: $S_2=S_1\times \frac{P_2}{P_1}=5.6\times {10}^{-4}\times \frac{4.0}{0.80}=2.8\times {10}^{-3}$ mol/L.

Determine Amount of N2 Dissolved at 4.0 atm

Multiply the solubility $S_2$ by the total blood volume $V=5.0$ L to find the moles of $N_2$ dissolved at 4.0 atm: $extmolesof{N}_2=2.8\times {10}^{-3}\times 5.0=0.014$ moles.

Determine Amount of N2 Dissolved at 0.8 atm

Calculate the moles of $N_2$ dissolved at the surface pressure using $S_1$: $extmolesof{N}_2=5.6\times {10}^{-4}\times 5.0=2.8\times {10}^{-3}$ moles.

Calculate Moles of N2 Released

Subtract the moles of $N_2$ at 0.8 atm from the moles at 4.0 atm to determine the moles released: $0.014-2.8\times {10}^{-3}=0.0112$ moles.

Volume of N2 Released at Surface Conditions

Use the ideal gas law $PV=nRT$ to find the volume of $N_2$ at 1 atm and ${37}^{\circ }\mathrm{C}$ (310 K), taking $R=0.0821$ L⋅atm/mol⋅K: $V=\frac{nRT}{P}=\frac{0.0112\times 0.0821\times 310}{1}\approx 0.285$ L.

Key concepts

Gas Solubility

Gas solubility refers to the amount of gas that can dissolve in a liquid. This is a crucial concept for understanding how gases behave when they come into contact with liquids, such as blood in the human body. Solubility depends on several factors like temperature and the pressure of the gas above the liquid.

In this context, Henry's Law provides a formula to quantify this relationship: $S=k\cdot P$, where $S$ is the solubility, $k$ is the Henry's Law constant, and $P$ is the partial pressure. The law tells us that solubility increases when the pressure of the gas increases.

• This means if you increase the pressure, the gas becomes more soluble.
• Conversely, when the pressure decreases, the gas solubility decreases.

This is why divers have to be cautious when they return to the surface; the change in pressure can lead to dissolved gases releasing from their bodies, a phenomenon linked to decompression sickness.

Partial Pressure

Partial pressure is the pressure of a single type of gas in a mixture of gases. It's crucial in calculating gas solubility using Henry's Law. In a mixture, each gas contributes to the total pressure exerted by the mixture.

For instance, when a diver breathes compressed air, the partial pressure of \ ${\text{N}}_2$ is elevated, increasing its solubility in blood.

• If \ ${\text{N}}_2$ partial pressure is 4.0 atm in the diver's blood at depth, more \ ${\text{N}}_2$ will dissolve than at the surface, where partial pressure returns to 0.8 atm.
• Understanding this concept helps predict how much gas will be released from the solution when pressure decreases.

When the diver returns to the surface, it is crucial to account for the change in partial pressure to avoid problems like the bends, caused by rapid gas release.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the volume, pressure, temperature, and moles of a gas using the equation $PV=nRT$. Here $P$ is the pressure, $V$ is the volume, $n$ represents the moles of gas, $R$ is the ideal gas constant, and $T$ is the temperature in Kelvin.

This law is particularly useful for converting the amount of gas released from a liquid into a measurable volume under standard conditions. In our case, it helps calculate the volume of nitrogen gas ${\text{N}}_2$ released when a diver surfaces.

• The law simplifies solving for the unknown volume by knowing the moles of gas, temperature, and pressure.
• It assumes gases behave ideally, making it a vital tool in chemistry for understanding gas behavior under varying conditions.

By using the Ideal Gas Law, it's possible to compute the volume of nitrogen gas released accurately, ensuring divers can safely manage pressure changes.