Question

The volume of blood in the body of a certain deep-sea diver is about \(6.00 \mathrm{~L}\). Blood cells make up about \(55 \%\) of the blood volume, and the remaining \(45 \%\) is the aqueous solution called plasma. What is the maximum volume of nitrogen measured at \(1.00 \mathrm{~atm}\) and \(37^{\circ} \mathrm{C}\) that could dissolve in the diver's blood plasma at a depth of \(93 \mathrm{~m}\), where the pressure is \(10.0\) atm? (This is the volume that could come out of solution suddenly, causing the painful and dangerous condition called the bends, if the diver were to ascend too quickly.) Assume that Henry's constant for nitrogen at \(37^{\circ} \mathrm{C}\) (body temperature) is \(5.8 \times 10^{-4} \mathrm{~mol} \cdot \mathrm{L}^{-1}\)-atm \({ }^{-1}\).

Short answer

The maximum volume of nitrogen that could dissolve in the diver's blood plasma at 93 meters depth is 0.405 L.

Step-by-step solution

Calculate the volume of plasma

To find the volume of plasma, calculate 45% of the total blood volume. Since the total blood volume is 6.00 L, the plasma volume is: \[ \text{Plasma volume} = 0.45 \times 6.00 \text{ L} = 2.70 \text{ L} \]

Apply Henry's Law

Henry's Law states that the amount of gas dissolved in a liquid is directly proportional to the pressure of the gas above the liquid. The formula is:\[ C = kP \]where \( C \) is the concentration of the gas, \( k \) is Henry's constant, and \( P \) is the pressure. Use this law to find the maximum volume of nitrogen that can dissolve in the diver's blood plasma at a pressure of 10.0 atm.

Calculate the amount of nitrogen that can dissolve

Using the given Henry's constant for nitrogen, \( k = 5.8 \times 10^{-4} \mathrm{~mol/L/atm} \), and the pressure at depth, \( P = 10.0 \mathrm{~atm} \), the maximum amount of nitrogen that could dissolve in the diver's plasma can be calculated using:\[ C = k \times P \]\[ C = (5.8 \times 10^{-4} \mathrm{~mol/L/atm}) \times (10.0 \mathrm{~atm}) \]\[ C = 5.8 \times 10^{-3} \mathrm{~mol/L} \]

Calculate the volume of nitrogen gas at STP using the ideal gas law

The ideal gas law states that \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume of gas, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. Convert the body temperature to Kelvin (310 K), use the calculated concentration for \( n \) (moles per liter of plasma), and because we're asked for the volume at 1 atm and 310 K, the volume can be calculated as:\[ V = \frac{nRT}{P} \]Using \( R = 0.0821 \mathrm{~L \cdot atm/mol \cdot K} \) and rearranging for \( V \):\[ V = \frac{C \times \text{Plasma volume} \times R \times T}{P} \]\[ V = \frac{5.8 \times 10^{-3} \mathrm{~mol/L} \times 2.70 \mathrm{~L} \times 0.0821 \mathrm{~L \cdot atm/mol \cdot K} \times 310 \mathrm{~K}}{1 \mathrm{~atm}} \]\[ V = 0.405 \mathrm{~L} \]

Interpret the result

The maximum volume of nitrogen that could dissolve in the diver's plasma at 93 m depth and potentially cause the bends upon rapid ascent is 0.405 L.

Key concepts

Blood Plasma

To understand the health risks associated with deep-sea diving, such as decompression sickness, we must first understand the role of blood plasma. Blood plasma is the liquid component of blood that makes up about 45% of the total blood volume. It is primarily composed of water, but it also contains dissolved salts, glucose, hormones, and various proteins. More importantly for divers, plasma is the medium through which gases like oxygen and nitrogen are transported and dissolved.

When we breathe, these gases enter our bloodstream, with oxygen being utilized by our cells and nitrogen remaining dissolved in our blood plasma under the pressure exerted by the surrounding environment, which at the surface is 1 atmosphere (atm). During a dive, as pressure increases, the amount of nitrogen that can dissolve in the plasma also increases, following Henry's Law. This can become a potential health issue for divers ascending too quickly since the excess nitrogen may form bubbles in the bloodstream, leading to decompression sickness. To mitigate this risk, divers must ascend gradually, allowing nitrogen to safely leave the body through exhalation.

Ideal Gas Law

In dealing with the solubility of gases in blood plasma, the ideal gas law becomes a useful tool. This mathematical equation describes the behavior of an ideal gas by relating its pressure (P), volume (V), temperature (T), and amount in moles (n) using the formula: \[ PV = nRT \]

Here, R denotes the ideal gas constant. In the context of the exercise regarding the diver's dissolved nitrogen, we adhere to the ideal gas law to calculate the volume of nitrogen that could be released from the blood plasma at a standard pressure and temperature after ascending. When gases dissolve in blood plasma, they do so under the conditions of the body and environment but can be related back to standard conditions using this law. It's essential for interpreting how the volume of a dissolved gas changes with pressure and temperature, which is directly applicable to understanding how the volume of nitrogen changes in a diver's body.

Solubility of Gases

When a diver descends into the depths of the ocean, the solubility of gases in their blood becomes a critical factor to consider. The solubility of gases in a liquid, such as blood plasma, is dependent on both the nature of the gas and the liquid, as well as the temperature and the pressure of the environment. Henry's Law precisely quantifies this relationship by stating that the concentration of a gas in a liquid is directly proportional to the partial pressure of that gas over the liquid's surface, provided the temperature remains constant.

The formula derived from Henry's Law is: \[ C = kP \]where C is the concentration, k is Henry's constant for the gas in the particular liquid, and P is the partial pressure of the gas. An increase in pressure, like during a deep-sea dive, increases the gas concentration that can dissolve in the plasma. This is why nitrogen becomes more soluble in a diver's blood under high pressure, potentially leading to health issues if the diver ascends too quickly for the excess nitrogen to be released safely.

Decompression Sickness

When discussing the solubility of gases in diver's blood plasma, one health issue of paramount importance is decompression sickness (DCS), also known as 'the bends.' Decompression sickness occurs when dissolved gases, mainly nitrogen, form bubbles inside the body due to a rapid decrease in surrounding pressure when a diver ascends too quickly. These bubbles can cause joint pain, dizziness, paralysis, and even death if not treated promptly.

Symptoms of DCS arise because nitrogen bubbles can disrupt blood flow and damage tissue. As pressure decreases during ascent, nitrogen that was dissolved under high pressure in the diver's blood begins to undissolve. If ascent is too rapid, this process outpaces the body's ability to safely expel the gas through respiration, leading to bubble formation. The key to preventing DCS lies in a slow, controlled ascent, allowing nitrogen to diffuse out of plasma and be expelled from the body at a rate that avoids bubble formation.