Problem 13 - College Physics

Question

Effect of diving on blood. It is reasonable to assume that the bulk modulus of blood is about the same as that of water ( \(2.2 \mathrm{GPa}\) ). As one goes deeper and deeper in the ocean, the pressure increases by \(1.0 \times 10^{4} \mathrm{~Pa}\) for every meter below the surface. (a) If a diver goes down \(33 \mathrm{~m}\) (a bit over \(100 \mathrm{ft}\) ) in the ocean, by how much does each cubic centimeter of her blood change in volume? (b) How deep must a diver go so that each drop of blood compresses to half its volume at the surface? Is the ocean deep enough to have this effect on the diver?

Step-by-Step Solution

Answer

Volume decreases by \(-1.5 \times 10^{-4} \; \text{cm}^3\) at 33 m. The ocean is not deep enough for blood to compress to half its volume.

Step 1: Understanding the Bulk Modulus Concept

The bulk modulus is a measure of a substance's resistance to uniform compression. In this problem, the bulk modulus of blood is given as that of water, which is approximately \( 2.2 \times 10^9 \; \text{Pa} \). The formula for the bulk modulus is:\[K = -V \frac{\Delta P}{\Delta V}\]where \( K \) is the bulk modulus, \( \Delta P \) is the change in pressure, \( V \) is the original volume, and \( \Delta V \) is the change in volume.

Step 2: Calculating Pressure Increase at 33 Meters Depth

The pressure increase per meter of depth is \(1.0 \times 10^{4} \; \text{Pa/m} \). Therefore, at \(33 \; \text{m}\), the change in pressure is:\[\Delta P = 33 \times 1.0 \times 10^{4} = 3.3 \times 10^5 \; \text{Pa}\]This additional pressure will be applied to each cubic centimeter of blood.

Step 3: Determining Volume Change for Blood at 33 Meters

Using the bulk modulus formula:\[2.2 \times 10^9 = -1 \frac{3.3 \times 10^5}{\Delta V}\]We solve for \( \Delta V \) (given \( V = 1 \; \text{cm}^3 = 1 \; \text{ml} \), the original volume):\[\Delta V = \frac{-3.3 \times 10^5}{2.2 \times 10^9} \approx -1.5 \times 10^{-4} \; \text{cm}^3\]This indicates a small decrease in volume.

Step 4: Finding Depth for Blood Volume to Halve

To find the depth where blood compresses to half its volume, we need \( \Delta V = -0.5 \; \text{cm}^3 \). Using the bulk modulus:\[2.2 \times 10^9 = -1 \frac{P}{-0.5}\]Solving for \( P \):\[P = 2.2 \times 10^9 \times 0.5 = 1.1 \times 10^9 \; \text{Pa}\]The pressure increase per meter is \(1.0 \times 10^4 \; \text{Pa/m} \).\[\text{Depth} = \frac{1.1 \times 10^9}{1.0 \times 10^4} = 1.1 \times 10^5 \; \text{m}\]

Step 5: Final Evaluation of Ocean Depths

Considering the calculation, \(1.1 \times 10^5 \; \text{m} \) is much deeper than any ocean on Earth, as average ocean depth is about 4,000 meters. Thus, it is not possible for the ocean to compress blood to half its volume under normal conditions.

Key Concepts

Pressure Increase in Diving

When diving deep into the ocean, a diver experiences an increase in pressure due to the weight of the water above them. Water is much heavier than air, and as you dive deeper, the amount of water and, consequently, the pressure increases. This increase happens in a very predictable way: every meter deeper you go, the pressure increases by approximately 10,000 pascals (Pa). This means at 33 meters underwater, the pressure is 330,000 Pa greater than at the surface.
This additional pressure can affect different aspects of a diver's physiology, particularly influencing how their body handles internal and external pressure changes during a dive.

Compression of Fluids

Fluids, like blood, have an interesting property known as "bulk modulus," which describes how compressible a substance is under pressure. The bulk modulus of blood is similar to that of water, set at approximately 2.2 GPa.
This measure indicates blood's resistance to being compressed. Utilizing the bulk modulus formula, we can calculate how much a given volume of blood (or any liquid) changes under certain pressures. For example, if a diver goes to a depth of 33 meters, the increase in pressure leads to a tiny decrease in the volume of each cubic centimeter of blood—specifically, about 0.00015 cm³ per cm³.
This change in volume is slight, but it's significant for understanding fluid behaviors under pressure.

Effects of Depth on Volume

As the diver descends deeper beneath the ocean surface, the effect of increasing pressure becomes more pronounced, especially on the volume of compressible fluids like blood. However, it's crucial to note that the changes are small unless extreme depths are reached.
To halve the volume of blood, the pressure needed is enormous. Calculations show that this would require diving to around 110,000 meters deep. Since no part of the Earth's oceans are that deep (they max out around 11,000 meters), such compression to half its volume is practically impossible in natural conditions.
Understanding these effects is vital for divers and people studying fluid dynamics as it addresses safety and physiological impacts on the body in underwater environments.