It is reasonable to assume that the bulk modulus of blood is about the same as that of water (2.20 GPa). As one goes deeper and deeper in the ocean, the pressure increases by 1.0*10^4 Pa for every meter below the surface.
If a diver goes down 33.0 m (a bit over 100 ft) in the ocean, by how much does each cubic centimeter of her blood change in volume? 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?
Compute the pressure change at depth of 33.0 m. It's about 3.3 times higher than the pressure at the surface
(density change)/density = change in pressure)/E
Where E is the bulk modulus. It's going to be a very small number: roughly 10^-5.
For the last part of the question, set
(density change)/density = 0.5 = P/E, and solve for the required pressure and ocean depth.
To find the change in volume per cubic centimeter of blood, we can use the formula:
Change in volume = Bulk modulus * Change in pressure / Initial volume
Given that the bulk modulus of blood (assumed to be similar to water) is 2.20 GPa (which is 2.20 x 10^9 Pa) and the change in pressure per meter is 1.0 x 10^4 Pa, we can calculate the change in volume per cubic centimeter:
Change in volume = (2.20 x 10^9 Pa) * (1.0 x 10^4 Pa) / (1 cm^3)
Calculating this, we find that each cubic centimeter of blood changes in volume by:
Change in volume = 2.20 x 10^5 cm^3
Therefore, each cubic centimeter of blood changes in volume by 2.20 x 10^5 cm^3 when the diver goes down 33.0 m in the ocean.
To find the depth at which each drop of blood compresses to half its volume at the surface, we can use the fact that the change in volume is proportional to the change in pressure. Since we want the volume to be halved, we need the change in pressure to be twice the initial change in pressure.
Let's assume the initial volume of a drop of blood at the surface is V. To halve its volume, we need a change in volume of V/2.
Using the formula again:
V/2 = Bulk modulus * (2 * Change in pressure) / Initial volume
Simplifying:
V/2 = (2.20 x 10^9 Pa) * (2 * 1.0 x 10^4 Pa) / V
Simplifying further, we obtain:
V^2 = 2.20 x 10^9 * 2 * 1.0 x 10^4 * 2
V^2 = 8.80 x 10^17
Taking the square root of both sides:
V = √(8.80 x 10^17)
V ≈ 2.97 x 10^8 cm^3
This is the volume of a drop of blood at the surface.
To find the depth at which each drop compresses to half its volume, we can use the formula for change in pressure:
Change in pressure = 1.0 x 10^4 Pa/m
For the volume to change from V to V/2, the change in pressure should be twice the initial change in pressure:
2 * 1.0 x 10^4 = (1.0 x 10^4 Pa/m) * Depth
Depth = 2 meters
Therefore, the diver must go down 2 meters for each drop of blood to compress to half its volume.
Now, considering the depth the diver went in the ocean, which is 33.0 m, we can see that each cubic centimeter of blood changes in volume by 2.20 x 10^5 cm^3.
To halve the volume of a drop of blood, they need to go down 2 meters. Since 33.0 m is greater than 2 meters, we can conclude that the ocean is deep enough to have this effect on the diver.
To determine the change in volume of each cubic centimeter of blood as the diver descends, we need to use the bulk modulus formula:
ΔV/V = -BΔP/P
where:
ΔV/V is the relative change in volume
B is the bulk modulus
ΔP is the change in pressure
P is the initial pressure
Given that the bulk modulus of blood is assumed to be the same as water at 2.20 GPa, we can convert this to Pascals (Pa) by multiplying by 10^9. Therefore, B = 2.20 * 10^9 Pa.
The change in pressure (ΔP) can be calculated by multiplying the pressure change per meter (1.0 * 10^4 Pa/m) by the depth the diver goes (33.0 m).
ΔP = 1.0 * 10^4 Pa/m * 33.0 m
Now, we can calculate the relative change in volume (ΔV/V) for each cubic centimeter of blood:
ΔV/V = -BΔP/P
Before we calculate the exact value, we need to know the initial pressure (P) of the blood. Since this information is not given, we cannot proceed with the precise calculation. However, we can make an approximation by assuming the initial pressure of blood to be the atmospheric pressure, which is approximately 101,325 Pa.
Using this value for P, we can find the relative change in volume:
ΔV/V = -BΔP/P
ΔV/V = -(2.20 * 10^9 Pa) * (1.0 * 10^4 Pa/m * 33.0 m) / (101,325 Pa)
By solving this equation, you can determine the relative change in volume for each cubic centimeter of blood as the diver descends 33.0 m.
Now, let's move on to the second part of the question. We need to find the depth at which each drop of blood compresses to half its volume at the surface.
To determine this, we can set up the following equation:
ΔV/V = -BΔP/P
Here, ΔV/V represents the relative change in volume, which is equal to -0.5 (since we want the volume to halve) and ΔP is the change in pressure. P represents the initial pressure of the blood.
Rearranging the equation, we have:
ΔP = -(ΔV/V)(P/B)
Since we want ΔP to be equal to the atmospheric pressure (101,325 Pa) to calculate the depth, we can substitute these values into the equation:
101,325 Pa = -(0.5)(P)/(2.20 * 10^9 Pa)
By solving this equation, you can find the depth at which each drop of blood compresses to half its volume at the surface.
Finally, compare the depth obtained from the second part of the question with the depth the diver goes (33.0 m) to determine if the ocean is deep enough to have this effect on the diver.