If deep-sea divers rise to the surface too quickly, nitrogen bubbles in their blood can expand and prove fatal.
This phenomenon is known as the bends. If a scuba diver rises quickly from a depth of 25.0 m in Lake Michigan (which is fresh water), what will be the volume at the surface of an bubble that occupied 1.00 mm^3 in his blood at the lower depth?(Assume that the pressure difference is due only to the changing water pressure, not to any temperature difference, an assumption that is reasonable, since we are warm-blooded creatures.)
V=?? mm^3
Does it seem that this difference is large enough to be a problem?
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3.2
To solve this problem, we need to use Boyle's Law, which states that the volume of a gas is inversely proportional to the pressure applied to it, assuming the temperature remains constant.
Boyle's Law can be expressed as:
P1 * V1 = P2 * V2
Where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
First, let's determine the initial pressure (P1) and volume (V1). The pressure at a depth of 25.0m is caused by the weight of the water above, which is given by the hydrostatic pressure formula:
P1 = ρ * g * h
Where:
ρ is the density of water (approximately 1000 kg/m^3 for fresh water)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the depth in meters
P1 = 1000 kg/m^3 * 9.8 m/s^2 * 25.0 m
P1 ≈ 245,000 Pa (pascal)
Now, let's calculate the final pressure (P2). Since the diver is rising to the surface, the final pressure will be the atmospheric pressure, which is approximately 101,325 Pa at sea level. However, we should note that this might differ depending on the location and altitude, so it's important to consider the accurate atmospheric pressure for Lake Michigan at the time.
With the pressure values defined, we can rearrange Boyle's Law equation to find the final volume (V2):
V2 = (P1 * V1) / P2
V2 = (245,000 Pa * 1.00 mm^3) / 101,325 Pa
Now we need to convert the volume from mm^3 to m^3 to match the unit of pressure:
V2 = (245,000 Pa * 1.00 * 10^(-9) m^3) / 101,325 Pa
Simplifying further:
V2 ≈ 2.42 * 10^(-12) m^3
To answer whether this difference is large enough to be a problem, it's important to consult medical professionals who specialize in diving-related issues. They can provide accurate information about the risk of getting the bends and the acceptable volume change for a specific individual.