A scuba diver must decompress after a deep dive to allow excess nitrogen to exit safely from his bloodstream. The length of time required for decompression depends on the total change in pressure that the diver experienced. Find this total change in pressure for a diver who starts at a depth of d = 18.9 m in the ocean (density of seawater = 1024 kg/m^3) and then travels aboard a small plane (with an unpressurized cabin) that rises to an altitude of h = 4600 m above sea level.
The pressure at the depth d
p1= ρ•g•d + p (atm) =1024•9.8•18.9 + 101325 =2.91•10 ⁵ Pa,
The pressure at the height h
p=p₀exp{-μgh/RT)
Since we don’t know the temperature, we use another expression.
Air pressure above sea level can be calculated as
p = 101325 (1 - 2.25577 10⁻⁵ h)⁵‧²⁶ , where
p = air pressure (Pa)
h = altitude above sea level (m)
p2=101325(1-2.26•10⁻⁵•4600)^5.26=5.68•10⁴ Pa
ΔP=2.91•10 ⁵-5.68•10⁴=2.34•10⁵ Pa
thanks!
To find the total change in pressure, we need to calculate the hydrostatic pressure at each location - the initial depth in the ocean and the final altitude in the plane.
The hydrostatic pressure at a given depth or altitude is given by the equation:
P = ρgh
Where:
P is the pressure,
ρ is the density of the fluid (seawater),
g is the acceleration due to gravity (9.8 m/s^2),
h is the depth or altitude.
First, let's calculate the pressure at the initial depth in the ocean:
P1 = ρ * g * d
Given:
ρ = 1024 kg/m^3 (density of seawater)
g = 9.8 m/s^2
d = 18.9 m
P1 = 1024 * 9.8 * 18.9
Next, let's calculate the pressure at the final altitude in the plane:
P2 = ρ * g * h
Given:
ρ = 1024 kg/m^3 (density of seawater)
g = 9.8 m/s^2
h = 4600 m
P2 = 1024 * 9.8 * 4600
Finally, the total change in pressure is the difference between P2 and P1:
Total change in pressure = P2 - P1
Let's calculate it.
To find the total change in pressure for the scuba diver, we first need to calculate the pressure at each location - underwater and at the altitude.
Step 1: Calculate the pressure underwater:
The pressure underwater is given by the hydrostatic pressure formula:
P1 = ρ * g * d
where P1 is the pressure underwater, ρ is the density of seawater, g is the acceleration due to gravity, and d is the depth.
Given values:
Density of seawater (ρ) = 1024 kg/m^3
Depth (d) = 18.9 m
Acceleration due to gravity (g) = 9.8 m/s^2
Substituting the values into the formula:
P1 = 1024 kg/m^3 * 9.8 m/s^2 * 18.9 m
P1 = 192883.2 Pa
Step 2: Calculate the pressure at altitude:
The pressure at altitude is given by the atmospheric pressure formula:
P2 = P0 * (1 - (L * h / T0))
where P0 is the atmospheric pressure at sea level, L is the temperature lapse rate, h is the altitude above sea level, and T0 is the temperature at sea level.
Given values:
Atmospheric pressure at sea level (P0) ≈ 101325 Pa
Temperature lapse rate (L) ≈ -0.0065 K/m
Altitude (h) = 4600 m (above sea level)
Temperature at sea level (T0) ≈ 288 K
Substituting the values into the formula:
P2 = 101325 Pa * (1 - (-0.0065 K/m * 4600 m) / 288 K)
P2 = 101325 Pa * (1 + 0.2993)
P2 = 132943 Pa
Step 3: Find the total change in pressure:
The total change in pressure is the difference between the pressure at altitude and the pressure underwater:
ΔP = P2 - P1
ΔP = 132943 Pa - 192883.2 Pa
ΔP ≈ -59940.2 Pa
Note: The negative sign indicates a decrease in pressure during ascent.
Therefore, the total change in pressure experienced by the scuba diver is approximately -59940.2 Pa.