Problem 32 - University Physics with Modern Physics

Question

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 \(d=20.0 \mathrm{~m}\) in the ocean (density of seawater \(\left.=1024 \mathrm{~kg} / \mathrm{m}^{3}\right)\) and then travels aboard a small plane (with an unpressurized cabin) that rises to an altitude of \(h=5000 . \mathrm{m}\) above sea level.

Step-by-Step Solution

Answer

Answer: The total change in pressure experienced by the diver is -248159 N/m².

Step 1: Calculate initial pressure at depth

We first determine pressure at the initial depth due to the water above the diver. To do this, we need to use the hydrostatic pressure formula: \(P = \rho g h\). Here, \(P\) is the pressure, \(\rho\) is the density of seawater, \(g\) is the gravitational acceleration (approximately \(9.81 \mathrm{m/s^2}\)), and \(h\) is the depth. \(P_{water} = (1024\,\mathrm{kg/m^3}) \times (9.81\,\mathrm{m/s^2}) \times (20.0\,\mathrm{m})\) \(P_{water} = 200960\,\mathrm{N/m^2}\) Next, we also need to add atmospheric pressure at sea level (approximately \(101325\, \mathrm{N/m^2}\)) to the initial pressure. \(P_{initial} = P_{water} + P_{atm}\) \(P_{initial} = 200960\,\mathrm{N/m^2} + 101325\,\mathrm{N/m^2}\) \(P_{initial} = 302285\,\mathrm{N/m^2}\)

Step 2: Calculate final pressure at altitude

Now we need to find the atmospheric pressure at the end altitude (5000 meters above sea level). We know that the atmospheric pressure decreases with increasing altitude. We can use the barometric formula to find the pressure at the given altitude: \(P_{altitude} = P_{0} e^{(-\frac{mg}{kT} h)}\) Here, \(P_{0}\) is the sea level standard atmospheric pressure (\(101325\,\mathrm{N/m^2}\)), \(m\) is the molar mass of Earth's air (\(0.029\,\mathrm{kg/mol}\)), \(g\) is the gravitational acceleration (\(9.81\,\mathrm{m/s^2}\)), \(k\) is the Boltzmann constant (\(1.381\times10^{-23}\,\mathrm{J/K}\)), \(T\) is the temperature (assuming \(293\,\mathrm{K}\)), and \(h\) is the altitude. \(P_{altitude} = 101325 \times e^{(-\frac{0.029 \times 9.81}{1.381\times10^{-23} \times 293} \times 5000)}\) \(P_{altitude} \approx 54126\,\mathrm{N/m^2}\)

Step 3: Calculate the total change in pressure

Now we can find the total change in pressure by subtracting the initial pressure from the final pressure: \(\Delta P = P_{altitude} - P_{initial}\) \(\Delta P = 54126\,\mathrm{N/m^2} - 302285\,\mathrm{N/m^2}\) \(\Delta P = -248159\,\mathrm{N/m^2}\) The total change in pressure is -\(248159\,\mathrm{N/m^2}\). Since the change in pressure is negative, this means that the total pressure experienced by the diver decreased from the starting depth to the end altitude.

Key Concepts

Scuba Diving Decompression

Understanding scuba diving decompression is crucial for any diver venturing into the depths of the ocean. Decompression is the process divers use to safely adapt to the changes in pressure as they ascend from deep waters. When a diver submerges, water pressure increases, and gases like nitrogen dissolve into their bloodstream at higher rates. If the diver ascends too quickly, these gases can form bubbles in the body, a condition known as decompression sickness or 'the bends.' To prevent this, divers must follow a decompression schedule, which allows nitrogen to be released slowly and safely from their body.

In our example, the scuba diver experiences a significant pressure change from a depth of 20 meters to an altitude of 5000 meters above sea level. This extreme shift necessitates a careful decompression strategy, taking into account both the underwater pressure and the decreased atmospheric pressure at high altitudes. By calculating the total pressure change, the diver can determine the necessary decompression stops and duration required to avoid the bends.

Barometric Formula

The barometric formula is a key component in understanding how atmospheric pressure decreases with increasing altitude. This equation expresses the relationship between the pressure and the height above sea level. It is based on the principle that the atmosphere becomes less dense, and thus less pressurized, as altitude increases.

Barometric Formula Explanation:

At sea level, we experience standard atmospheric pressure, but as we ascend, say in a small plane with an unpressurized cabin, the pressure decreases exponentially. The reduction in pressure at a specific altitude is calculated using the formula:\[P_{altitude} = P_{0} e^{(-\frac{mg}{kT} h)}\]where the variables represent sea-level atmospheric pressure \(P_{0}\), molar mass of air \(m\), gravitational acceleration \(g\), Boltzmann constant \(k\), temperature \(T\), and altitude \(h\). This formula helps us understand how atmospheric conditions change, which is particularly critical for activities such as aviation, mountaineering, and indeed, for planning the decompression phases in scuba diving.

Pressure Change Calculation

Calculating the change in pressure that a scuba diver experiences is a pivotal part of planning a safe dive and successful ascent. To figure out this change, we first need to determine the hydrostatic pressure caused by the weight of water above the diver at a certain depth. This is done using the formula:\[P = \rho g h\]where \(P\) is the pressure, \(\rho\) is the density of the fluid (seawater in this scenario), \(g\) is the acceleration due to gravity, and \(h\) is the depth underwater.

Once we've established the pressure underwater, we need to calculate the pressure at the altitude the diver will ascend to. For this, we can employ the barometric formula discussed above. These two values give us the initial and final pressures, which we can then compare to deduce the total change in pressure as:\[\text{Pressure change, } \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \underline{\phantom{xxx}} \text: \underline{\phantom{xxx}} \underline{\phantom{xxx}} \text{ equal to the final pressure minus the initial pressure.\]