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A diver observes a bubble of air rising from the bottom of a lake (where the absolute pressure is \(3.50 \mathrm{~atm}\) ) to the surface (where the pressure is 1.00 atm \() .\) The temperature at the bottom is \(4.0^{\circ} \mathrm{C},\) and the temperature at the surface is \(23.0^{\circ} \mathrm{C}\). (a) What is the ratio of the volume of the bubble as it reaches the surface to its volume at the bottom? (b) Would it be safe for the diver to hold his breath while ascending from the bottom of the lake to the surface? Why or why not?

Short Answer

Expert verified
(a) The volume ratio of the bubble at the surface to its volume at the bottom of the lake is 3.27. (b) No, it would not be safe for a diver to hold his breath while ascending due to risk of pulmonary barotrauma.

Step by step solution

01

Calculate the ratio of the absolute pressures

We start by calculating the ratio of the absolute pressures at the bottom and surface of the lake, respectively. The absolute pressure is the atmospheric pressure at sea level (1 atm) plus the pressure under the water. It is given that the pressure at the surface is 1 atm and the pressure at the bottom is 3.5 atm. The pressure ratio \(P1/P2\) is thus \(1.00 / 3.50 = 0.286\).
02

Convert temperatures to the Kelvin scale

The temperatures given are in the Celsius scale, we need to convert them into the absolute temperature scale (Kelvin) to use in the ideal gas law. The conversion is done using the formula: \(Kelvin = degrees Celsius + 273\). So, the temperature at the bottom is \(277 K\) and at the surface is \(296 K\). The temperature ratio \(T1/T2\) is \(296 / 277 = 1.069\).
03

Calculate the volume ratio

Now, we can calculate the volume ratio using the relation \(P1/P2 = V2/V1 * T1/T2\). By substituting the already computed values of \(P1/P2\) and \(T1/T2\), we can solve for \(V2/V1 = P1/P2 * T2/T1 = 0.286 * 1.069 = 0.306\) which means that the volume of the air bubble is about 3.27 times larger at the surface than at the bottom of the lake.
04

Discuss safety issues

The increase in volume of a gas bubble from the bottom to the top of a lake seen in step 3 can be dangerous for a diver if he holds his breath. Trapped air in the diver's lungs will also expand as he ascends, and if the breath is held this can lead to rupture of the lung, an injury known as pulmonary barotrauma. Therefore, it would not be safe for the diver to hold his breath while ascending.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Pressure
Understanding absolute pressure is critical in underwater activities such as diving. The concept of absolute pressure plays a vital role in calculating how gas volumes change with depth. Absolute pressure is the total pressure experienced by an object, including atmospheric pressure and the pressure exerted by the water above it.

To calculate absolute pressure at a certain depth, you add the atmospheric pressure to the pressure exerted by the weight of the water. For instance, at the bottom of the lake, where the absolute pressure is given as 3.50 atm, this includes both the atmospheric pressure (1 atm at sea level) and the pressure from the water column above the point of interest. The pressure decreases as the diver ascends, equating to 1 atm (atmospheric pressure) at the surface.

In diving, understanding the changes in absolute pressure is essential for safety and proper planning of the dive profile to avoid decompression sickness or barotrauma due to rapid changes in pressure.
Temperature Conversion Kelvin
When studying the effects of pressure and volume changes under water, such as when a diver observes an air bubble rising, you must use the Kelvin scale for temperature. While Celsius is a common scale for everyday temperature measurements, Kelvin is the SI unit of thermodynamic temperature and is used in scientific equations, including those for gas laws.

To convert Celsius to Kelvin, add 273.15. For example, the lake�s bottom temperature, given as 4.0°C, is converted to Kelvin by calculating it to be 277.15 K. Similarly, the surface temperature of 23.0°C is 296.15 K. These Kelvin temperatures are used in calculations involving the ideal gas law because it is an absolute scale where 0 K, or absolute zero, represents the point where no more thermal energy can be removed from a substance.

Importance in Diving:

Using Kelvin allows divers and scientists to calculate how gas volumes will respond to temperature changes, crucial for understanding how gases behave under the varying conditions encountered in diving.
Pulmonary Barotrauma
Pulmonary barotrauma is a significant concern in diving and underwater activities; it refers to the damage to lung tissue caused by expansion of trapped air in the lungs while ascending. When a diver breath-holds and ascends, the air within the lungs expands due to the decrease in ambient pressure (according to Boyle's Law) which can lead to the rupture of lung tissue.

Dangers:

The increasing volume of an air bubble, as seen in the example of the bubble rising from the bottom to the surface of the lake, directly mirrors what can happen to a diver's lungs if they hold their breath while ascending. This highlights why divers are taught to never hold their breath and to always ascend slowly and breathe continuously, to allow the air in their lungs to safely expand and escape.

Preventive Measures:

To mitigate the risk of pulmonary barotrauma, divers must understand and follow proper ascent procedures. This includes continuous breathing, controlling ascent rate, and using properly maintained equipment. Understanding the potential hazards, such as those suggested by the exercise where the volume of a bubble increases significantly while ascending, is crucial for diving safety.

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