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Chapter 18: Problem 14

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锟絪 bottom temperature, given as 4.0掳C, is converted to Kelvin by calculating it to be 277.15 K. Similarly, the surface temperature of 23.0掳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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Most popular questions from this chapter

The vapor pressure is the pressure of the vapor phase of a substance when it is in equilibrium with the solid or liquid phase of the substance. The relative humidity is the partial pressure of water vapor in the air divided by the vapor pressure of water at that same temperature, expressed as a percentage. The air is saturated when the humidity is \(100 \%\). (a) The vapor pressure of water at \(20.0^{\circ} \mathrm{C}\) is \(2.34 \times 10^{3} \mathrm{~Pa}\). If the air temperature is \(20.0^{\circ} \mathrm{C}\) and the relative humidity is \(60 \%,\) what is the partial pressure of water vapor in the atmosphere (that is, the pressure due to water vapor alone)? (b) Under the conditions of part (a), what is the mass of water in \(1.00 \mathrm{~m}^{3}\) of air? (The molar mass of water is \(18.0 \mathrm{~g} / \mathrm{mol}\). Assume that water vapor can be treated as an ideal gas.)

A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is \(500.0 \mathrm{~m}^{3}\) and the surrounding air is at \(15.0^{\circ} \mathrm{C}\) what must the temperature of the air in the balloon be for it to lift a total load of \(290 \mathrm{~kg}\) (in addition to the mass of the hot air)? The density of air at \(15.0^{\circ} \mathrm{C}\) and atmospheric pressure is \(1.23 \mathrm{~kg} / \mathrm{m}^{3}\).

A balloon of volume \(750 \mathrm{~m}^{3}\) is to be filled with hydrogen at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{~Pa}\right) .\) (a) If the hydrogen is stored in cylinders with volumes of \(1.90 \mathrm{~m}^{3}\) at a gauge pressure of \(1.20 \times 10^{6} \mathrm{~Pa}\), how many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if both the gas in the balloon and the surrounding air are at \(15.0^{\circ} \mathrm{C} ?\) The molar mass of hydrogen \(\left(\mathrm{H}_{2}\right)\) is \(2.02 \mathrm{~g} / \mathrm{mol} .\) The density of air at \(15.0^{\circ} \mathrm{C}\) and atmospheric pressure is \(1.23 \mathrm{~kg} / \mathrm{m}^{3} .\) See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass \(4.00 \mathrm{~g} / \mathrm{mol}\) ) instead of hydrogen, again at \(15.0^{\circ} \mathrm{C} ?\)

A parcel of air over a campfire feels an upward buoyant force because the heated air is less dense than the surrounding air. By estimating the acceleration of the air immediately above a fire, one can estimate the fire's temperature. The mass of a volume \(V\) of air is \(n M_{\text {air }},\) where \(n\) is the number of moles of air molecules in the volume and \(M_{\text {air }}\) is the molar mass of air. The net upward force on a parcel of air above a fire is roughly given by \(\left(m_{\text {out }}-m_{\text {in }}\right) g,\) where \(m_{\text {out }}\) is the mass of a volume of ambient air and \(m_{\text {in }}\) is the mass of a similar volume of air in the hot zone. (a) Use the ideal-gas law, along with the knowledge that the pressure of the air above the fire is the same as that of the ambient air, to derive an expression for the acceleration \(a\) of an air parcel as a function of \(\left(T_{\text {out }} / T_{\text {in }}\right),\) where \(T_{\text {in }}\) is the absolute temperature of the air above the fire and \(T_{\text {out }}\) is the absolute temperature of the ambient air. (b) Rearrange your formula from part (a) to obtain an expression for \(T_{\text {in }}\) as a function of \(T_{\text {out }}\) and \(a\). (c) Based on your experience with campfires, estimate the acceleration of the air above the fire by comparing in your mind the upward trajectory of sparks with the acceleration of falling objects. Thus you can estimate \(a\) as a multiple of \(g .\) (d) Assuming an ambient temperature of \(15^{\circ} \mathrm{C}\), use your formula and your estimate of \(a\) to estimate the temperature of the fire.

Modern vacuum pumps make it easy to attain pressures of the order of \(10^{-13}\) atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of \(9.00 \times 10^{-14}\) atm and an ordinary temperature of \(300.0 \mathrm{~K}\), how many molecules are present in a volume of \(1.00 \mathrm{~cm}^{3} ?\) (b) How many molecules would be present at the same temperature but at 1.00 atm instead?
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