Problem 14 - University Physics with Modern Physics

Question

A diver observes a bubble of air rising from the bottom of a lake (where the absolute pressure is 3.50 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?

Step-by-Step Solution

Answer

(a) The volume ratio is approximately 3.74. (b) It is unsafe to hold breath while ascending due to expanding lung air.

Step 1: Understand the Given Parameters

We are given the absolute pressures at the bottom and surface of the lake, which are 3.50 atm and 1.00 atm, respectively. The temperatures given are 4.0°C at the bottom and 23.0°C at the surface. We need to find the ratio of the volume of the bubble at the surface to its volume at the bottom.

Step 2: Convert Temperatures to Kelvin

To apply the ideal gas law, we need to convert the temperatures from Celsius to Kelvin. - Temperature at the bottom: 4°C = 4 + 273.15 = 277.15 K. - Temperature at the surface: 23°C = 23 + 273.15 = 296.15 K.

Step 3: Apply the Combined Gas Law

The pressure-volume-temperature relationship for a gas is given by the combined gas law: \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \).- Let's denote: - \( P_1 = 3.50 \) atm, \( T_1 = 277.15 \) K. - \( P_2 = 1.00 \) atm, \( T_2 = 296.15 \) K.- We need to find the ratio \( \frac{V_2}{V_1} \).

Step 4: Rearrange for the Volume Ratio

Rearrange the equation to find the volume ratio:\[ \frac{V_2}{V_1} = \frac{P_1 \times T_2}{P_2 \times T_1} \]. Substitute the known values:\[ \frac{V_2}{V_1} = \frac{3.50 \times 296.15}{1.00 \times 277.15} \].

Step 5: Calculate the Ratio

Calculate the values:\[ \frac{V_2}{V_1} = \frac{3.50 \times 296.15}{277.15} \approx 3.74 \]. This is the ratio of the bubble's volume at the surface to its volume at the bottom.

Step 6: Safety Considerations for Holding Breath

It would not be safe for the diver to hold his breath while ascending. As the diver rises, the pressure decreases, causing the air in the lungs to expand. This expansion could potentially lead to lung over-expansion injuries if the diver holds his breath.

Key Concepts

Pressure and Volume

When a gas, like the air in a bubble from a diver, moves from one place to another under different pressure conditions, its volume changes. This is all about the relationship between pressure and volume in gases.
When the bubble is at the bottom of the lake, it experiences a higher pressure of 3.50 atm. As it rises to the surface, the pressure decreases to 1.00 atm.
Because of this, the volume of the bubble will increase, as there is less pressure compressing it.
You'll find this pressure-volume relationship in Boyle's Law, which we'll explore further in another section.

• As pressure decreases, volume increases.
• As pressure increases, volume decreases.

Temperature Conversion

Understanding temperatures in the context of gas laws requires converting from Celsius to Kelvin. This is crucial since Kelvin is the temperature scale used in gas law calculations.
In our problem, at the bottom of the lake, the temperature is 4°C, which converts to 277.15 K by adding 273.15. At the surface, the temperature is 23°C, which is 296.15 K.

• Always add 273.15 to convert Celsius to Kelvin.
• Using Kelvin helps in deriving accurate volume or pressure values in the combined gas law.

Converting temperatures ensures accuracy and consistency when applying gas laws like the Combined Gas Law.

Boyle's Law

Boyle's Law is a crucial principle in understanding how gases behave under pressure changes. It states that the volume of a gas is inversely proportional to its pressure, as long as the temperature remains constant.
This means if you increase the pressure on a gas, its volume decreases. Conversely, if you reduce the pressure, the volume increases.

• This principle helps predict how the bubble's volume changes as it ascends.
• While the temperature also plays a role, Boyle's Law provides clear insight into the basic pressure-volume aspect.

Using Boyle's Law, we can easily understand the expansion of the air bubble. However, remember that in real situations, temperature changes are also significant, thus, the combined gas law is often used.

Safety in Diving

Diving safety is crucial and understanding gas behavior underwater can prevent potential hazards. When a diver ascends, the pressure decreases, causing the air in their lungs or equipment to expand.
It is essential for divers not to hold their breath while surfacing. If a diver holds their breath, the expanding air could cause damage to the lungs, also known as over-expansion injuries or even result in burst lungs.

• Never hold your breath while ascending in water.
• Keep breathing normally to allow expanding air to escape safely.

Paying attention to these safety aspects could mean the difference between a safe dive and a hazardous situation.