∙∙A diver observes a bubble of air rising from the bottom of...

$\bullet$ $\bullet$ 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 $23.0^{\circ} \mathrm{C}$ . $$\begin{array}{l}{\text { (a) What is the ratio of the volume of the bubble as it reaches }} \\ {\text { the surface to its volume at the bottom? (b) Would it be safe }} \\ {\text { for the diver to hold his breath while ascending from the bottom of the lake to the surface? Why or why not? }}\end{array}$$

$\bullet$ $\bullet$ 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 $23.0^{\circ} \mathrm{C}$ .
$$\begin{array}{l}{\text { (a) What is the ratio of the volume of the bubble as it reaches }} \\ {\text { the surface to its volume at the bottom? (b) Would it be safe }} \\ {\text { for the diver to hold his breath while ascending from the bottom of the lake to the surface? Why or why not? }}\end{array}$$

Key Concepts

Diver Safety and Pulmonary Barotrauma

In diving, the risks associated with changes in pressure are critical, particularly regarding lung expansion. If a diver holds their breath while ascending, the decreasing ambient pressure can cause the air in the lungs to expand excessively, potentially leading to lung rupture. This concept of pulmonary barotrauma highlights the importance of proper breathing techniques during ascent.

Temperature-Volume Relationship (Charles’ Law)

Charles’ Law states that, at constant pressure, the volume of a gas is directly proportional to its absolute temperature. This concept is important because the temperature of the bubble changes as it ascends, affecting its volume enlargement, and is incorporated in the Combined Gas Law application.

Combined Gas Law

The Combined Gas Law relates the pressure, volume, and temperature of a fixed amount of gas using the equation (P1 × V1) / T1 = (P2 × V2) / T2. In scenarios where a gas bubble changes altitude, both pressure and temperature vary, making the combined law essential to calculate the new volume of the bubble as it rises.

Pressure-Volume Relationship (Boyle’s Law)

Boyle’s Law states that for a constant temperature, the volume of a gas is inversely proportional to its pressure. Although temperature is not held constant in this scenario, understanding Boyle's Law helps elucidate how a decrease in pressure leads to an increase in volume, which is a central concept in the problem.

Transcript

00:01 This is chapter 15, problem number 12.

00:03 We have a diver who is observed in a bubble of air rising from the bottom of the lake, where we have the pressure.

00:12 Let me use the index b to refer to the bottom.

00:17 It's 3 .5 atm atmosphere, right? and the temperature at the bottom is given to us as 4 celsius.

00:25 And the pressure on the surface is one atmosphere.

00:29 And the temperature on the surface is 23 celsius.

00:34 Now we're asked the ratio of the volume on the surface with respect to the bottom in part a.

00:43 So we are asked the surface over the bottom, this ratio.

00:50 So in order to address this question, since we have the pressure and the temperature in the bottom and the pressure and temperature on the surface, and in the meantime, inside the bubble, there is always the same amount of air as it's ascending from the bottom towards the surface of the lake.

01:13 If you write the ideal gas law for these two cases on the bottom at the surface, then we would do pb vb equals nrtb, right? this is for the bottom.

01:28 So let's try to see what remains constant here.

01:31 Like we said, the amount of air is constant inside the bubble, right? so the number of moles is constant, and as you know, the ideal gas constant is constant.

01:40 So if we divide both sides by the temperature at the bottom, we get on the left -hand side of the equation equals to n times r.

01:49 Then if we write the ideal gas law for when the bubble is on the surface of the lake, then it would be p surface, v surface over the temperature at the surface, right? they have to be equal to each other...

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