A diver named jacques observes a bubble of air rising from...

A diver named Jacques 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 4.00 ∘C, and the temperature at the surface is 23.0 ∘C. Part A: What is the ratio of the volume of the bubble as it reaches the surface (Vs) to its volume at the bottom (Vb)? Part B: If Jacques were to hold his breath the air in his lungs would be kept at a constant temperature. Would it be safe for Jacques to hold his breath while ascending from the bottom of the lake to the surface?

Transcript

00:01 Hello students in this question we are given that a diver named jagus offers a bubble of air rising from the bottom of a lake where the absolute pressure is 3 .5 atmosphere to the surface where pressure is 1 atmosphere.

00:13 The temperature at the bottom is 4 degrees celsius so temperature at bottom is given 4 .0 degrees celsius and temperature at the surface is 23 .0 degrees celsius.

00:28 So we have to find what is the ratio of the volume of the bubble as it reaches the surface with the volume at the bottom.

00:36 So first of all here the temperature if we convert into kelvin, so this is 277 kelvin and this is 2 .96 kelvin.

00:53 So now the air in the bubble is to be treated as an ideal gas.

01:00 So we write the ideal gas equation for the air at the surface of the lake.

01:05 So at surface, at surface we write ps multiply vs is nrts.

01:16 This is our first equation where vs is the volume of air in the bubble and ps is the pressure of air in the bubble at the surface.

01:25 T s is the temperature of air bubble at the surface of the lake and it is the number of mole of air and r is.

01:32 Is the ideal gas constant.

01:34 Now if we write at this same equation for bottom, so at bottom we have p b multiplied by vb is equal to n r t b.

01:47 This is our second equation over here because here we have p b which is the pressure of air in the bubble at the bottom of the lake volume vb is the volume of air in the bubble at the bottom of the leg and is the number of moles of air are idle gas constant is the temperature of the air in the bubble at the bottom of the lake.

02:07 Now if we divide equation 1 by equation 2, so dividing 1 by 2, we get, so here we get, p s multiplied vs over pb multiplied vb is equal to nrts over nrt s over nr t, so we get here ps multiply by vs over p b multiply by vb is equal to t s over t b now now if we rearrange in the equation to get the ratio of v s over vb so vs over vb is equal to t s p b over this is t b p s now the we substitute the values so we have v s over v b is equal to the temperature at the surface is 296 multiply by pressure at the bottom is 3 .5 atmosphere over here over the temperature at bottom is 277 kelvin multiply with the pressure at the surface is 180m so we have vs over vb is equal to 3 .74...

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