A scuba diver creates a spherical bubble with a radius of 2.5 cm at a depth of 30.0 m where the total pressure (including atmospheric pressure) is 4.00 atm. What is the radius of the bubble when it reaches the surface of the water? (Assume that the atmospheric pressure is 1.00 atm and the temperature is 298 K.)

Video transcript

Hello everyone today. We are being presented with the following problem At 40 m deep underwater where the total pressure is five atmospheres, a dolphin produces a bubble with the diameter of seven cm. We are being asked to calculate the radius of the bubble when it reaches 10 m below the surface and the total pressure is two atmospheres. So the first thing I'm gonna recount is the fact that we are going to be using boils lot, which states that the pressure, initial pressure times the initial volume is equal to the final pressure times the final volume. Notice how we also have radius. So we're gonna have to account for the volume of a sphere. And so we're gonna use volume is equal to 4/3 times pi times the radius cubed. We see that our density as seven centimeters. And so by default our radius is half of that 3. cm. We simply need to plug in our value to find the initial volume. So we say V one is equal to 4/ pi r cubed. Which then translates to 4 3rd pi 3. cm cubed. This gives us an initial volume of 179.6 cm cubed. We then use our initial Boyle's law and we saw for V two. V two is equal to p one times p Times V one over P two plugging in our values for the initial pressure. We have five atmospheres Times our initial volume which we just sold for 179.6 cm cubed. And we Put that all over two atmospheres, Pressure units cancel. And we're left with a final volume of .98 cm cubed. We then plug this into our initial volume equation that we're solving for a sphere. So we see that 4/3 pirate cubed and then we plug in our values. So the V. Two that we use, we use that value for this equation. So you say 448.98 cm cubed Is equal to 4/3 pi r cubed. Rearranging this equation, we end up with a final radius of 4. cm. I hope this helped. And until next time.