A submarine has run into trouble and is stuck at the bottom of the ocean. Several people are on board and must make their way to the surface without any diving gear. The air pressure aboard the submarine is 3.04 atm. The air temperature inside the submarine is 12.1 ¢XC and you can take body temperature (inside the lungs) to be 36.11 ¢XC
(a) The first person to leave takes a breath as deep as possible by exhaling as far as possible (leaving a volume of 1.12 L in their lungs), and then slowly inhaling to increase their lung volume by 4.61 L
This person, unwisely, holds their breath all the way to the surface. What would the volume of their lungs be (if they didn't burst)? [For interest, note that human lung capacities range from 4 to 6 L]
To find the volume of the person's lungs when they reach the surface, we need to consider the change in pressure during the ascent.
The ideal gas law can be used to relate the initial and final volumes and pressures:
P1 * V1 / T1 = P2 * V2 / T2
Where:
P1 = initial pressure
V1 = initial volume
T1 = initial temperature
P2 = final pressure (atmospheric pressure at the surface)
V2 = final volume (what we need to find)
T2 = final temperature
Let's plug in the given values and solve for V2.
P1 = 3.04 atm
V1 = 1.12 L
T1 = 36.11 ¢XC = 36.11 + 273.15 K = 309.26 K
P2 = 1 atm (atmospheric pressure at the surface)
T2 = unknown
We can rearrange the equation to solve for V2:
V2 = V1 * P1 * T2 / (P2 * T1)
Substituting in the given values:
V2 = (1.12 L) * (3.04 atm) * (T2) / (1 atm * 309.26 K)
Now, we need to find the final temperature, T2. Since the person is ascending to the surface, we can assume that the temperature remains constant. So, T2 will be the same as the initial temperature, T1.
V2 = (1.12 L) * (3.04 atm) * (309.26 K) / (1 atm * 309.26 K)
Simplifying the equation:
V2 = 3.43 L
Therefore, if the person holds their breath all the way to the surface, the volume of their lungs would be 3.43 L (if they didn't burst).
Note: It's important to emphasize that holding your breath while ascending from a deep dive is extremely dangerous and can lead to lung injuries. It's recommended to exhale continuously while ascending to release the expanding air in the lungs.