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 3.29 atm. air temperature inside the submarine is 13 ¢XC. A person leaves and takes a breath as deep as possible by exhaling as far as possible (leaving a volume of 1.11 L in their lungs), and then slowly inhaling to increase their lung volume by 4.77 L. His body temperature is 37.42 ¢XC.
(i) How many particles of air do their lungs contain after inhaling?
(ii) This person breathes out all the way to the surface in order to maintain a constant lung volume. How many moles of gas remain in the lungs?
To answer these questions, we can use the ideal gas law equation, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
To convert the given temperature from Celsius to Kelvin, we can use the equation Kelvin = Celsius + 273.15.
(i) To find the number of particles of air in the person's lungs after inhaling, we can use the ideal gas law equation.
First, we need to convert the temperature from Celsius to Kelvin:
T = 13 ¢XC + 273.15 = 286.15 K.
We can assume that the pressure inside the lungs is the same as the pressure inside the submarine, which is 3.29 atm.
Using the ideal gas law equation, we have:
P * V = n * R * T
3.29 atm * (1.11 L + 4.77 L) = n * (0.0821 L * atm/ K * mol) * 286.15 K.
Simplifying the equation gives:
3.29 atm * 5.88 L = n * 23.7001 L * atm / K * mol * 286.15 K.
Further simplification gives:
19.329 L * atm = n * 6797.76413 L * atm / K * mol.
Now, we can cancel out the units of liter, atm, and K, giving:
19.329 = n * 6797.76413.
To solve for n, we divide both sides of the equation by 6797.76413:
n = 19.329 / 6797.76413.
Calculating this equation gives an approximate value of n = 0.002845 moles.
Therefore, after inhaling, the person's lungs contain approximately 0.002845 moles of air particles.
(ii) To calculate the number of moles of gas remaining in the person's lungs after exhaling all the way to the surface, we use the ideal gas law equation again.
We already know that the pressure inside the submarine (and the lungs) is 3.29 atm, the volume is 1.11 L, and the temperature is 13 ¢XC + 273.15 K = 286.15 K.
Using the ideal gas law equation, we have:
P * V = n * R * T
3.29 atm * 1.11 L = n * (0.0821 L * atm / K * mol) * 286.15 K.
Simplifying the equation gives:
3.29 atm * 1.11 L = n * 23.7001 L * atm / K * mol * 286.15 K.
Further simplification gives:
3.6449 L * atm = n * 6797.76413 L * atm / K * mol.
Now, we can cancel out the units of liter, atm, and K, giving:
3.6449 = n * 6797.76413.
To solve for n, we divide both sides of the equation by 6797.76413:
n = 3.6449 / 6797.76413.
Calculating this equation gives an approximate value of n = 0.000537 moles.
Therefore, after exhaling all the way to the surface, approximately 0.000537 moles of gas remains in the person's lungs.