During a transatlantic flight, you drink a bottle of water and tightly close its lid again. Back on the ground during taxiing to the terminal, you notice that the thin plastic bottle is crushed, and you eyeball it to be only about 80 percent of its original volume (the photo shows the actual bottle). If the pressure on the ground is about 112 kPa, what was the cabin pressure up in the air? The cabin temperature stayed a comfortable 19 degrees Celsius at all times. Approximate air as an ideal gas
P1V1=P2V2 Constant temp...
p1=P2V2/V1
To find the cabin pressure up in the air during the transatlantic flight, we can use the ideal gas law, which states:
PV = nRT
Where:
P is the pressure
V is the volume
n is the number of moles of the gas
R is the ideal gas constant
T is the temperature in Kelvin
First, let's convert the temperature from Celsius to Kelvin:
T = 19°C + 273.15 = 292.15 K
Since you noticed the bottle being crushed and its volume reduced to approximately 80% of its original volume, we can assume that the volume V is reduced by 20%. So the volume V' in the air would be 80% of the original volume.
Now, let's denote the original pressure as P' and the original volume as V'. We have:
P' * V' = nRT
Since we want to find the cabin pressure (P) up in the air, we can use the new volume (V') and solve for P:
P * V' = nRT
Now we can compare the two equations:
1) P' * V' = nRT
2) P * V' = nRT
Dividing equation 2) by equation 1), we have:
(P * V') / (P' * V') = (nRT) / (nRT)
Simplifying, the nRT terms cancel out:
P / P' = 1
Therefore, the cabin pressure (P) up in the air is equal to the original pressure (P') on the ground, which is approximately 112 kPa.