dT/dz=−α for z≤11 km
where α=6.5 K/km (Kelvin per km) and z is the height above the sea level. The temperature stays then approximately constant between 11 km and 20 km above sea level.
Assume a temperature of 5 ∘C and a pressure of 1 atm at sea level (1 atm = 1.01325 ×105 N/m^2). Furthermore, take the molecular weight of the air to be (approximately) 29 g/mol. The universal gas constant is R=8.314 JK−1mol−1 and the acceleration due to gravity is g=10 m/s2 (independent of altitude). Assume that air can be treated as an ideal gas.
(a) Under the assumptions above, calculate the atmospheric pressure p (in atm) at z= 10 km above sea level for the case of a linear temperature drop.
p=
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(b) The cruising altitude of a commercial aircraft is about 33'000 ft (or 10 km). Assume that the cabin is pressurized to 0.8 atm at cruising altitude. What is the minimal force Fmin (in Newton) per square meter that the walls have to sustain for the cabin not to burst? Use the atmospheric pressure found in (a).
Fmin=
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(c) We close a plastic bottle full of air inside the cabin when the aircraft is at cruising altitude of z= 10 km. The volume of the bottle is V1, the pressure and temperature inside the cabin are 0.8 atm and T1=27 ∘C, respectively. Assume that at sea level the atmospheric pressure is 1 atm, and the temperature is decreased by 15 Kelvin with respect to the cabin's temperature.
What is the magnitude of the percentage change in volume of the air inside the bottle when it is brought to sea level? (Enter the magnitude of the percentage change in volume in
∣∣∣ΔV/V1∣∣∣×100=
To solve these problems, we need to use the ideal gas law, which states:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
(a) First, let's calculate the temperature at z = 10 km using the linear temperature drop given in the problem. We know that the temperature at sea level is 5 degrees Celsius, which is equivalent to 278 Kelvin.
dT/dz = -α (given)
Integrating both sides of the equation gives us:
∫dT = -α∫dz
T = -αz + C
Using the boundary condition that T = 5 degrees Celsius (278 Kelvin) at z = 0, we can solve for C:
278 = -α(0) + C
C = 278
So the equation for temperature becomes:
T = -αz + 278
Now we can substitute this equation into the ideal gas law to find the pressure at z = 10 km:
P * V = n * R * T
We can assume the volume remains constant for this problem, so we can write:
P = (n * R * T) / V
Substituting the equation for temperature, we get:
P = (n * R * (-αz + 278)) / V
Now we can calculate the pressure at z = 10 km by plugging in the given values and solving for P.
(b) To find the minimal force per square meter that the cabin walls have to sustain, we need to consider the pressure difference between the inside and outside of the cabin.
The pressure inside the cabin is given as 0.8 atm, and we calculated the pressure at z = 10 km in part (a). The pressure outside the cabin is higher due to the higher altitude, so we subtract the cabin pressure from the atmospheric pressure to find the pressure difference:
ΔP = Poutside - Pinside
The force per square meter is equal to the pressure difference multiplied by the surface area of the cabin walls:
Fmin = ΔP * A
We can substitute the pressure difference and solve for Fmin using the values provided.
(c) To find the percentage change in volume of the air inside the bottle when brought to sea level, we can use the ideal gas law again.
For this problem, we are given the initial pressure (0.8 atm) and temperature (27 degrees Celsius or 300 Kelvin) inside the cabin at cruising altitude. We are also given the pressure (1 atm) and a decrease in temperature by 15 Kelvin at sea level.
We can calculate the initial volume inside the bottle at cruising altitude using the ideal gas law:
P1 * V1 = n * R * T1
Rearranging the equation, we get:
V1 = (n * R * T1) / P1
Similarly, we can calculate the final volume at sea level using the new temperature (T2 = T1 - 15 K) and pressure (P2 = 1 atm):
V2 = (n * R * T2) / P2
The percentage change in volume can be calculated using the formula:
|ΔV/V1| * 100
Substituting the values, we can evaluate this expression to find the magnitude of the percentage change in volume.