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 70 percent of its original volume (the photo shows the actual bottle). If the pressure on the ground is about 108 kPa, what was the cabin pressure up in the air? The cabin temperature stayed a comfortable 20 degrees Celsius at all times. Approximate air as an ideal gas.
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To determine the cabin pressure up in the air, we can use the ideal gas law, which states:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature in Kelvin
Since we are given the following information:
- The bottle is tightly closed throughout the flight, which means the number of moles of gas remains constant.
- The initial volume of the bottle is 100% (or 1) of its original volume.
- The volume of the bottle on the ground is 70% (or 0.70) of its original volume.
- The pressure on the ground is 108 kPa.
- The temperature is 20 degrees Celsius.
To find the cabin pressure up in the air, we need to calculate the initial pressure (P_initial) when the bottle was at its original volume.
First, let's convert the temperature to Kelvin:
T = 20 degrees Celsius + 273.15 = 293.15 K.
Next, rearrange the ideal gas law equation to solve for the initial pressure:
P_initial = (n * R * T) / V_initial
We know that the final volume (V_final) is 0.70 V_initial, which means the initial volume (V_initial) is V_final / 0.70.
Substituting this back into the equation, we have:
P_initial = (n * R * T) / (V_final / 0.70)
Now, we can substitute the given values and constants into the equation:
P_initial = (n * 8.314 J/(mol·K) * 293.15 K) / (V_final / 0.70)
Here, we need to be careful with the units. The ideal gas constant (R) has units of J/(mol·K), so we must ensure that our temperature is in Kelvin and our pressure is in pascals (Pa) instead of kilopascals (kPa):
P_initial = (n * 8.314 J/(mol·K) * 293.15 K) / ((V_final / 0.70) * 1000) [1 kPa = 1000 Pa]
Now, we can substitute the given value for the final volume and solve for the initial pressure.
Please provide the number of moles of gas (n) to complete the calculation.