Calculate the pressure and composition of air on the top of Mt. Everest, assuming that the atmosphere has a temperature of 0oC independent of altitude (h = 29,141 ft). Assume that air at sea level is 20% O2 and 80% N2.
To calculate the pressure and composition of air on the top of Mt. Everest, we can use the ideal gas law and the Dalton's law of partial pressures.
Step 1: Convert the altitude to meters
Given that the altitude is h = 29,141 ft, we need to convert it to meters. 1 ft is approximately 0.3048 meters, so we can calculate:
h = 29,141 ft * 0.3048 m/ft = 8,848 m.
Step 2: Determine the temperature
The problem states that the temperature is 0°C. However, it's important to note that the temperature of the atmosphere generally decreases with increasing altitude. If we assume a constant temperature of 0°C, it is a simplification, but it will serve as an approximate estimate.
Converting the temperature to Kelvin:
0°C + 273.15 = 273.15 K.
Step 3: Calculate the pressure
Using the ideal gas law, we have the equation:
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.
Since we are interested in the pressure, we can rearrange the equation:
P = (nRT) / V.
At the top of Mt. Everest, the volume will be the same as at sea level (assuming negligible changes at this scale). Therefore, we can set up a ratio to find the pressure at the top compared to sea level:
Ptop / Psea level = (nRT)top / (nRT)sea level.
Since n, R, and T are constant, we can assume they cancel out from the equation, leading to:
Ptop / Psea level = Vtop / Vsea level.
Using the fact that Vtop = Vsea level, because we are assuming a constant volume, we can simplify the equation to:
Ptop = Psea level.
Thus, the pressure at the top of Mt. Everest will be equal to the pressure at sea level.
Step 4: Determine the composition
The composition of air at sea level is given as 20% O2 and 80% N2. Assuming ideal behavior, we can use Dalton's law of partial pressures to find the partial pressure of each component.
The partial pressure of O2 (PO2) is equal to the total pressure (Psea level) multiplied by the mole fraction of O2 in air:
PO2 = Psea level * (mole fraction of O2).
Likewise, the partial pressure of N2 (PN2) is equal to the total pressure (Psea level) multiplied by the mole fraction of N2 in air:
PN2 = Psea level * (mole fraction of N2).
Given that the mole fraction of O2 is 20% and the mole fraction of N2 is 80%, we can substitute these values into the equations.
Step 5: Calculate the pressure and composition
Since the pressure at the top of Mt. Everest is equal to the pressure at sea level, we can conclude that the pressure is the same regardless of altitude.
Therefore, the pressure on the top of Mt. Everest will be the same as the pressure at sea level.
As for the composition, assuming ideal behavior and using the given mole fractions, we can calculate the partial pressures of O2 and N2 at sea level:
PO2 = Psea level * (mole fraction of O2)
= Psea level * 0.20.
PN2 = Psea level * (mole fraction of N2)
= Psea level * 0.80.
Since the mole fractions and the pressure at sea level are given, you can substitute these values to find the partial pressures of O2 and N2.