Calculate the pressure and composition of air on the top of Mt. Everest, assuming that the
atmosphere has a temperature of −20°C independent of altitude (h = 29141 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 Mount Everest, we need to use the barometric formula and Dalton's law of partial pressures.
1. Calculate the pressure at sea level (p0):
At sea level, the average atmospheric pressure is approximately 101325 Pa.
2. Convert the altitude (h) from feet to meters:
h = 29141 ft * 0.3048 m/ft = 8,871.76 meters
3. Calculate the pressure at the given altitude (p):
The barometric formula states that the pressure decreases exponentially with altitude. It can be written as:
p = p0 * (1 - (L * h / T0))^((g * M) / (R * L))
Where:
- p0 is the pressure at sea level
- L is the temperature lapse rate (approximately 0.0065 K/m)
- h is the altitude
- T0 is the temperature at sea level
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- M is the molar mass of air (approximately 0.02896 kg/mol)
- R is the ideal gas constant (approximately 8.314 J/(mol K))
In this case, the temperature is given as -20°C (-20 + 273.15 K = 253.15 K).
4. Substitute the values into the formula and solve for p:
p = 101325 * (1 - (0.0065 * 8871.76 / 253.15))^((9.8 * 0.02896) / (8.314 * 0.0065))
5. Calculate the mole fractions of oxygen (XO2) and nitrogen (XN2):
At sea level, air is composed of 20% O2 and 80% N2. We can convert these percentages into mole fractions by dividing by the molar masses of O2 and N2.
XO2 = (20 / 100) / (32 / 28.96) = 0.2857
XN2 = (80 / 100) / (28 / 28.96) = 0.7143
6. Calculate the partial pressures of oxygen (pO2) and nitrogen (pN2):
According to Dalton's law of partial pressures, the total pressure (p) is the sum of the partial pressures of each component.
pO2 = p * XO2
pN2 = p * XN2
7. Provide the final values:
The pressure at the top of Mount Everest is p, and the composition is 28.57% O2 and 71.43% N2.
Please note that these calculations are based on assumptions and ideal gas behavior. The actual pressure and composition may vary in real-world conditions.
To calculate the pressure and composition of air on the top of Mt. Everest, we can start by using the barometric formula, which relates the pressure of the atmosphere to the altitude:
P = P0 * exp(-M * g * h / (R * T))
Where:
P is the pressure at the given altitude
P0 is the pressure at sea level
M is the average molar mass of air
g is the acceleration due to gravity
h is the altitude
R is the gas constant
T is the temperature
We also need to calculate the partial pressures of oxygen (O2) and nitrogen (N2) at sea level, and then use these partial pressures to find their compositions at the given altitude.
First, let's calculate the pressure at sea level (P0) using the temperature given (-20°C) and converting it to Kelvin:
T0 = -20 + 273.15 = 253.15 K
Next, we need to calculate the partial pressures of oxygen (PO2) and nitrogen (PN2) at sea level. Given that air at sea level is 20% oxygen and 80% nitrogen, we can use the known pressure at sea level to derive these partial pressures:
PO2 = 20% * P0
PN2 = 80% * P0
Now, let's calculate the pressure at the top of Mt. Everest using the given altitude (h = 29141 ft) and plugging in the known values into the barometric formula:
P = P0 * exp(-M * g * h / (R * T))
The average molar mass of air (M) is approximately 28.97 g/mol, the acceleration due to gravity (g) is approximately 9.81 m/s^2, and the gas constant (R) is approximately 8.314 J/(mol*K).
After calculating P, we can calculate the partial pressures of oxygen (P2) and nitrogen (P2) at the given altitude using the known percentages and the calculated pressure.
P2O2 = 20% * P
P2N2 = 80% * P
Now, we have the pressure and composition of air on the top of Mt. Everest.