Mount Kilimanjaro is the highest peak on the African continent and rises to a height of
approximately 5900m above sea level.Take the gas constant to be 287 J/kg K and the temperature
lapse rate to be 0.0065 K/m.
It is known that altitude sickness can occur to human beings when the absolute pressure of the
atmosphere drops below 75 kPa.Determine if normal human beings can scale half of the mountain
without feeling uncomfortable if the pressure and temperature at ground level is 101 kPa and 25°C
To determine if normal human beings can scale half of Mount Kilimanjaro without feeling uncomfortable, we need to calculate the pressure and temperature at half of the mountain's height.
Given:
Height of Mount Kilimanjaro: 5900m
Gas constant (R): 287 J/kg K
Temperature lapse rate (L): 0.0065 K/m
Pressure at ground level (P1): 101 kPa
Temperature at ground level (T1): 25°C
To calculate the pressure and temperature at half of the mountain's height (2950m), we can use the following formulas:
Pressure at height h (P2) = P1 * (1 - (L * h) / T1)^(g * M / (R * L))
Temperature at height h (T2) = T1 - L * h
where:
h is the height above ground level (2950m in this case)
P1 is the pressure at ground level (101 kPa)
L is the temperature lapse rate (0.0065 K/m)
T1 is the temperature at ground level (25°C or 298 K)
g is the acceleration due to gravity (approximately 9.81 m/s^2)
M is the molar mass of Earth's air (approximately 0.02896 kg/mol)
Let's calculate the pressure and temperature at half of Mount Kilimanjaro's height:
Using the given values:
h = 2950m
P1 = 101 kPa
L = 0.0065 K/m
T1 = 298 K
g = 9.81 m/s^2
M = 0.02896 kg/mol
Calculating the pressure at height h (P2):
P2 = P1 * (1 - (L * h) / T1)^(g * M / (R * L))
= 101 * (1 - (0.0065 * 2950) / 298)^(9.81 * 0.02896 / (287 * 0.0065))
Calculating the temperature at height h (T2):
T2 = T1 - L * h
= 298 - 0.0065 * 2950
Now we can calculate the values:
P2 ≈ 75.628 kPa
T2 ≈ 278.525 K
Therefore, at half of Mount Kilimanjaro's height (2950m), the approximate pressure is 75.628 kPa and the temperature is 5.525°C (278.525 K).
Since the pressure at this height is below the threshold of 75 kPa, it is likely that normal human beings would start to feel uncomfortable while scaling half of Mount Kilimanjaro due to altitude sickness.
To determine if normal human beings can scale half of Mount Kilimanjaro without feeling uncomfortable, we need to calculate the pressure and temperature at half of the mountain's height.
First, let's find the temperature at half of the mountain by using the temperature lapse rate of 0.0065 K/m. Since the mountain's height is 5900m, half of the height would be 2950m.
Temperature at half of the mountain = Temperature at ground level - (Temperature lapse rate * Height at half of the mountain)
Temperature at half of the mountain = 25°C - (0.0065 K/m * 2950m)
Now, let's calculate the temperature at half of the mountain:
Temperature at half of the mountain = 25°C - (0.0065 K/m * 2950m)
Temperature at half of the mountain ≈ 25°C - 19.175°C
Temperature at half of the mountain ≈ 5.825°C
Next, we need to calculate the pressure at half of the mountain using the ideal gas law.
Pressure = Density * Gas constant * Temperature
Since we don't have the density, we can use the relationship between pressure and density in the hydrostatic equation:
Pressure_1 / Density_1 = Pressure_2 / Density_2
Where Pressure_1 is the pressure at ground level (101 kPa) and Density_1 is the initial density at ground level. We can assume a constant density since the temperature and pressure are not changing drastically over half of the mountain's height.
Therefore, Pressure_2 (pressure at half of the mountain) = Pressure_1 * (Density_2 / Density_1)
Since the density is constant, we can rewrite the equation as:
Pressure_2 = Pressure_1 * (Height_2 / Height_1)
Where Height_2 is half of the mountain's height (2950m) and Height_1 is the total height of the mountain (5900m).
Now, let's calculate the pressure at half of the mountain:
Pressure at half of the mountain = 101 kPa * (2950m / 5900m)
Pressure at half of the mountain ≈ 50.5 kPa
Now that we have the pressure and temperature at half of the mountain, we can check if it drops below the threshold for altitude sickness.
The absolute pressure at half of the mountain is 50.5 kPa, which is below the threshold of 75 kPa. Therefore, normal human beings may feel uncomfortable from altitude sickness when scaling half of Mount Kilimanjaro.
Please note that altitude sickness is a complex condition influenced by various factors, and this calculation only considers the pressure and temperature changes. Professional guidance and acclimatization are necessary for mountaineering at high altitudes.