The 15th floor meteorological observatory of the O&M building is 188 ft from the level of the main entrance. Assume that the temperature at the entrance is 25 degrees Celcius, that station pressure is 1000 mb, and the mixing ration is .010 kg/kg. What is the actual (station) pressure at a baramoter in the observatory?
Use the mixing ratio to calculate the density of the air. In this case, it means that it is 1% water vapor. The mean molecular weight of the air will be about 28.9 g/mole.
Then use the fact that pressure decreases with altitude according to
delta P = -Integral of (density) g dy.
Assume the temperature stays the same with height for 188 feet from the ground, even though this may not be true. It is close enough in this case. You can even ignore the change of density with height and get a good value for the integral.
The density of the 25 C air with mean molecular weight of 28.9 will be about
28.9 g/(24.45 l) = 28.9*10^-3 kg/24.45*10^-3 m^3 = 1.182 kg/m^3
188 ft = 57.3 m
delta P = -(1.182)(9.8)(57.3) = -664 N/m^2 (approximately)
You should convert the pressure decrease to millibars when you are done.
The 1% mixing ratio does not make a lot of difference.
To find the actual (station) pressure at the barometer in the observatory, we need to take into account the change in pressure due to the difference in height between the main entrance and the observatory.
We can use the barometric formula, which states that the pressure decreases with increasing height in the atmosphere. The formula is given by:
P2 = P1 * e^(-h / H),
Where:
- P1 is the initial pressure at the main entrance
- P2 is the pressure at a different height (observatory)
- h is the difference in height between the two locations
- H is the scale height of the atmosphere
First, we need to determine the scale height (H). The scale height indicates the vertical distance over which the pressure decreases by a factor of e (approximately 2.718). It can be calculated using the formula:
H = (RT) / (mg),
Where:
- R is the ideal gas constant (8.314 J/(mol*K))
- T is the temperature in Kelvin (25 + 273.15)
- m is the molar mass of air (approximately 0.029 kg/mol)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Let's calculate the value of H:
H = (8.314 J/(mol*K) * (25 + 273.15) K) / ((0.029 kg/mol) * (9.8 m/s^2)) = approximately 7242 m.
Now, we can calculate the pressure at the observatory (P2):
P2 = 1000 mb * e^(-188 ft / 7242 m),
To convert the height from feet to meters, we need to multiply it by the conversion factor: 1 ft = 0.3048 m.
P2 = 1000 mb * e^(-188 ft * 0.3048 m/ft / 7242 m),
P2 = 1000 mb * e^(-57.3024 / 7242).
Using a scientific calculator or computational tool, we can evaluate the exponential term and calculate P2.