At an altitude of 50.00 km, the average atmospheric temperature is essentially 0 degrees C. What is the average number of air molecules per cubic centimeter of air at this altitude?
0.001935
To find the average number of air molecules per cubic centimeter of air at an altitude of 50.00 km, we can use the ideal gas law. The ideal gas law equation is:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature in Kelvin
First, we need to convert the temperature from Celsius to Kelvin. To do this, we add 273.15 to the temperature:
T = 0 + 273.15 = 273.15 K
Next, we need to calculate the pressure at an altitude of 50.00 km. The pressure decreases with altitude, and we can use the barometric formula to determine the pressure at this height. The barometric formula is:
P = P0 * exp(-M * g * h / (RT))
Where:
P0 = standard pressure at sea level (usually 1013.25 hPa)
M = molar mass of air (28.97 g/mol)
g = acceleration due to gravity (9.8 m/s^2)
h = altitude in meters
First, let's convert the altitude from kilometers to meters:
h = 50.00 km * 1000 = 50000 meters
Next, we can substitute the values into the barometric formula to find the pressure at this altitude:
P = 1013.25 * exp(-0.02897 * 9.8 * 50000 / (8.314 * 273.15))
After calculating this value, we can substitute the obtained values for pressure (P), volume (V), and temperature (T) in the ideal gas law equation:
n = (P * V) / (R * T)
To find the volume, we can assume the air behaves as an ideal gas and use the ideal gas law again:
V = (n * R * T) / P
Substituting the obtained values, we can solve for the volume.
Finally, we can calculate the average number of air molecules per cubic centimeter using Avogadro's number (6.022 x 10^23 molecules/mol) and the volume:
Number of molecules = (n * Avogadro's number) / Volume
By following these steps, you can find the average number of air molecules per cubic centimeter of air at an altitude of 50.00 km.