what is the altitude which you'll find 99% of the atmosphere's mass? Assume P=1 atm, atmosphere T=290 K and the atmosphere is mostly N2.
To find the altitude which contains 99% of the atmosphere's mass, we need to consider the distribution of gases in the atmosphere based on their molecular weight.
The atmosphere is composed mostly of nitrogen (N2), which has a molecular weight of approximately 28 atomic mass units (AMU).
We can use the barometric formula to determine the relationship between pressure and altitude:
P = P0 * e^(-M * g * h / (R * T))
where:
- P is the pressure at altitude h
- P0 is the pressure at sea level (1 atm)
- M is the molar mass of air (28.97 g/mol)
- g is the acceleration due to gravity (9.8 m/s^2)
- R is the ideal gas constant (0.0821 L·atm/(mol·K))
- T is the temperature
Since we know that we need to find the altitude at which 99% of the atmosphere's mass is contained, we can assume that at that altitude the pressure is approximately 1% of the sea level pressure (0.01 atm).
Setting P = 0.01 atm in the barometric formula and solving for h:
0.01 atm = 1 atm * e^(-28.97 g/mol * 9.8 m/s^2 * h / (0.0821 L·atm/(mol·K) * 290 K))
Simplifying the equation:
e^(-28.97 g/mol * 9.8 m/s^2 * h / (0.0821 L·atm/(mol·K) * 290 K)) = 0.01
Taking the natural logarithm of both sides:
-28.97 g/mol * 9.8 m/s^2 * h / (0.0821 L·atm/(mol·K) * 290 K) = ln(0.01)
Solving for h:
h = (ln(0.01) * (0.0821 L·atm/(mol·K) * 290 K)) / (-28.97 g/mol * 9.8 m/s^2)
Calculating h:
h ≈ 8,523 meters
Therefore, at an altitude of approximately 8,523 meters above sea level, you will find 99% of the atmosphere's mass, assuming a pressure of 1 atm, an atmosphere temperature of 290 K, and that the atmosphere is mostly composed of N2.