If a city has an atmospheric pressure of 12.27 psi, what is its altitude? (Recall that 1 mi = 5,280 ft. Round your answer to the nearest foot.)
To determine the altitude of a city based on its atmospheric pressure, we can use a mathematical relationship known as the barometric formula. The formula is given by:
P = P₀ * (1 - (L * h) / T₀)^(gM / RL)
Where:
- P is the atmospheric pressure at the given altitude
- P₀ is the atmospheric pressure at sea level (standard pressure), which is 14.7 psi (pounds per square inch)
- L is the temperature lapse rate, which is the rate at which temperature decreases with increasing altitude. For this calculation, we assume L = 0.0065 K/ft.
- h is the height or altitude above sea level that we want to calculate
- T₀ is the standard temperature at sea level, which is 288.15 K (Kelvin)
- g is the acceleration due to gravity, which is approximately 32.17 ft/s²
- M is the molar mass of Earth's air, which is approximately 0.0289644 kg/mol
- R is the specific gas constant for air, which is approximately 8.314 J/(mol·K)
- RL is the product of R and L
To determine the altitude, we need to rearrange the barometric formula to solve for h:
h = (T₀ / L) * (1 - (P / P₀)^((RL) / (gM)))
Let's plug in the provided values and calculate the altitude:
P = 12.27 psi
P₀ = 14.7 psi
L = 0.0065 K/ft
T₀ = 288.15 K
g = 32.17 ft/s²
M = 0.0289644 kg/mol
R = 8.314 J/(mol·K)
RL = R * L
Now we can substitute these values into the formula and solve for h.