A summer day in Narragansett, Rhode Island, the air temperature is 70 degrees F and the barometric pressure is 15.5 lbf/in^2 estimate the air density in kg/m^3
The Correct Answer and Explanation is:
To estimate the air density at a temperature of 70 degrees Fahrenheit and a barometric pressure of 15.5 lbf/in², we can use the Ideal Gas Law and convert the values appropriately. The Ideal Gas Law is given by the formula:
[
PV = nRT
]
Where:
- ( P ) is the pressure in pascals (Pa),
- ( V ) is the volume in cubic meters (m³),
- ( n ) is the number of moles of the gas,
- ( R ) is the universal gas constant (approximately ( 8.314 \, \text{J/(mol·K)} )),
- ( T ) is the temperature in Kelvin (K).
Step 1: Convert Pressure from lbf/in² to Pascals
First, we need to convert the barometric pressure from lbf/in² to pascals.
1 lbf/in² = 6894.76 Pa
So,
[
P = 15.5 \, \text{lbf/in²} \times 6894.76 \, \text{Pa/lbf/in²} = 106,721.78 \, \text{Pa}
]
Step 2: Convert Temperature from Fahrenheit to Kelvin
The conversion from Fahrenheit to Kelvin is done using the formula:
[
T(K) = \frac{(T(°F) – 32) \times 5}{9} + 273.15
]
Applying the values:
[
T = \frac{(70 – 32) \times 5}{9} + 273.15 = 294.26 \, K
]
Step 3: Calculate Air Density
Using the Ideal Gas Law rearranged to find density (( \rho )):
[
\rho = \frac{P}{RT}
]
Where ( R ) for dry air is approximately ( 287.05 \, \text{J/(kg·K)} ).
Substituting in the values:
[
\rho = \frac{106,721.78 \, \text{Pa}}{287.05 \, \text{J/(kg·K)} \times 294.26 \, K}
]
Calculating:
[
\rho \approx \frac{106,721.78}{84,464.45} \approx 1.26 \, \text{kg/m³}
]
Conclusion
Thus, the estimated air density on this summer day in Narragansett, Rhode Island, is approximately 1.26 kg/m³.
This calculation illustrates how changes in temperature and pressure influence air density, crucial for applications in meteorology, aviation, and various engineering fields. The lower air density at higher temperatures reflects the behavior of gases where increased thermal energy causes expansion, reducing density, while higher pressure increases density by compressing air molecules closer together.