To calculate the density and specific volume of air at a given temperature and pressure, you can use the ideal gas law, which relates the pressure, volume, and temperature of a gas. The ideal gas law is expressed as:
PV = nRT
Where: P = Pressure V = Volume n = Number of moles of gas R = Ideal gas constant (8.314 J/(mol·K)) T = Temperature
To calculate the density (ρ), you can use the equation:
ρ = (molar mass of air * P) / (R * T)
To calculate the specific volume (v), you can use the equation:
v = 1 / ρ
The molar mass of air is approximately 28.97 grams/mole.
Substituting the values into the equations: Temperature (T) = 293 K Pressure (P) = 1 Pa Molar mass of air = 28.97 g/mol Ideal gas constant (R) = 8.314 J/(mol·K)
First, we need to convert the pressure from Pascals (Pa) to atmospheres (atm), as the ideal gas constant has units of J/(mol·K) when pressure is in atm:
1 atm = 101325 Pa (approximately)
P = 1 Pa / 101325 Pa/atm ≈ 9.87 x 10^(-6) atm
Now we can calculate the density:
ρ = (28.97 g/mol * 9.87 x 10^(-6) atm) / (8.314 J/(mol·K) * 293 K) ≈ 1.18 x 10^(-3) g/cm^3
To convert the density to kg/m^3, multiply by 1000:
ρ = 1.18 kg/m^3
Finally, we can calculate the specific volume:
v = 1 / ρ = 1 / 1.18 ≈ 0.847 m^3/kg
Therefore, at a temperature of 293 K and a pressure of 1 Pa, the density of air is approximately 1.18 kg/m^3, and the specific volume is approximately 0.847 m^3/kg.