Determine the specific gravity N2 gas at 75 °F and 765 torr compared to air at a) standard temperature and pressure b) 75 °F and 765 torr
The Correct Answer and Explanation is:
To determine the specific gravity (SG) of nitrogen gas (N₂) compared to air, we use the formula: Specific Gravity (SG)=Density of N2Density of air\text{Specific Gravity (SG)} = \frac{\text{Density of N}_2}{\text{Density of air}}
Specific gravity is dimensionless and typically calculated at the same conditions (temperature and pressure) for both substances.
Step 1: Molar Masses
- Molar mass of N₂ = 28.02 g/mol
- Molar mass of air ≈ 28.97 g/mol (approximate mixture of N₂, O₂, Ar, CO₂, etc.)
a) SG of N₂ at 75 °F and 765 torr compared to air at STP (0 °C, 1 atm)
This is a comparison at different conditions, so we must calculate the actual gas densities using the ideal gas law: ρ=PMRT\rho = \frac{PM}{RT}
Where:
- ρ\rho = density (g/L)
- PP = pressure (atm)
- MM = molar mass (g/mol)
- RR = ideal gas constant = 0.08206 L·atm/mol·K
- TT = temperature in Kelvin
Convert units:
- 75 °F = (75 – 32) × 5/9 + 273.15 = 297.04 K
- 765 torr = 765 / 760 = 1.00658 atm
- STP: T = 273.15 K, P = 1 atm
Density of N₂ at 75 °F and 765 torr: ρN2=(1.00658)(28.02)(0.08206)(297.04)=1.148 g/L\rho_{N_2} = \frac{(1.00658)(28.02)}{(0.08206)(297.04)} = 1.148 \, \text{g/L}
Density of air at STP: ρair=(1.0)(28.97)(0.08206)(273.15)=1.293 g/L\rho_{\text{air}} = \frac{(1.0)(28.97)}{(0.08206)(273.15)} = 1.293 \, \text{g/L} SGa=1.1481.293≈0.89\text{SG}_{a} = \frac{1.148}{1.293} \approx \boxed{0.89}
b) SG of N₂ at 75 °F and 765 torr compared to air at 75 °F and 765 torr
Since both gases are at the same conditions, we can simply use molar masses: SGb=28.0228.97≈0.97\text{SG}_{b} = \frac{28.02}{28.97} \approx \boxed{0.97}
Explanation
Specific gravity (SG) compares the density of a substance to that of a reference—in this case, nitrogen gas (N₂) compared to air. It’s important to ensure both substances are under the same conditions when determining SG, as gas density changes with temperature and pressure.
In part (a), we compare N₂ at 75 °F and 765 torr to air at standard temperature and pressure (STP: 0 °C and 1 atm). Since the gases are under different conditions, we must calculate their actual densities using the ideal gas law. The density formula ρ=PMRT\rho = \frac{PM}{RT} tells us how pressure, temperature, and molar mass influence density. Calculations show N₂ has a density of about 1.148 g/L at 75 °F and 765 torr, while air has a density of 1.293 g/L at STP. The specific gravity is the ratio of these densities: SG ≈ 0.89. This indicates that under these conditions, N₂ is lighter than air at STP.
In part (b), both gases are at the same conditions (75 °F and 765 torr), allowing us to simply compare their molar masses. Nitrogen’s molar mass is 28.02 g/mol, while air’s is approximately 28.97 g/mol. The SG is then 28.02 / 28.97 ≈ 0.97. This means that at identical conditions, nitrogen is slightly less dense than air, but very close in value—just 3% lighter.
In summary, specific gravity varies with comparison conditions. When comparing gases at different states, their physical densities must be calculated. But when at the same state, SG simplifies to a ratio of molar masses.
