A vessel of capacity 3 m³ contains 1 kg mole of Nitrogen at 90°C. Characteristic gas constant, R for nitrogen (molecular weight, M 28) is 296.9 J/ kg K. The pressure and specific volume of the gas will be:

Concept:

We use the ideal gas law to determine the pressure and specific volume of nitrogen gas under given conditions.

Given:

• Volume of vessel, \( V = 3 \, \text{m}^3 \)
• Amount of gas, \( n = 1 \, \text{kg mole} \)
• Temperature, \( T = 90^\circ \text{C} = 363.15 \, \text{K} \)
• Characteristic gas constant for nitrogen, \( R = 296.9 \, \text{J/kg K} \)
• Molecular weight of nitrogen, \( M = 28 \, \text{kg/kmol} \)

Step 1: Calculate the universal gas constant (R_u)

The universal gas constant is related to the characteristic gas constant by:

\( R_u = R \times M = 296.9 \times 28 = 8313.2 \, \text{J/kmol K} \)

Step 2: Determine the pressure using ideal gas law

The ideal gas law is given by:

\( P V = n R_u T \)

Substituting the values:

\( P = \frac{n R_u T}{V} = \frac{1 \times 8313.2 \times 363.15}{3} \)

\( P = 1,006,000 \, \text{Pa} = 10.06 \, \text{bar} \)

Step 3: Calculate the specific volume

Specific volume (\( v \)) is the volume per unit mass:

\( v = \frac{V}{m} \)

Mass of nitrogen:

\( m = n \times M = 1 \times 28 = 28 \, \text{kg} \)

Thus:

\( v = \frac{3}{28} = 0.107 \, \text{m}^3/\text{kg} \)