Question

Consider a rubber plate that is in contact with nitrogen gas at $298 \mathrm{~K}\( and \)250 \mathrm{kPa}$. Determine the molar and mass densities of nitrogen in the rubber at the interface. Answers: $0.0039 \mathrm{kmol} / \mathrm{m}^{3}, 0.1092 \mathrm{~kg} / \mathrm{m}^{3}$

Short answer

Answer: The molar density of nitrogen gas at the interface is \(0.0039\,\mathrm{kmol/m^3}\), and the mass density is \(0.1092\,\mathrm{kg/m^3}\).

Step-by-step solution

Write the ideal gas equation for the given conditions.

For nitrogen gas at 298 K and 250 kPa, we will use the ideal gas equation: PV = nRT Where P is the pressure (250 kPa), V is the volume (unknown), n is the amount of substance (unknown), R is the universal gas constant (8.314 kPa·L/(mol·K)), and T is the temperature (298 K). To find the molar density (in \(\mathrm{kmol/m^3}\)), we will first need to find the volume per mole, which can be done by rearranging the equation above: V = nRT/P

Plug the values into the equation and calculate the volume per mole of nitrogen gas.

Substitute the given values into the equation: V = n (8.314) (298) / 250 V = n (2470.612) / 250 V = n (9.882448) Now, divide both sides by the amount of substance (n) to get the volume per mole: V/n = 9.882448 L/mol We need to convert this to a molar density in \(\mathrm{kmol/m^3}\). By taking the reciprocal and converting the units, we get the molar density: \(\frac{1\,\mathrm{mol}}{9.882448\,\mathrm{L}} \cdot \frac{1000\,\mathrm{L}}{1\,\mathrm{m^3}} \cdot \frac{1\,\mathrm{kmol}}{1000\,\mathrm{mol}} = 0.0039\,\mathrm{kmol/m^3}\) The molar density of nitrogen gas at the interface is \(0.0039\,\mathrm{kmol/m^3}\).

Calculate the mass density using the molar density and molar mass of nitrogen gas.

Now that we have the molar density, we can calculate the mass density by multiplying by the molar mass (28.0134 g/mol): \(0.0039\,\mathrm{kmol/m^3} \cdot \frac{28.0134\,\mathrm{g}}{1\,\mathrm{mol}} \cdot \frac{1\,\mathrm{kg}}{1000\,\mathrm{g}} \cdot \frac{1000\,\mathrm{mol}}{1\,\mathrm{kmol}} = 0.1092\,\mathrm{kg/m^3}\) The mass density of nitrogen gas at the interface is \(0.1092\,\mathrm{kg/m^3}\). The molar density and mass density of Nitrogen in the rubber at the interface are \(0.0039\,\mathrm{kmol/m^3}\) and \(0.1092\,\mathrm{kg/m^3}\), respectively.