Question

Large amounts of nitrogen gas are used in the manufacture of ammonia, principally for use in fertilizers. Suppose \(120.00 \mathrm{~kg}\) of \(\mathrm{N}_{2}(g)\) is stored in a \(1100.0-\mathrm{L}\) metal cylinder at \(280^{\circ} \mathrm{C}\). (a) Calculate the pressure of the gas, assuming ideal-gas behavior. (b) By using the data in Table 10.3 , calculate the pressure of the gas according to the van der Waals equation. (c) Under the conditions of this problem, which correction dominates, the one for finite volume of gas molecules or the one for attractive interactions?

Short answer

The pressure of nitrogen gas using the ideal gas law is calculated as \(P = \frac{nRT}{V}\), after converting the mass of nitrogen gas to moles and temperature to Kelvin. The pressure using the van der Waals equation is calculated as \(P = \frac{nRT}{V-nb} - a\frac{n^2}{V^2}\), where a and b are the van der Waals constants for nitrogen gas. Comparing the calculated pressures from both equations allows us to determine whether the finite volume term \((n^2a/V^2)\) or the attractive interaction term \((nb)\) correction dominates under these conditions.

Step-by-step solution

(Step 1: Convert mass of nitrogen gas to moles)

To calculate the pressure of nitrogen gas, we first need to convert the given mass (120.00 kg) to moles. We can do this using the molar mass of nitrogen gas, which is 28.02 g/mol. Remember to convert kg to g. Moles of nitrogen gas = \(\frac{120.00 kg \times 1000 \frac{g}{kg}}{28.02 \frac{g}{mol}}\)

(Step 2: Convert Celsius to Kelvin)

Pressure calculations require temperature to be expressed in Kelvin. Convert the given temperature from Celsius to Kelvin like this: \(T (K) = T (°C) + 273.15\) \(T (K) = 280 + 273.15\)

(Step 3: Calculate pressure using the ideal gas law)

We can now calculate the pressure of nitrogen gas using the ideal gas law: \(PV = nRT\) where: P = pressure (unknown) V = volume = 1100.0 L n = moles of nitrogen gas (calculated in Step 1) R = gas constant = 0.0821 \(\frac{L \cdot atm}{mol \cdot K}\) T = temperature in Kelvin (calculated in Step 2) Solve for pressure (P): \(P = \frac{nRT}{V}\)

(Step 4: Calculate pressure using the van der Waals equation)

Now, we'll calculate the pressure using the van der Waals equation instead: \(P = \frac{nRT}{V-nb} - a\frac{n^2}{V^2}\) where: P = pressure (unknown) n = moles of nitrogen gas (calculated in Step 1) R = gas constant = 0.0821 \(\frac{L \cdot atm}{mol \cdot K}\) T = temperature in Kelvin (calculated in Step 2) V = volume = 1100.0 L a and b = van der Waals constants for nitrogen gas, consult Table 10.3 for their values Solve for pressure (P):

(Step 5: Compare the two pressures and determine the dominant correction)

Compare the calculated pressures from Steps 3 and 4 to determine which correction, finite volume or attractive interactions, dominates under these conditions. This can be done by analyzing the differences in pressure values calculated using the ideal gas law and the van der Waals equation. If the difference is largely due to the finite volume term \((n^2a/V^2)\) or the attractive interaction term \((nb)\), the correction that dominates can be determined.

Key concepts

Van der Waals Equation

The Van der Waals Equation is a modified version of the Ideal Gas Law. It accounts for the real behavior of gases by considering two main factors: the finite volume occupied by gas molecules and the attractive forces between them. While the Ideal Gas Law, expressed as \( PV = nRT \), assumes that gas molecules have no volume and do not attract each other, these assumptions do not hold true in real-life scenarios.

In the Van der Waals equation, expressed as \( P = \frac{nRT}{V - nb} - a\frac{n^2}{V^2} \), two correction factors are added:

• Volume correction \( nb \): This accounts for the physical space that gas molecules occupy, reducing the volume of the container effectively available to the gas.
• Pressure correction \( a\frac{n^2}{V^2} \): This accounts for intermolecular forces, which cause the gas molecules to attract each other, reducing the pressure they exert on the container walls.

These adjustments make the Van der Waals equation helpful for calculating the properties of gases under non-ideal conditions, such as high pressures or low temperatures, where deviations from ideal behavior become significant.

Gas Pressure Calculation

Gas pressure calculation is crucial in understanding how gases behave under different conditions. The pressure of a gas is determined by how frequently and forcefully gas molecules collide with the walls of their container.

In ideal conditions, we use the Ideal Gas Law to calculate pressure. The formula \( P = \frac{nRT}{V} \) relates pressure (\( P \)) to:

• Moles of gas (\( n \)): The amount of gas present.
• Temperature (\( T \)): Expressed in Kelvin, it reflects the energy of the gas molecules.
• Volume (\( V \)): The space in which the gas is contained.
• Gas constant (\( R \)): A constant that relates the energy scale to the temperature scale.

Calculating the gas pressure accurately helps in applications like industrial gas storage, as in ammonia production, ensuring safety and efficiency in gas usage.

Molar Mass

Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). In the context of gases, understanding molar mass is essential for converting between the mass of a gas and the number of moles, which is necessary for pressure and volume calculations.

For nitrogen gas, represented as \( \text{N}_2 \), the molar mass is \( 28.02 \frac{g}{mol} \). This means that 28.02 grams of nitrogen correspond to one mole of nitrogen molecules. To convert a given mass of nitrogen gas to moles, use the formula:

• Moles = \( \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \)

For example, to convert 120 kg of nitrogen to moles, convert kilograms to grams, then divide by the molar mass:
\[ \frac{120,000 \text{ g}}{28.02 \frac{\text{g}}{\text{mol}}} \]

Understanding molar mass allows students to link chemical compositions with physical measurements, crucial for tasks such as calculating gas pressures in a given volume.