Question

The density of nitrogen at 0.913 atm and \(18^{\circ} \mathrm{C}\) is \(1.07 \mathrm{g} / L .\) Explain how this shows that the formula of nitrogen is \(\mathrm{N}_{2}\) rather than just \(\mathrm{N}\).

Short answer

Nitrogen has a molar mass of about 28 g/mol, indicating the formula is \(N_2\), as opposed to \(N\) with 14 g/mol.

Step-by-step solution

Understand the Relationship

To determine the molecular formula of nitrogen, we need to relate the density, pressure, and temperature of nitrogen gas to its molar mass using the Ideal Gas Law. The density given is 1.07 g/L, the pressure is 0.913 atm, and the temperature is 18°C.

Convert Temperature to Kelvin

Convert the temperature from Celsius to Kelvin using the formula: \[ \text{Temperature in Kelvin} = 18 + 273.15 = 291.15 \text{ K} \]

Apply the Ideal Gas Law

The Ideal Gas Law is given by: \[ PV = nRT \] Where: - \( P = 0.913 \text{ atm} \)- \( V \) is the volume in liters- \( n \) is the number of moles- \( R = 0.0821 \text{ L atm/mol K} \) is the ideal gas constant- \( T = 291.15 \text{ K} \)We aim to find the molar mass (M) using density \( D \) in the equation: \[ M = \frac{D \cdot R \cdot T}{P} \]

Calculate the Molar Mass of Nitrogen

Substitute the values into the equation:\[ M = \frac{1.07 \text{ g/L} \times 0.0821 \text{ L atm/mol K} \times 291.15 \text{ K}}{0.913 \text{ atm}} \approx 28 \text{ g/mol} \]This result is close to the known molar mass of nitrogen gas \(N_2\), which is approximately 28 g/mol.

Compare with Atomic Nitrogen

The atomic mass of elemental nitrogen (\(N\)) is approximately 14 g/mol. Since the calculated molar mass is about 28 g/mol, this indicates the molecular formula is \(N_2\), rather than just \(N\).

Key concepts

Molecular Formula Determination

Understanding the molecular formula of nitrogen requires a firm grasp of the relationship between its physical properties and chemical composition. In this exercise, we're tasked with showing that the molecular formula of nitrogen is \( \mathrm{N}_2 \) instead of just \( \mathrm{N} \). We use the Ideal Gas Law and the concept of density to make this determination. Given the density, pressure, and temperature of nitrogen gas, we start by converting all measurements into compatible units. The temperature is converted to Kelvin for consistency. We then apply the Ideal Gas Law to establish a link between density and molar mass. Once we calculate the molar mass to be close to \( 28 \text{ g/mol} \), a figure that matches the known molar mass of \( \mathrm{N}_2 \), we confirm that nitrogen gas exists as diatomic \( \mathrm{N}_2 \) molecules at the given conditions. This shows that nitrogen pairs up in nature, bonding together to form molecules with a molar mass of approximately 28 g/mol rather than existing singly as single atoms.

Molar Mass Calculation

Molar mass calculation is crucial for identifying the molecular formula of a gas. In this context, knowing how to derive it using the Ideal Gas Law is essential. The Ideal Gas Law equation \( PV = nRT \) helps to understand this. Here, \( P \) stands for pressure, \( V \) for volume, \( n \) for moles, \( R \) is the ideal gas constant, and \( T \) is temperature. For molar mass calculation, we actually use a derived form of the Ideal Gas Law:

• \( M = \frac{D \cdot R \cdot T}{P} \)

where \( M \) is molar mass, and \( D \) is density. In the example given, substituting the values provided—density of 1.07 g/L, pressure of 0.913 atm, and temperature of 291.15 K—into the formula results in a calculated molar mass of approximately 28 g/mol. This confirms that the substance in question is not atomic nitrogen \( \mathrm{N} \) (which has a molar mass of about 14 g/mol) but rather molecular nitrogen \( \mathrm{N}_2 \).

Gas Density Analysis

Gas density offers valuable insight when determining chemical identities and molecular formulas. Density is mass per unit volume and it varies for different gases under given conditions. To analyze gas density in this problem, we relate it to the molar mass using the Ideal Gas Law. For example, the equation \( M = \frac{D \cdot R \cdot T}{P} \) helps connect these values. By plugging in given data, the result leads to the determination of molar mass, further confirming the molecular structure of the gas.

• At a density of 1.07 g/L, nitrogen's behavior aligns with expectations for \( \mathrm{N}_2 \) under these circumstances.
• By confirming the calculated molar mass around 28 g/mol, we deduce nitrogen gas is \( \mathrm{N}_2 \), not \( \mathrm{N} \).

Evaluating why nitrogen presents as \( \mathrm{N}_2 \), we delve into molecular stability. Nitrogen naturally forms strong triple bonds with itself, ensuring it remains diatomic (\( \mathrm{N}_2 \)), which is reflected in its density against elemental nitrogen.