Question

What is the density of a sample of nitrogen gas \(\left(N_{2}\right)\) that exerts a pressure of 5.30 \(\mathrm{atm}\) in a \(3.50-L\) container at \(125^{\circ} C ?\)

Short answer

The density of the nitrogen gas sample under the given conditions is approximately \(1.96\;\text{g/L}\).

Step-by-step solution

Write down the Ideal Gas Law and the formula for density.

The Ideal Gas Law relates the pressure, volume, and temperature of a gas to the number of moles of the gas (n): \[PV = n\;RT \] Where P is pressure, V is volume, and T is temperature in Kelvin (K). The formula for density (ρ) is defined as mass (m) divided by volume (V): \[\rho = \frac{m}{V} \]

Convert the given temperature to Kelvin and write down the given values.

First, we need to convert the given temperature from Celsius (°C) to Kelvin (K) using the formula: \[T (K)=T (°C)+273.15\] Now we can write the values given in the problem: P = 5.30 atm V = 3.50 L T = 125 °C ⟹ T = 125 + 273.15 = 398.15 K

Rewriting the Ideal Gas Law equation to include the molecular mass and density.

In order to get the mass of the gas in the Ideal Gas Law equation, we want to express the number of moles (n) as mass (m) divided by the molecular mass (M). So, the equation becomes: \[PV = \frac{m}{M} \; RT\] Now, we rewrite the equation in terms of density (ρ) using the formula \(\rho = \frac{m}{V}\): \[P = \rho \frac{RT}{M}\] Since we are looking for the density (ρ), we rearrange the equation to isolate ρ: \[\rho = \frac{PM}{RT}\]

Plug in the values and find the density of the gas.

As we know, Nitrogen gas has a molecular mass of M = 28.02 g/mol. Now, we plug in the known values into our formula: \[\rho = \frac{5.30\;\text{atm}\cdot 28.02\;\text{g/mol}}{(0.0821\;\text{L}\cdot\text{atm} / \text{mol}\cdot\text{K}) \cdot 398.15\;\text{K}}\] After calculating the expression, we find the density (ρ): \[\rho = 1.96\;\text{g/L}\] So, the density of the nitrogen gas sample under these conditions is approximately 1.96 g/L.

Key concepts

Gas Density

Gas density is a measure of how much mass of the gas is present in a specific volume. It is typically expressed in units such as grams per liter (g/L). Understanding the density of gases helps in many practical applications, such as calculating the weight of the gas in a container or comparing it to the density of other gases.The formula for density is straightforward:

• Density, denoted as \( \rho \), equals mass (m) divided by volume (V): \(\rho = \frac{m}{V}\).

By manipulating this formula using the concepts of the Ideal Gas Law, you can replace mass with known variables to calculate the density. It allows us to address the density of gases without directly measuring their mass. Always remember, density can change with temperature and pressure, which is why the Ideal Gas Law can be particularly useful.

Temperature Conversion

When working with gases in chemistry, converting temperature from Celsius to Kelvin is crucial because the Ideal Gas Law uses an absolute temperature scale. Celsius is not suitable because it can go below zero, making it impossible to calculate meaningful gas volume or pressure relationships.To convert:

• Add 273.15 to the Celsius temperature to convert to Kelvin. For example: \[ T (\text{K}) = T (\text{°C}) + 273.15 \]

In the given exercise, converting \(125\;^{\circ}C\) to Kelvin results in a temperature of \(398.15\;K\). Using the Kelvin scale ensures we avoid negative temperatures, providing consistent, usable data in calculations.

Molecular Mass of Nitrogen

Understanding the molecular mass of nitrogen is essential for calculations involving this gas. Molecular mass (or molar mass) is the mass of one mole of a substance and is expressed in grams per mol (g/mol).Nitrogen typically exists as a diatomic molecule,\( N_2 \). This means that a nitrogen molecule is made up of two nitrogen atoms. The atomic mass of a single nitrogen atom is approximately 14.01 g/mol. Therefore, the molecular mass of nitrogen gas is:

• \( M = 2 \times 14.01 \;\text{g/mol} = 28.02 \;\text{g/mol} \).

Knowing this value is crucial for density calculations involving nitrogen gas, as it helps to replace the number of moles in the formulas with mass-related variables.

Pressure-Volume Relationship

The pressure-volume relationship is a fundamental concept in the study of gases, often explored through the Ideal Gas Law. The law itself is a comprehensive guide to understanding how gas particles behave under different conditions of temperature, volume, and pressure.The Ideal Gas Law is stated as:

• \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature (in Kelvin).

In practice, this relationship tells us how, for a given amount of gas (fixed moles), volume and pressure are inversely proportional if the temperature is constant. This means, if you increase the volume, the pressure decreases, and vice versa. The formula underpins many applications, such as determining gas density or predicting how gases will respond to changes in environmental conditions.