Question

(a) Calculate the density of dinitrogen tetroxide gas \(\left(\mathrm{N}_{2} \mathrm{O}_{4}\right)\) at \(111.5 \mathrm{kPa}\) and \(0{ }^{\circ} \mathrm{C}\). (b) Calculate the molar mass of a gas if 2.70 g occupies \(0.97 \mathrm{~L}\) at \(134.7 \mathrm{~Pa}\) and \(100^{\circ} \mathrm{C}\).

Short answer

The density of N2O4 at 111.5 kPa and 0°C is \(4.56 g/L\). The molar mass of the unknown gas is \(64.23 g/mol\).

Step-by-step solution

a) Calculating the density of dinitrogen tetroxide gas

First, let's convert the given temperature and pressure to Kelvin and Pascals, respectively: Temperature = \(0^{\circ}C + 273.15 = 273.15 K\) Pressure = \(111.5 kPa * 1000 = 111500 Pa\) Next, we need to find the molar mass of dinitrogen tetroxide (N2O4). Using the periodic table, we can find the atomic masses of nitrogen (N) and oxygen (O): Molar mass of nitrogen (N) = 14.01 g/mol Molar mass of oxygen (O) = 16.00 g/mol So, the molar mass of N2O4 will be: Molar mass of N2O4 = 2(14.01) + 4(16.00) = 28.02 + 64.00 = 92.02 g/mol Now, let's rearrange the ideal gas law formula to solve for the density (\(ρ\)) where density = mass / volume: \(ρ = \frac{n \times M}{V}\) \(ρ = \frac{P \times M}{R \times T}\) (density of the gas) Plug in the values: \(ρ = \frac{111500 \times 92.02}{8.314 \times 273.15}\) Calculate the density: \(ρ = \frac{10254711}{2250.88} = 4.56 g/L\) The density of N2O4 at 111.5 kPa and 0°C is \(4.56 g/L\).

b) Calculating the molar mass of the unknown gas

First, let's convert the given temperature and pressure to Kelvin and Pascals, respectively: Temperature = \(100^{\circ} C + 273.15 = 373.15 K\) Pressure = \(134.7 Pa\) (already given in Pascals) Now, let's use the ideal gas law to calculate the number of moles (n): \(PV = nRT\) Solve for n: \(n = \frac{PV}{RT}\) Plug in the values: \(n = \frac{134.7 \times 0.97}{8.314 \times 373.15}\) Calculate the number of moles: \(n = \frac{130.459}{3102.47} = 0.04204 \, mol\) Now, let's use the number of moles (n) to calculate the molar mass (M) of the unknown gas. We know that the mass of the gas is 2.70 g. So: Molar mass (M) = mass / moles Solve for M: \(M = \frac{2.70}{0.04204}\) Calculate the molar mass: \(M = 64.23 g/mol\) The molar mass of the unknown gas is \(64.23 g/mol\).

Key concepts

Density Calculation

Density is a measure of mass per unit volume and is a crucial concept when dealing with gases. For gases, it's essential to remember that density is not constant and depends on the conditions of temperature and pressure. In the context of the Ideal Gas Law, we often utilize the equation \( \rho = \frac{PM}{RT} \), where \( \rho \) is the density, \( P \) is the pressure, \( M \) is the molar mass, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. Let's break it down:

• Pressure \( (P) \): Ensure that pressure is in Pascals when using the Ideal Gas Law. For conversion, multiply kPa by 1000.
• Molar Mass \( (M) \): Calculate using atomic masses from the periodic table. In our example, the molar mass of \( \text{N}_2\text{O}_4 \) was calculated as \( 92.02 \text{ g/mol} \).
• Ideal Gas Constant \( (R) \): Standard value \( 8.314 \text{ J/mol K} \).
• Temperature \( (T) \): Always convert to Kelvin by adding 273.15 to the Celsius temperature.
• Substitute these values correctly to find the gas's density under the given conditions.

Calculating density for gases allows us to predict how much space a certain mass of gas will occupy, which is highly relevant in real-world applications like balloon filling or chemical reactions in containers.

Molar Mass Determination

Determining the molar mass of a gas can help identify the gas. It's a crucial step in understanding what a gas is composed of, especially when dealing with unknown samples. To find out the molar mass, you can rearrange the Ideal Gas Law:Here's a step-by-step breakdown:

• Molar Mass \( (M) \): Derived from the equation \( M = \frac{m}{n} \), where \( m \) is the mass of the gas in grams and \( n \) is the number of moles.
• Number of Moles \( (n) \): Calculate this using the equation \( n = \frac{PV}{RT} \).
• Pressure \( (P) \): Keep in Pascals.
• Volume \( (V) \): Should be in liters.
• Ideal Gas Constant \( (R) \): \( 8.314 \text{ J/mol K} \).
• Temperature \( (T) \): Converted from degrees Celsius to Kelvin.

In practice, you first calculate the number of moles using the gas's pressure, volume, and temperature. Then, use the calculated moles and the given mass to determine the molar mass. This entire process is vital in chemical engineering, lab experiments, and quality control where it's paramount to confirm the identity of gaseous substances.

Gas Laws

Gas laws are the mathematical relationships that describe the behavior of gases. They help predict how gases will behave under different sets of conditions. The Ideal Gas Law, represented by \( PV = nRT \), is central to these calculations. Here’s a quick guide to understanding the key components:

• Pressure \( (P) \): Represents the force the gas exerts on its container. It's typically measured in Pascals (Pa).
• Volume \( (V) \): The space that the gas occupies, usually measured in liters (L).
• Moles \( (n) \): Indicates the amount of gas, given in moles.
• Ideal Gas Constant \( (R) \): A constant value, typically \( 8.314 \text{ J/mol K} \).
• Temperature \( (T) \): Measure of the thermal energy, converted to Kelvin for calculations.

The relationships defined by these variables allow predictions of a gas’s behavior:

• Boyle's Law: Pressure is inversely proportional to volume when temperature and moles are constant.
• Charles's Law: Volume is directly proportional to temperature when pressure and moles are constant.
• Avogadro's Law: Volume is directly proportional to the number of moles when temperature and pressure are constant.

These laws are essential for solving real-world problems like calculating the amount of gas needed for a reaction, analyzing engine efficiency, and even understanding how balloon animals are affected by altitude changes.