Question

\begin{equation}\begin{array}{l}{\text { (a) Calculate the density of } \mathrm{NO}_{2} \text { gas at } 0.970 \text { atm and } 35^{\circ} \mathrm{C} \text { . }} \\ {\text { (b) Calculate the molar mass of a gas if } 2.50 \mathrm{g} \text { occupies } 0.875} \\ {\text { L at } 685 \text { torr and } 35^{\circ} \mathrm{C} \text { . }}\end{array}\end{equation}

Short answer

The density of NO2 gas at 0.970 atm and 35°C is approximately 1.85 g/L. The molar mass of the unknown gas is approximately 77.64 g/mol.

Step-by-step solution

(Part a - Step 1: Convert temperature to Kelvin)

Replace Celsius with Kelvin for the temperature. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. Temperature in Kelvin: \(T = 35^{\circ}\text{C} + 273.15 = 308.15 \text{K}\)

(Part a - Step 2: Use the Ideal Gas Law formula)

Use the Ideal Gas Law formula to find the density. The Ideal Gas Law can be expressed as: \(PV = nRT\) Where: P = Pressure in atm V = Volume of the gas in liters n = Number of moles of the gas R = Ideal Gas constant (0.0821 L atm/mol K) T = Temperature in Kelvin To find the density, we'll rearrange the formula to solve for n/V \(\frac{n}{V} = \frac{P}{RT}\)

(Part a - Step 3: Compute the density of NO2 gas)

Insert the values for pressure and temperature, and the Ideal Gas constant into the formula, then calculate the number of moles per volume: \(\frac{n}{V} = \frac{0.970 \text{ atm}}{(0.0821 \text{ L atm/mol K})(308.15 \text{ K})} = 0.0403 \text{ mol/L}\) Since the density of the gas is the mass of the gas divided by its volume, and the molecular mass of NO2 is 46.01 g/mol, we can calculate the density as follows: Density = \((0.0403 \text{ mol/L}) (46.01 \text{ g/mol})\) = \(1.85 \text{ g/L}\) Thus, the density of NO2 gas at 0.970 atm and 35°C is approximately 1.85 g/L.

(Part b - Step 1: Convert temperature and pressure to appropiate units)

Convert temperature from Celsius to Kelvin just like part (a). Temperature in Kelvin: \(T = 35^{\circ}\text{C} + 273.15 = 308.15 \text{K}\) Convert pressure from torr to atm: Pressure in atm: \(P = 685 \text{ torr} \times \frac{1 \text{ atm}}{760 \text{ torr}} = 0.901 \text{ atm}\)

(Part b - Step 2: Use Ideal Gas Law to find the number of moles)

Insert the values for pressure, volume, temperature and the Ideal Gas constant into the Ideal Gas Law formula, then calculate the number of moles: \(PV = nRT\) \(n = \frac{PV}{RT} = \frac{(0.901 \text{ atm})(0.875 \text{ L})}{(0.0821 \text{ L atm/mol K})(308.15 \text{ K})} = 0.0322 \text{ mol}\)

(Part b - Step 3: Calculate the molar mass of the gas)

Now that we have the number of moles and the mass of the gas (2.50 g), we can calculate the molar mass by dividing the mass by the number of moles: Molar mass = \(\frac{2.50 \text{ g}}{0.0322 \text{ mol}} = 77.64 \text{ g/mol}\) Thus, the molar mass of the unknown gas is approximately 77.64 g/mol.

Key concepts

Density Calculation using Ideal Gas Law

Calculating the density of a gas using the Ideal Gas Law is a critical skill in chemistry. The gas density (\( \text{Density} = \frac{\text{mass}}{\text{volume}} \)) can be determined by rearranging the Ideal Gas Law as:\[ PV = nRT \]Where:

• \( P \) is the pressure in atmospheres (atm)
• \( V \) is the volume in liters (L)
• \( n \) is the number of moles
• \( R \) is the Ideal Gas constant (0.0821 L atm/mol K)
• \( T \) is the temperature in Kelvin (K)

To find the number of moles per unit volume (\( \frac{n}{V} \)), rearrange the formula:\[ \frac{n}{V} = \frac{P}{RT} \]Once you have \( \frac{n}{V} \), you can multiply by the molar mass of the gas to find its density:\[ \text{Density} = \left( \frac{n}{V} \right) \times M \]Where \( M \) is the molar mass. For \( \text{NO}_2 \), this yields the density at given conditions.

Molar Mass Calculation of a Gas

To determine the molar mass of a gas, it is essential to comprehend the relationship between mass, moles, and the Ideal Gas Law. Begin with the formula:\[ M = \frac{\text{mass}}{n} \]Where \( M \) is the molar mass, and \( n \) is the number of moles. The Ideal Gas Law (\( PV = nRT \)) helps us find \( n \):\[ n = \frac{PV}{RT} \]Substitute this into the molar mass equation:\[ M = \frac{\text{mass} \times RT}{PV} \]In the exercise example, the mass given is 2.50 g. The pressure and volume must be converted to appropriate units, ensuring to use atm for pressure and L for volume after conversion.By calculating \( n \), we used it along with the given gas mass to find the molar mass, arriving at a final value of approximately 77.64 g/mol.

Gas Pressure and Temperature Conversions

It's essential to convert temperature and pressure into the correct units for calculations involving the Ideal Gas Law. **Temperature Conversion:**- Convert Celsius to Kelvin by adding 273.15. For example: \[ T = 35^{\circ}\text{C} + 273.15 = 308.15 \text{K} \]This ensures that the correct units are used in the Ideal Gas Law formula.**Pressure Conversion:**- Convert pressure from torr to atm using: \[ P = \frac{\text{torr value}}{760} \]This conversion is crucial because the Ideal Gas constant \( R \) is given in units of L atm/mol K, requiring pressure to be in atm. In the provided exercise, converting 685 torr to atm yielded 0.901 atm. These conversions are vital in obtaining accurate results in subsequent calculations using the Ideal Gas Law.