Problem 47 - Chemistry

Question

A 2.10-L vessel contains 4.65 g of a gas at 1.00 atm and \(27.0^{\circ} \mathrm{C}\). (a) Calculate the density of the gas in grams per liter. (b) What is the molar mass of the gas?

Step-by-Step Solution

Answer

The density of the gas is 2.21 g/L and the molar mass of the gas is 54.7 g/mol.

Step 1: Convert Temperature from Celsius to Kelvin

Using the conversion \(K = °C + 273.15\), the provided temperature value of \(27.0 °C\) becomes \(27.0 °C + 273.15 = 300.15 K\).

Step 2: Find Gas Density

The gas density formula is \(D = \frac{m}{V}\), where \(D\) is density, \(m\) is the mass of gas (4.65 g), and \(V\) is the volume of gas (2.10 L). So, \(D = \frac{4.65 g}{2.10 L} = 2.21 g/L\).

Step 3: Determine Molar Mass using the Ideal Gas Law

The ideal gas law formula is \(PV = nRT\), where \(P\) is pressure (1.00 atm), \(V\) is volume (2.10 L), \(n\) is number of moles, \(R\) is gas constant (0.0821 \(L \cdot atm / (K \cdot mol\))), and \(T\) is temperature (300.15 K). To calculate the number of moles \(n\), it's rearranged to \(n = \frac{PV}{RT}\). Substituting, \(n = \frac{1.00 atm \cdot 2.10 L}{0.0821 L \cdot atm / (K \cdot mol) \cdot 300.15 K} = 0.085 mol\). The formula for molar mass is \(MM = \frac{m}{n}\), where \(m\) is the mass of gas (4.65 g) and \(n\) is number of moles (0.085 mol). Substituting the values, \(MM = \frac{4.65 g}{0.085 mol} = 54.7 g/mol\).

Key Concepts

Density Calculation of a Gas

Calculating the density of a gas involves using a simple formula. The density \(D\) of a gas is determined by dividing its mass \(m\) by the volume \(V\) it occupies. In the ideal gas law problem given, you have a gas with a mass of 4.65 grams, contained in a 2.10-liter vessel. Applying the formula:

• \(D = \frac{m}{V}\)

Substituting in the numbers:

• \(D = \frac{4.65 \text{ g}}{2.10 \text{ L}}\)
• \(D \approx 2.21 \text{ g/L}\)

This calculation gives you the density of the gas in grams per liter. It basically tells you how much mass of the gas is present in each liter of volume it occupies.

Molar Mass Determination

Determining the molar mass of a gas in this context uses the ideal gas law, which is a cornerstone equation in chemistry. The ideal gas law is \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. To find the number of moles of the gas, rearrange the equation to \(n = \frac{PV}{RT}\).

• Given: \(P = 1.00 \text{ atm}\), \(V = 2.10 \text{ L}\), \(R = 0.0821 \text{ L} \cdot \text{atm} / (\text{K} \cdot \text{mol})\), and \(T = 300.15 \text{ K}\).
• \(n = \frac{1.00 \text{ atm} \times 2.10 \text{ L}}{0.0821 \text{ L} \cdot \text{atm} / (\text{K} \cdot \text{mol}) \times 300.15 \text{ K}}\)
• \(n \approx 0.085 \text{ mol}\)

With the number of moles determined, calculate the molar mass \(MM\) using \(MM = \frac{m}{n}\), where \(m\) is mass and \(n\) is moles.

• \(MM = \frac{4.65 \text{ g}}{0.085 \text{ mol}}\)
• \(MM \approx 54.7 \text{ g/mol}\)

This tells you the mass of one mole of the gas.

Temperature Conversion to Kelvin

Converting temperature to Kelvin is straightforward yet essential in calculations involving gases, as the Kelvin scale is used in gas equations to maintain proportional relationships. The conversion formula from Celsius to Kelvin is:

• \(K = °C + 273.15\)

For the given problem, the initial temperature is 27.0°C. Applying the conversion:

• \(K = 27.0 \degree C + 273.15\)
• \(K = 300.15\)

The Kelvin temperature is now suitable for use in the ideal gas law and other thermodynamic equations. This process ensures that all temperature measurements align with the absolute scale, which starts at absolute zero, making it ideal for scientific calculations.