Question

The density (in \(\mathrm{g} /\) l) of an equimolar mixture of methane and ethane at 1 atm and \(0^{\circ} \mathrm{C}\) is (a) \(1.03\) (b) \(2.05\) (c) \(0.94\) (d) \(1.25\)

Short answer

g/L

Step-by-step solution

1. Determine the molar mass of each gas

To start, we need to know the molar mass of both methane (CH4) and ethane (C2H6). From the periodic table, the molar mass of carbon is 12, and hydrogen is 1. Molar mass of methane (CH4) = 12 + (4 × 1) = 16 g/mol Molar mass of ethane (C2H6) = (2 × 12) + (6 × 1) = 24 + 6 = 30 g/mol

2. Find the molar mass of the equimolar mixture

Since the mixture contains equal moles of methane and ethane, the average molar mass of the mixture can be found by taking the arithmetic mean of the molar masses of both gases. Mixture molar mass = (Molar mass of CH4 + Molar mass of C2H6) / 2 Mixture molar mass = (16 g/mol + 30 g/mol) / 2 = 23 g/mol

3. Use the ideal gas equation to find the density of the mixture

The ideal gas equation is given by: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. To find the density (mass/volume), we can rearrange this equation: Density = mass/volume = (n × Mixture molar mass) / V Since PV = nRT, we can substitute for n: Density = (P × Mixture molar mass) / (R × T) Given: P = 1 atm, T = \(0^{\circ} \mathrm{C}\) = 273.15 K, and R = 0.0821 L × atm / (mol × K) Density = (1 atm × 23 g/mol) / (0.0821 L × atm / (mol × K) × 273.15 K)

4. Calculate the density and match with the given options

Now, we can calculate the density of the equimolar mixture: Density = (23.0 g/mol) / (0.0821 L × atm / (mol × K) × 273.15 K) = 1.03 g/L Comparing the calculated density with the given options, we find that the correct answer is: (a) \(1.03\)

Key concepts

Ideal Gas Law

The Ideal Gas Law is a crucial concept in understanding how gases behave under different conditions. It is generally expressed by the equation \( PV = nRT \), where:

• \(P\) stands for pressure.
• \(V\) is the volume.
• \(n\) represents the number of moles.
• \(R\) is the universal gas constant \((0.0821 \text{ L} \,\text{{atm}} / (\text{{mol}} \,\text{K}))\).
• \(T\) is the absolute temperature in Kelvin.

This law helps to relate the different properties of a gas. By manipulating the equation, we can derive important information, such as gas density or molar mass.
In the context of this problem, we rearranged the equation to calculate density. By substituting known values, we convert the equation to find density: \[ \text{Density} = \frac{P \times \text{Mixture molar mass}}{R \times T}\] Understanding and manipulating the Ideal Gas Law is central to solving many gas-related problems.

Molar Mass Calculation

Calculating molar mass is a fundamental step in finding gas density. It's useful for understanding the composition of gases. The molar mass of a molecule is determined by adding up the atomic masses of its elements. You can find these values on the periodic table.
For example,

• The molar mass of methane \((\text{CH}_4)\) is calculated as \(12 \text{ g/mol (for C)} + (4 \times 1 \text{ g/mol (for H)}) = 16 \text{ g/mol}\).
• The molar mass of ethane \((\text{C}_2\text{H}_6)\) is \{(2 \times 12 \text{ g/mol}) + (6 \times 1 \text{ g/mol}) = 30 \text{ g/mol}\}.

These calculations are crucial in determining the overall molar mass of any mixture formed by different gases. The molar mass will influence other properties, like density, especially in mixtures of gases.

Equimolar Mixture

An equimolar mixture contains equal amounts of moles of different substances. For the problem, equimolar refers to equal moles of methane and ethane. Calculating properties like density requires an understanding of how these substances contribute equally to the mixture.
To find the molar mass of such a mixture, take the mean of individual molar masses:\[ \text{Mixture molar mass} = \frac{\text{Molar mass of Methane} + \text{Molar mass of Ethane}}{2}\]In this case, it results in \(\frac{16\, ext{g/mol} + 30\, ext{g/mol}}{2} = 23\, ext{g/mol}\). This averaged molar mass is then used to determine the density of the gas mixture under the conditions specified.

Gas Density Calculation

Gas density is essentially mass per unit volume of gas. To calculate it using the Ideal Gas Law, you need the molar mass of the gas or mixture, the pressure, temperature, and the universal gas constant. The specific formula rearranges the Ideal Gas Law to \(\text{Density} = \frac{P \times \text{Mixture molar mass}}{R \times T}\), where:

• \(P = 1 \text{ atm}\)
• \(R = 0.0821 \text{ L atm / mol K}\)
• \(T = 273.15 \text{ K}\)

In the problem, you use these values along with the molar mass of the mixture \((23 \text{ g/mol})\) to solve for density. Carrying out the calculation, \(\text{Density} = \frac{1 \text{ atm} \times 23 \text{ g/mol}}{0.0821 \text{ L atm / mol K} \times 273.15 \text{ K}} = 1.03 \text{ g/L}\).
This calculation determines how compact the gas molecules are within a given space, based on the conditions at 0°C and 1 atm.