Problem 62 - A Complete Resource Book in Chemistry for JEE Main

Question

The density of a gas is \(1.964 \mathrm{~g} \mathrm{dm}^{-3}\) at \(273 \mathrm{~K}\) and \(76 \mathrm{~cm} \mathrm{Hg}\). The gas is (a) \(\mathrm{CH}_{4}\) (b) \(\mathrm{C}_{2} \mathrm{H}_{6}\) (c) \(\mathrm{CO}_{2}\) (d) \(\mathrm{Xe}\)

Step-by-Step Solution

Answer

The gas is \(\text{CO}_2\).

Step 1: Understand the Relationship

To find out which gas it is, we'll use the ideal gas law concept: the molar mass of the gas can be found using the density formula for an ideal gas, given as \(\text{density} = \frac{PM}{RT}\), where \(P\) is the pressure, \(M\) is the molar mass, \(R\) is the gas constant, and \(T\) is the temperature.

Step 2: Convert Units

Pressure is given in \(\text{cm Hg}\) and needs to be converted to \(\text{Pa}\). Use the conversion \(76 \text{ cm } \text{Hg} = 76 \times 133.322 \text{ Pa} = 10132.472 \text{ Pa}\). Temperature is in Kelvins and does not need conversion.

Step 3: Insert Values and Solve for Molar Mass

Now substitute the known values into the density formula: \[1.964 = \frac{10132.472 \times M}{8.314 \times 273}\]. Solving for \(M\) yields: \[M = \frac{1.964 \times 8.314 \times 273}{10132.472} \approx 44.0 \text{ g/mol}\].

Step 4: Identify the Gas

Compare the calculated molar mass to known molar masses: \(\text{CH}_4\) is approximately \(16 \text{ g/mol}\), \(\text{C}_2\text{H}_6\) is approximately \(30 \text{ g/mol}\), \(\text{CO}_2\) is approximately \(44 \text{ g/mol}\), and \(\text{Xe}\) is approximately \(131 \text{ g/mol}\). The molar mass closely matches \(\text{CO}_2\).

Key Concepts

Density of Gas

Gas density is a crucial property that expresses how much mass of a gas occupies a specific volume. Density can be defined as the mass per unit volume, commonly represented as \( \frac{\text{g}}{\text{dm}^3} \). In terms of ideal gases, density has a direct relationship with pressure, molar mass, and temperature. To calculate the density of a gas under ideal conditions, we can use the formula:

• \( \text{density} = \frac{P M}{R T} \)

This formula shows that density is proportional to the pressure \( P \) and molar mass \( M \), and inversely proportional to the gas constant \( R \) and temperature \( T \).
This relationship is extremely practical because once the density of a gas at certain conditions is known, one can calculate its molar mass if the pressure and temperature are given. This aids in identifying the type of gas. The exercise demonstrates this by comparing calculated molar mass values against known gas molar masses.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, typically expressed in \( \text{g/mol} \). Knowing the molar mass is fundamental for identifying gases and performing stoichiometric calculations in chemistry.
To determine the molar mass of a gas using the ideal gas law substitute in the known values for pressure, temperature, and density into the rearranged version of the ideal gas law formula:

• \( M = \frac{\text{density} \times R \times T}{P} \)

Using this equation allows you to solve for molar mass \( M \). In this exercise, the calculated molar mass was approximately \( 44.0 \text{ g/mol} \), which matches the molar mass of carbon dioxide \( \text{CO}_2 \). Different gases have distinct molar masses; for example:

• \( \text{CH}_4 \approx 16 \text{ g/mol} \)
• \( \text{C}_2\text{H}_6 \approx 30 \text{ g/mol} \)
• \( \text{CO}_2 \approx 44 \text{ g/mol} \)
• \( \text{Xe} \approx 131 \text{ g/mol} \)

Calculating the molar mass using density can provide a strong indication of which gas you are dealing with.

Gas Constant

The gas constant, \( R \), is an essential component of the ideal gas law equation \( PV = nRT \). It provides a bridge between the macroscopic and microscopic properties of gases. The value of \( R \) is consistent, making it a dependable factor when calculating gas-related properties. The common value used in many calculations is:

• \( 8.314 \text{ J/mol K} \)

This constant represents energy per temperature increment per mole. When you use \( R \) in calculations, it's crucial to make sure your pressure, volume, and temperature units are consistent, which usually means converting them to match those used in \( R \). For example, if you're using \( R \) with Joules, ensure the pressure is in Pascals and the volume in cubic meters.
In the exercise, the gas constant \( R \) was intrinsic in calculating the molar mass from the provided gas density. Converting pressure from \( \text{cm Hg} \) to Pascals allowed the use of \( R \) to seamlessly solve the problem. Understanding how to apply \( R \) effectively is key to mastering thermodynamics involving gases.