Problem 74 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the density (in \(\mathrm{g} \mathrm{L}^{-1}\) ) of \(\mathrm{CO}_{2}\) at \(400 \mathrm{~K}\) and exerting a pressure of \(0.0821 \mathrm{~atm} ?(\mathrm{R}=0.0821 \mathrm{~L} \mathrm{~atm}\) \(\left.\mathrm{mol}^{-1} \mathrm{~K}^{-1}\right)\) (a) \(0.01\) (b) \(0.11\) (c) \(2.5\) (d) 44

Step-by-Step Solution

Answer

The density of \(\mathrm{CO}_2\) under the given conditions is \(0.11 \ \mathrm{g/L}\).

Step 1: Understand the Ideal Gas Law

The ideal gas law is given by \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.

Step 2: Rewrite the Ideal Gas Law

To find density, which is mass per unit volume, first recognize that \( n = \frac{m}{M} \), where \( m \) is mass and \( M \) is molar mass. Rewriting the ideal gas law, we have: \( P = \frac{m}{M} \cdot \frac{RT}{V} \). Rearranging, \( m = \frac{PM}{RT} \cdot V \).

Step 3: Calculate Molar Mass of \(\mathrm{CO}_2\)

The molar mass \( M \) of \(\mathrm{CO}_2\) is the sum of the atomic masses of C and O: \( M = 12 + 2(16) = 44 \ \mathrm{g/mol} \).

Step 4: Derive Density Formula

Density \( d \) is \( \frac{m}{V} \). By substituting for \( m \) using the equation from Step 2, \( d = \frac{PM}{RT} \).

Step 5: Substitute Given Values

Substitute \( P = 0.0821 \ \mathrm{atm} \), \( M = 44 \ \mathrm{g/mol} \), \( R = 0.0821 \ \mathrm{L \ atm \ mol^{-1} \ K^{-1}} \), and \( T = 400 \ \mathrm{K} \) into the density formula: \[ d = \frac{0.0821 \times 44}{0.0821 \times 400} \].

Step 6: Simplify the Expression

Notice that \( 0.0821 \) cancels out in the numerator and the denominator, simplifying to \[ d = \frac{44}{400} = 0.11 \ \mathrm{g/L} \].

Key Concepts

Density calculation

Density is defined as mass per unit volume. In scientific fields, calculating the density of gases is a fundamental concept that helps us understand properties of various substances under different conditions.
To calculate density of a gas, we often use the Ideal Gas Law, which connects pressure, volume, temperature, and the number of moles in a system.

The Ideal Gas Law is expressed as:

• \(PV = nRT\)

Where:

• \(P\) stands for pressure
• \(V\) represents volume
• \(n\) is the number of moles
• \(R\) is the gas constant
• \(T\) is the temperature

To find the density, recognize that the number of moles \(n\) can be expressed as \(n = \frac{m}{M}\), where \(m\) is mass and \(M\) is the molar mass.
Substituting, we manipulate the formula to \(d = \frac{PM}{RT}\), which allows us to find the density directly when pressure, molar mass, temperature, and the gas constant are known.

Carbon dioxide properties

Carbon dioxide (CO₂) is a colorless and odorless gas that plays a crucial role in Earth's atmosphere. It is a molecule made up of one carbon atom covalently double bonded to two oxygen atoms. This compound is a significant greenhouse gas due to its ability to trap heat in the atmosphere.
Carbon dioxide has several applications, including:

• Usage in carbonation of beverages
• As a refrigerant
• For fire extinguishers

More importantly, in the context of chemical calculations, the molar mass of CO₂ needs to be taken into account.
The molar mass of CO₂ is calculated by adding the atomic mass of carbon (12 g/mol) and twice the atomic mass of oxygen (16 g/mol each), which equals to 44 g/mol.
This molar mass is essential for solving problems using the Ideal Gas Law.

Gas constant

The gas constant, often symbolized by \( R \), is a crucial component in the Ideal Gas Law equation.
It acts as a bridge connecting the macroscopic measurements of pressure and volume with the microscopic constants like molar mass and temperature.

The gas constant \(R\) has a consistent value of \(0.0821 \ \mathrm{L \ atm \ mol^{-1} \ K^{-1}}\), which is used in scenarios where pressure is measured in atmospheres and volume in liters.

When applying the Ideal Gas Law, knowing the appropriate units for \(R\) is necessary.

• For instance, if pressure is in atmospheres and volume is in liters, the given \(R\) value is perfectly suited.
• Ensure all other units in the equation are compatible too to maintain consistency in the equation.

This consistency ensures accurate calculations, whether you are finding the density of a gas or some other related property.