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 \(0.0821\) atm? \((R=0.0821 \mathrm{~L}\) 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 \( \mathrm{CO}_2 \) is \( 0.11 \mathrm{~g/L} \).

Step 1: Understand the Problem

We need to find the density of carbon dioxide \( \mathrm{CO}_2 \) under given conditions. Density is mass per unit volume (\( \mathrm{g/ L} \)). Given the ideal gas law \( PV = nRT \), we can relate \( n = \frac{m}{M} \) where \( m \) is the mass and \( M \) is molar mass.

Step 2: Use Ideal Gas Law

The ideal gas law is \( PV = nRT \). Rearrange this equation to solve for density: \( \frac{m}{V} = \frac{PM}{RT} \), where \( m/V \) is density and \( M \) is the molar mass of \( \mathrm{CO}_2 \), approximately \( 44 \mathrm{~g/mol} \).

Step 3: Plug in the Values

Plug in the known values into the rearranged equation: \( P = 0.0821 \mathrm{~atm} \), \( M = 44 \mathrm{~g/mol} \), \( R = 0.0821 \mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} \), and \( T = 400 \mathrm{~K} \). Calculate the density as \( \frac{0.0821 \times 44}{0.0821 \times 400} \) \( \mathrm{g/L} \).

Step 4: Simplify the Expression

Cancel out \( 0.0821 \) and solve the fraction: \( \frac{44}{400} \).

Step 5: Final Calculation

Perform the division: \( 44/400 = 0.11 \mathrm{~g/L} \).

Key Concepts

Density Calculation

Density is a measure of how much mass is contained in a given volume. In scientific terms, it is usually expressed in grams per liter (g/L) for gases. To find the density of a gas, like carbon dioxide, under specific conditions of temperature and pressure, we apply the ideal gas law, which helps relate pressure, volume, temperature, and the amount of gas in moles. Here's how you can calculate the density of a gas:

• Use the ideal gas law formula: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.
• Density can be formulated as mass per unit volume: \( \text{density} = \frac{m}{V} \).
• Through the relationship \( n = \frac{m}{M} \) (where \( m \) is mass and \( M \) is molar mass of the gas), we can adjust the gas law to solve for density: \( \frac{m}{V} = \frac{PM}{RT} \).

By inputting the values given and performing the calculation, you ascertain the density of the gas under specific conditions, recognizing how these factors influence the result.

Carbon Dioxide Molar Mass

The molar mass of a compound is the weight of one mole of that compound. For carbon dioxide (\( \mathrm{CO}_2 \)), which is composed of one carbon atom and two oxygen atoms, the calculation of its molar mass is fairly straightforward. Let's break it down:

• Carbon has a molar mass of approximately \( 12 \mathrm{~g/mol} \).
• Oxygen has a molar mass of approximately \( 16 \mathrm{~g/mol} \).
• Thus, the molar mass of \( \mathrm{CO}_2 \) is calculated as: \( 12 \mathrm{~g/mol} + 2 \times 16 \mathrm{~g/mol} = 44 \mathrm{~g/mol} \).

Knowing the molar mass is crucial when using the ideal gas law to determine the density of carbon dioxide under varied conditions. It allows you to connect the mass of the gas to its volume using the gas law. This connection illustrates why molar mass is a key factor in density calculations.

Gas Laws

Gas laws are the simple guidelines that explain how gases should behave under various conditions of temperature, volume, and pressure. One of the most prominent laws is the Ideal Gas Law, expressed as \( PV = nRT \). This equation shows:

• \( P \) is the pressure of the gas.
• \( V \) is the volume it occupies.
• \( n \) is the number of moles of the gas.
• \( R \) is the universal gas constant (\( 0.0821 \mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} \)).
• \( T \) is the temperature in Kelvin.

These laws help us predict and calculate how a gas will react when subjected to changes in the aforementioned variables. Whether you are measuring pressure in an enclosed system or tracking volume changes, the ideal gas law and supporting principles from Boyle's, Charles's, and Avogadro's laws help inform these predictions and calculations.

Temperature and Pressure Effects on Gases

Understanding how temperature and pressure affect gases is vital for interpreting real-world scenarios. As general rules:

• Increasing the temperature of a gas often causes its volume to increase if pressure is constant, aligning with Charles's Law \( V \propto T \) (at constant \( P \)).
• Increasing pressure typically decreases a gas's volume when temperature remains constant, as Boyle’s Law highlights \( PV = k \) (at constant \( T \)).

Applying these principles combined with the Ideal Gas Law \( PV = nRT \) allows for predicting the behavior of gases. For instance, in our context—calculating the density of \( \mathrm{CO}_2 \)—temperature and pressure dictate the configuration of molecules in a given space. As the pressure is directly proportional to density and temperature inversely proportional, adjusting these factors will inherently shift the density outcome when calculated using the ideal gas components.