Problem 37 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Calculate density of \(\mathrm{NH}_{3}\) at \(30{ }^{\circ} \mathrm{C}\) and 5 atm pressure.

Step-by-Step Solution

Answer

The density of NH₃ at 30°C and 5 atm is approximately 3.43 g/L.

Step 1: Write the Ideal Gas Law Equation

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

Step 2: Convert Temperature to Kelvin

The given temperature is \(30^{\circ} \mathrm{C}\). To convert this to Kelvin, use the formula \( T(K) = T(^{\circ}C) + 273.15 \). Thus, we have, \( 30 + 273.15 = 303.15 \ ext{K} \).

Step 3: Calculate Molar Volume using Ideal Gas Law

Since \( n = \frac{m}{M} \) and \( V = \frac{m}{\rho} \), substitute \( n = \frac{PV}{RT} \) to solve for \( V \): \[ V = \frac{nRT}{P} \]. Use \( R = 0.0821 \, L \, atm \, K^{-1} \, mol^{-1} \): \[ V = \frac{n \times 0.0821 \times 303.15}{5} \].

Step 4: Calculate Molecular Mass of \( \mathrm{NH}_{3} \)

The molar mass of \( \mathrm{NH}_{3} \) is calculated as follows: 1 Nitrogen = 14 g/mol, and 3 Hydrogen \( \times \ 1 \text{g/mol} = 3 \text{g/mol} \), so total molar mass = 14 + 3 = 17 \text{g/mol}.

Step 5: Calculate Density Using the Molar Volume

The density \( \rho \) is given by \( \rho = \frac{m}{V} \). Substituting \( m = M \times n \) gives \( \rho = \frac{P \times M}{R \times T} \). Thus, \( \rho = \frac{5 \times 17}{0.0821 \times 303.15} \approx 3.43 \text{g/L} \).

Key Concepts

Density Calculation

Density is a fundamental concept that measures the mass of a substance per unit volume. In simpler terms, it tells us how much "stuff" is packed into a given space. For gases, this is particularly important as it can vary with changes in pressure and temperature.
To calculate the density of a gas, we need to divide its mass by its volume. However, gas mass isn’t typically something we measure directly. Instead, we use the Ideal Gas Law to determine the volume and then calculate density by using the relation \( \rho = \frac{m}{V} \).

• Using the Pressure \(P\), Volume \(V\), and Temperature \(T\), we can solve for the volume that a given mass of gas will occupy.
• With Volume and Molar Mass (\(M\)), we can find the Density \(\rho\).

Knowing the density is useful in applications where the identification of the gas or prediction of its behavior under different conditions is needed.

Molar Mass Calculation

Molar Mass is the mass of one mole of a given substance and is typically expressed in grams per mole \((g/mol)\). For compounds, such as ammonia \((\mathrm{NH}_3)\), the molar mass is calculated by summing the atomic masses of all atoms in the molecule.

• Each element contributes to the overall mass according to the number of atoms present and their individual atomic masses.
• For \(\mathrm{NH}_3\), it consists of one nitrogen atom (14 \(g/mol\)) and three hydrogen atoms (3 \(g/mol\)), giving a total molar mass of 17 \(g/mol\).

This value is utilized in various calculations, such as determining how much of a compound is needed for a reaction or, as here, to find the density of a gas.

Gas Laws

Gas Laws describe the relationships between the pressure, volume, temperature, and amount of gas. They are critical in predicting how gases will behave in different conditions.
The Ideal Gas Law is a fundamental equation used, expressed as \(PV = nRT\), where:

• \(P\) is the pressure in atm
  
• \(V\) is the volume in liters
  
• \(n\) is the number of moles
  
• \(R\) is the ideal gas constant \(0.0821 \, L \, atm \, K^{-1} \, mol^{-1}\)
  
• \(T\) is the temperature in Kelvin
  

The Ideal Gas Law simplifies the approximation of real gases under various conditions, assuming ideal behavior: no interactions between gas molecules and occupying no volume themselves.
In real-life applications, adjustments might be needed, but for many conditions and calculations, this law provides a close enough approximation for practical use.