Problem 38 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The value of \(P V\) for \(5.6\) litres of an ideal gas is \(\ldots \ldots \ldots \ldots \ldots . R T\), at N.T.P.

Step-by-Step Solution

Answer

The value of \( PV \) is \( 0.25RT \).

Step 1: Understand N.T.P. Conditions

N.T.P. stands for Normal Temperature and Pressure, which is defined as a temperature of 0°C (or 273.15 K) and a pressure of 1 atm. These are the conditions for which standard gas calculations are made.

Step 2: Use the Ideal Gas Law

The ideal gas law is expressed as \( PV = nRT \), 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 in Kelvin.

Step 3: Calculate Number of Moles

To find \( n \), the number of moles, use the given volume of the gas. At N.T.P., 1 mole of an ideal gas occupies 22.4 litres. Therefore, the number of moles \( n \) is \( \frac{5.6}{22.4} = 0.25 \) moles.

Step 4: Substitute Values into the Ideal Gas Law

With \( n = 0.25 \), \( R = 0.0821 \text{ atm} \cdot \text{L} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \), and \( T = 273.15 \text{ K} \), substitute these values into the equation \( PV = nRT \):\[ PV = 0.25 \times 0.0821 \times 273.15 \].

Step 5: Solve for PV

Calculate \( PV \) by performing the multiplication: \( 0.25 \times 0.0821 \times 273.15 = 5.604 \) atm⋅L.

Step 6: Convert to Terms of R and T

Since the expression of \( PV \) in terms of \( RT \) is needed, write \( 5.604 \text{ atm} \cdot \text{L}\) in the form of \( xRT \), where \( x = n \). As calculated, \( n = 0.25 \).

Step 7: Conclusion: Express PV in Terms of R and T

Thus, \( PV = 0.25RT \).

Key Concepts

Normal Temperature and Pressure (NTP)

Normal Temperature and Pressure, often abbreviated as NTP, is a standard set of conditions for scientific measurements involving gases. This benchmark aids scientists in comparing various gas properties without external variables interfering. NTP is defined by two main parameters: a temperature of 0°C, which is equivalent to 273.15 Kelvin, and a pressure of 1 atmosphere (atm). These are the baseline conditions under which many calculations in chemistry, particularly involving gases, are simplified and standardized.

Understanding NTP is crucial for a few reasons:

• It provides a common ground for scientists worldwide to conduct consistent experiments.
• It simplifies the calculation of gas properties such as volume, pressure, and temperature when using the ideal gas law.

Being familiar with these conditions, especially when dealing with ideal gas calculations, helps ensure accurate and consistent results. By considering NTP as the starting point, adjustments can be made easily for any changes in the gas conditions during experiments or real-life applications.

Universal Gas Constant

The universal gas constant, symbolized as \( R \), is a fundamental component of the ideal gas law. In this context, \( R \) serves as a bridge connecting various properties of gases, such as pressure, volume, temperature, and moles. Its value is approximately 0.0821 atm·L/(mol·K), making it essential for calculations involving gases.

There are several key points to understand about the universal gas constant:

• It provides a consistent way to relate the typical conditions of gases to their physical behaviors.
• \( R \) appears in the formula \( PV = nRT \) which binds the four primary characteristics of gases together.
• Despite different units such as energy per temperature and mol (J/mol·K) in more advanced calculations, the essence of \( R \) remains the same across various scientific disciplines.

Whether you're calculating pressure changes, finding unknown temperatures, or predicting volume changes, \( R \) comes into play as the common factor that upholds the relationships between these properties in gases.

Moles Calculation

Calculating moles, symbolized as \( n \), is at the heart of the ideal gas law applications, being directly proportional to the volume of gas at a given set of conditions. Specifically, at NTP, 1 mole of an ideal gas occupies 22.4 litres. This conversion factor simplifies the process of determining the number of moles present in any given volume under NTP.

Here's how moles calculation works step by step:

• First, determine the volume of gas you have. For example, let's consider 5.6 litres as in the exercise.
• Use the conversion factor at NTP: 1 mole of gas = 22.4 litres.
• Simply divide the given volume by the volume per mole: \( n = \frac{5.6}{22.4} = 0.25 \) moles.

This simple division provides the number of moles of gas in your sample. Having this value is critical when using the ideal gas law, as it allows for solving for other variables depending on what you need to find or predict in your experiment or calculation.