Problem 141 - A Complete Resource Book in Chemistry for JEE Main

Question

A gas at \(350 \mathrm{~K} 15 \mathrm{~atm}\) has a molar volume 12 percent smaller than that calculated from the perfect gas law. Compressibility factor under these conditions can be expressed in scientific notation as \(88 \times 10^{-x} .\) The value of \(x\) is

Step-by-Step Solution

Answer

The value of \(x\) is 2.

Step 1: Understanding the Perfect Gas Law

The perfect gas law is given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is the temperature in Kelvin.

Step 2: Calculate the Ideal Molar Volume

First, calculate the molar volume (\( V_m \)) using the perfect gas law. Since we are given \( P = 15 \) atm and \( T = 350 \) K:\[V_m = \frac{RT}{P} = \frac{0.0821 \times 350}{15} \approx 1.915 \text{ L/mol}.\]

Step 3: Determine the Real Molar Volume

The real molar volume is 12% less than the ideal molar volume. Calculate this by multiplying the ideal molar volume by 88%.\[V_{real} = 1.915 \times 0.88 \approx 1.686 \text{ L/mol}.\]

Step 4: Calculate the Compressibility Factor (Z)

The compressibility factor \( Z \) is defined as the ratio of the molar volume of the real gas to the molar volume of the ideal gas:\[Z = \frac{V_{real}}{V_{ideal}} = \frac{1.686}{1.915} \approx 0.880.\]

Step 5: Express Z in Scientific Notation

Convert \( Z \) to scientific notation as \( 88 \times 10^{-x} \). Recognizing that 0.880 can be expressed as \( 8.8 \times 10^{-1} \) and similarly \( 88 = 8.8 \times 10^{1} \), we place it against the format given, leading us to deduce that:\[Z = 8.8 \times 10^{-1} = 88 \times 10^{-x} \Rightarrow x = 2.\]

Key Concepts

Perfect Gas Law

The perfect gas law is a fundamental equation used to describe the relationship between the pressure, volume, temperature, and amount of gas. This relationship is encapsulated in the formula \( PV = nRT \), where:

• \( P \) stands for the pressure of the gas.
• \( V \) is the volume that the gas occupies.
• \( n \) represents the number of moles of gas.
• \( R \) is the ideal gas constant, typically valued at 0.0821 L·atm/mol·K.
• \( T \) is the temperature measured in Kelvin.

This formula assumes ideal conditions where gases don't experience any intermolecular forces, and the individual gas molecules occupy no volume themselves.

Gases behave closest to this ideal under high temperature and low pressure. Under these conditions, the deviation between an ideal gas and a real gas is minimal, allowing accurate assumptions using the perfect gas law.

Molar Volume

Molar volume refers to the volume occupied by one mole of a gas under specific conditions of temperature and pressure. According to the perfect gas law, the molar volume can be calculated using the formula \( V_m = \frac{RT}{P} \). Here:

• \( R \) is the ideal gas constant.
• \( T \) is the temperature in Kelvin.
• \( P \) signifies the pressure in atm (atmospheres).

For example, by substituting in the values \( R = 0.0821 \ \text{L·atm/mol·K}, T = 350 \ \text{K}, \) and \( P = 15 \ \text{atm}, \) we calculate the ideal molar volume as approximately 1.915 L/mol.

It's essential to distinguish between the ideal molar volume and real molar volume. Real gases exhibit behavior that causes deviations from ideal calculations, resulting in a real molar volume different from the theoretical ideal. In our scenario, the real molar volume is given as 12% smaller than the ideal, illustrating this deviation.

Scientific Notation

Scientific notation is a beneficial method used to express very large or very small numbers in a concise form. It breaks numbers down into a product of a base and a power of ten, such as \( a \times 10^n \).

• \( a \) is a number greater than or equal to 1 but less than 10.
• \( n \) represents the exponent, indicating the power of 10.

It's particularly useful in scientific fields where precise computation involves numerous significant digits.

In the context of compressibility factors, scientific notation simplifies complex expressions. For instance, expressing 0.880 as \( 8.8 \times 10^{-1} \) provides clarity and maintains accuracy. This format is easier to calculate with, especially when software or precise devices are involved.

Ideal Gas Constant

The ideal gas constant, denoted as \( R \), is a crucial part of the perfect gas law equation. It serves as a bridge linking the macroscopic properties of gases (such as pressure and volume) to their microscopic behavior (like molecule interactions).

Its value, given as 0.0821 L·atm/mol·K, is derived from experimental data, making it consistent for use across a variety of conditions and with different gases. This value allows calculations involving temperature (in Kelvin), pressure (in atmospheres), and volume (in liters) to give results in consistent and standard units.

• \( R \) facilitates calculation of the molar volume and other critical gas metrics.
• Its universality means it applies to any ideal gas scenario, simplifying problem-solving techniques in chemistry and physics.

Understanding the ideal gas constant helps in pinpointing the differences when dealing with real gas behaviors. It lays the groundwork for more advanced concepts like van der Waals equations, which address the discrepancies present in real gas conditions.