Problem 128 - A Complete Resource Book in Chemistry for JEE Main

Question

A gas, which obeys Boyle's law, Charle's law, Avogadro's law etc., or ideal gas equation \(\mathrm{PV}=\mathrm{nRT}\) under all conditions of temperature and pressure, is called ideal gas. No gas is ideal. All gases are real gases. The real gas obeys these gas laws only when the temperature is high or pressure is low. The extent of derivations of a real gas form ideal behaviour is expressed in terms of compressibility factor \(Z\) defined as \(\mathrm{Z}=\frac{\mathrm{PV}}{\mathrm{nRT}}\) Real gases have characteristic temperatures like critical temperature, inversion temperature and Boyle temperature. These temperatures can be calculated using van der Waal constants. Density of two gases of same molecular weight are in the ratio \(1: 3\) and their temperatures are in the ratio \(3: 2 .\) The ratio of respective pressures is (a) \(2: 1\) (b) \(2: 3\) (c) \(3: 2\) (d) \(1: 2\)

Step-by-Step Solution

Answer

The ratio of the pressures is \( \frac{3}{2} \) or option (c) \(3: 2\).

Step 1: Understand the Relationships

Boyle's law is generally given as \( P \propto \frac{1}{V} \) for constant temperature, while Charle's law states \( V \propto T \) for constant pressure. Avogadro's law is given as \( V \propto n \). Together, these relate to the ideal gas law: \( PV = nRT \). Real gases, deviating from this, have a compressibility factor \( Z = \frac{PV}{nRT} \). Here, we are given the density ratios and temperature ratios of gases, requiring analysis of pressure ratios.

Step 2: Density and Molar Mass Relationship

Density (\( \rho \)) of a gas is given by \( \rho = \frac{m}{V} \). For ideal gases, it relates to molar mass (\( M \)), pressure (\( P \)), and temperature (\( T \)) as \( \rho = \frac{PM}{RT} \). The problem gives us \( \rho_1 : \rho_2 = 1 : 3 \) and \( T_1 : T_2 = 3 : 2 \). We aim to find \( P_1 : P_2 \).

Step 3: Express Densities in Terms of Pressures and Temperatures

From \( \rho = \frac{PM}{RT} \), deduce \( \rho_1 = \frac{P_1M}{RT_1} \) and \( \rho_2 = \frac{P_2M}{RT_2} \). The densities give us \( \frac{\rho_1}{\rho_2} = \frac{P_1/T_1}{P_2/T_2} \). Plug in the density ratio, \( \frac{1}{3} = \frac{P_1 \cdot T_2}{P_2 \cdot T_1} \).

Step 4: Insert Coefficients and Equate

Substitute \( T_1 = 3x \) and \( T_2 = 2x \) (assuming \( x \) is a scaling factor for clarity), into the equation: \( \frac{1}{3} = \frac{P_1 \times 2x}{P_2 \times 3x} \). The \( x\) factors cancel, simplifying to \( \frac{1}{3} = \frac{2P_1}{3P_2} \).

Step 5: Solve for Pressure Ratio

Cross-multiply the equation \( \frac{1}{3} = \frac{2P_1}{3P_2} \) to solve for \( P_1 : P_2 \). This gives \( 3 = 2 \frac{P_1}{P_2} \), leading to \( \frac{P_1}{P_2} = \frac{3}{2} \).

Key Concepts

Boyle's Law

Boyle's Law is one of the foundational gas laws that describes how the pressure of a gas varies inversely with its volume when temperature is held constant. In simple terms, if you decrease the volume of a gas, its pressure increases if the temperature does not change, and vice versa. This principle is mathematically represented as: \[ P \propto \frac{1}{V} \]where:

• P is the pressure of the gas
• V is the volume of the gas

The product of pressure and volume remains constant in this law, provided the temperature doesn’t change. In the context of real gases, Boyle's law holds ideally at low pressures because molecules are far apart, and their interactions are negligible. However, as pressure increases, real gases may start deviating from this behavior due to intermolecular forces. The ideal gas behavior serves as a useful approximation under many conditions but remember, it's not applicable in all situations.

Charles's Law

Charles's Law describes how gases tend to expand when heated. It establishes a direct relationship between the volume and temperature of a gas at constant pressure. This relationship can be expressed using the equation: \[ V \propto T \]Where:

• V refers to the volume of the gas
• T represents the temperature in Kelvin

Key Takeaway: When you increase the temperature of a gas, its volume increases, provided the pressure remains unchanged. This happens because heating the gas causes the molecules to move faster, requiring more space. In the real world, you notice that this law works perfectly for ideal gases. For real gases, deviations can occur at very high temperatures or pressures where molecular size and interactions can no longer be ignored. Despite these slight deviations, Charles's Law is instrumental in understanding how temperature influences gas volume in practical applications.

Avogadro's Law

Avogadro's Law is an essential principle that relates the volume of a gas to the amount of substance, usually measured in moles. It tells us that equal volumes of all gases, at the same temperature and pressure, contain an equal number of molecules. This can be expressed through the relation:\[ V \propto n \]where:

• V is the volume of the gas
• n is the number of moles of the gas

Avogadro's Law is crucial because it helps us understand that gas volume is directly proportional to the number of gas moles when temperature and pressure constants are maintained. It simplifies to the ideal gas equation when combined with Boyle and Charles's laws. This is particularly useful in chemical reactions involving gases as it allows stoichiometric calculations. For real gases, molecular interactions at very high pressures or low temperatures might prevent strict adherence to this law. Nevertheless, Avogadro's Law remains a cornerstone for understanding gaseous behavior in chemistry.

Real Gases

Real gases differ from ideal gases in how they behave at various temperatures and pressures due to intermolecular forces and the finite volume of gas molecules. While the ideal gas law, \( PV = nRT \), assumes no forces or volume, real gases sometimes exhibit deviations. These deviations become significant under conditions of high pressure or low temperature.For real gases, two major factors come into play: intermolecular attractions and repulsions between molecules. This means that at higher pressures, molecules have less space to move freely, and their volume cannot be ignored.Real gases can be better described using equations such as the Van der Waals equation, which accounts for these molecular factors:\[ \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT \]Understanding real gases is essential in industries and science for precise calculations and processes, such as in the design of chemical reactors and industrial gas pipelines.

Compressibility Factor

The Compressibility Factor, often denoted as Z, is a number that indicates how much a real gas deviates from ideal gas behavior. It helps to express how the equations for ideal gases (like the ideal gas law) need to be adjusted for real gases. The compressibility factor is given by:\[ Z = \frac{PV}{nRT} \]Where:

• P is the pressure of the gas
• V is the volume of the gas
• n is the number of moles
• R is the universal gas constant
• T is the temperature in Kelvin

Z = 1 for an ideal gas under all conditions. For real gases:

• If Z < 1, this indicates that the gas molecules are attracting each other more than expected.
• If Z > 1, this means that the gas molecules are repelling each other more than expected.

Understanding the compressibility factor is crucial for accurately modeling gaseous behaviors in scientific and engineering applications where precision is paramount.