Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(Z\) is a compressibility factor, van der Waals equation at low pressure can be written as: (a) \(\mathrm{Z}=1-\frac{\mathrm{Pb}}{\mathrm{RT}}\) (b) \(Z=1+\frac{P b}{R T}\) (c) \(\mathrm{Z}=1+\frac{\mathrm{RT}}{\mathrm{Pb}}\) (d) \(Z=1+\frac{a}{V R T}\)

Step-by-Step Solution

Answer

The correct answer is (b): \(Z=1+\frac{P b}{R T}\).

Step 1: Understand the Van der Waals Equation

The van der Waals equation is used to describe the behavior of real gases: \[ \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT \] where \(P\) is the pressure, \(V_m\) is the molar volume, \(R\) is the gas constant, \(T\) is the temperature, and \(a\) and \(b\) are van der Waals constants.

Step 2: Simplify for Low Pressure

At low pressure, the equation simplifies because the volume is very large, meaning \(b\) is negligible compared to \(V_m\) and the term \( \frac{a}{V_m^2} \) becomes very small compared to \(P\). Hence, the equation reduces to: \[ PV_m = RT \] which leads the compressibility factor \(Z\) to be: \( Z = \frac{PV}{RT} \).

Step 3: Derive the Compressibility Factor Z

For a real gas at low pressure, the van der Waals equation is: \[ Z = \frac{PV_m}{RT} = 1 + \frac{Pb}{RT} \] assuming the term \( \frac{a}{V_m^2} \) is negligible. This equation shows how the volume correction \(b\) slightly modifies the ideal gas behavior.

Step 4: Match with Given Options

Our derived equation \( Z = 1 + \frac{Pb}{RT} \) matches with option (b). The volume correction \(b\) shows a positive deviation from ideal behavior, which confirms the correct answer is option (b).

Key Concepts

Compressibility Factor

The compressibility factor, denoted by the letter \( Z \), is a useful dimensionless quantity in the study of gases. It helps us understand how a real gas deviates from the ideal gas behavior. For an ideal gas, \( Z \) is equal to 1.Real gases, however, often do not behave in this ideal manner due to intermolecular forces and finite molecular volume. Therefore, the compressibility factor is expressed as:
\[ Z = \frac{PV}{RT} \]
- **P** represents pressure.- **V** is molar volume.- **R** stands for the ideal gas constant.- **T** is temperature.When \( Z > 1 \), it indicates the effects of molecular volume are dominant, meaning the molecules occupy significant space. Meanwhile, if \( Z < 1 \), attractive forces between molecules are stronger, causing the gas to occupy less volume than predicted by the ideal gas law.

• So, understanding \( Z \) helps chemists and scientists refine gas predictions beyond the simplicity of the ideal model.

Real Gases

Unlike ideal gases, real gases have volume and experience intermolecular forces. These differences become noticeable under conditions of high pressure or low temperature. The van der Waals equation is often used to model these real gas behaviors. This equation incorporates two essential corrections:

• **Volume Correction (b):** Acknowledges that gas particles themselves take up space.
• **Pressure Correction (a):** Accounts for the intermolecular forces that reduce the actual pressure exerted by the gas.

At high pressures, real gases deviate from ideal behavior much more. The molecules are compressed into a smaller volume, and intermolecular forces become significant. At low temperatures, similar deviations occur because the reduced kinetic energy increases the effect of attractions between molecules. Thus, understanding real gases and their deviations from ideal conditions help in modeling and predicting the behavior of gases under various conditions.

Ideal Gas Behavior

In an ideal gas, the molecules do not interact with each other except for elastic collisions, and they occupy no volume. The ideal gas law, expressed as \( PV = nRT \), provides a simple linear relationship between the pressure (P), volume (V), and temperature (T) of a gas, with 'n' being the number of moles.
This law is based on assumptions:

• The gas consists of many small particles placed far apart relative to their size.
• There are no attractive or repulsive forces between the particles.
• The collisions between particles and with container walls are completely elastic.

The simplicity of this model makes ideal gases a useful approximation under many conditions; however, real gases often require more complex equations, like the van der Waals equation, to accurately describe their behavior.In natural settings, gases rarely meet all ideal conditions. Yet, the concept of ideal gas behavior lays an important foundational understanding, serving as a benchmark against which real gases can be studied.