Question

The van der Waals' equation of law of corresponding states for 1 mole of gas is : (a) \(\left(P_{r}+\frac{3}{V_{r}^{2}}\right)\left(3 V_{r}-1\right)=8 T_{r}\) (b) \(\left(P_{r}-\frac{3}{V_{r}^{2}}\right)\left(3 V_{r}-1\right)=8 T_{r}\) (c) \(\left(P_{r}+\frac{3}{V_{r}^{2}}\right)\left(3 V_{r}+1\right)=8 \pi T_{r}\) (d) \(\left(P_{r}+\frac{3}{V_{r}^{2}}\right)\left(3 V_{r}+1\right)=8\)

Short answer

The correct option is (a).

Step-by-step solution

Understanding the van der Waals' law of corresponding states

The law of corresponding states says that all gases at the same reduced temperature and pressure will have the same reduced volume. The van der Waals equation in terms of reduced properties (for one mole of gas) is commonly given by \( \left(P_r + \frac{3}{V_r^2}\right)(3V_r - 1) = 8T_r \) where \(P_r\), \(V_r\), and \(T_r\) are the reduced pressure, volume, and temperature, respectively.

Identify the correct form of equation

Among the given options, identify the one that matches the correct form of the van der Waals' law of corresponding states. Option (a) is the accurate representation of the law.

Key concepts

Physical Chemistry JEE

Physical Chemistry is a significant branch of chemistry that deals with the study of the physical properties and behavior of matter, particularly at the atomic and molecular levels. It is a subject that heavily relies on mathematics and physics to describe various chemical phenomena. For students preparing for competitive exams like the Joint Entrance Examination (JEE) in India, mastering the concepts of physical chemistry is essential.

Topics like thermodynamics, kinetics, and the study of intermolecular forces such as van der Waals' forces are integral to understanding chemical reactions and material properties. A critical aspect of JEE Physical Chemistry is the application of theoretical concepts to solve practical problems. Exercises like the van der Waals' equation and the law of corresponding states not only test students' understanding of the subject matter but also their ability to apply these concepts in various contexts.

Reduced Properties of Gases

The 'reduced properties' of gases refer to dimensionless quantities that help in comparing the behavior of different substances without the complication of using different units or scales. These properties are expressed in terms of a gas's critical temperature, volume, and pressure. The concept of reduced properties is pivotal in the understanding of the van der Waals' law of corresponding states.

Reduced properties are defined as follows:

• Reduced pressure (\( P_r \)) is the ratio of the pressure of the gas to its critical pressure (\( P_c \)).
• Reduced volume (\( V_r \)) is the ratio of the volume of the gas to its critical volume (\( V_c \)).
• Reduced temperature (\( T_r \)) is the ratio of the temperature of the gas to its critical temperature (\( T_c \)).

By using these reduced properties, the complex behaviors of gases under various conditions can be simplified, making it easier to analyze and predict the behavior of a substance in response to changes in temperature, pressure, or volume.

States of Matter

The states of matter describe how matter is organized based on the kinematics and interactions between particles. Traditionally, matter is observed in three fundamental states: solid, liquid, and gas, with plasma sometimes mentioned as a fourth. In gases, particles move rapidly and are spaced far apart, resulting in high compressibility and the ability to flow freely.

In the context of van der Waals' equation and the law of corresponding states, the focus is on the gaseous state. The van der Waals' equation modifies the ideal gas law to account for the non-ideal behavior observed in real gases, such as the forces between molecules and the finite volume they occupy. Understanding how real gases deviate from ideal behavior under different temperatures and pressures is key to mastering sections of physical chemistry relevant to the study of matter's states.