Problem 97 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A real gas obeying van der Waal's equation will resemble ideal gas if the constants (a) \(a\) and \(b\) are small (b) \(a\) is large and \(b\) is small (c) \(a\) is small and \(b\) is large (d) \(a\) and \(b\) are large

Step-by-Step Solution

Answer

A real gas will resemble an ideal gas if the constants 'a' and 'b' are small.

Step 1: Understand van der Waals Equation

The van der Waals equation for a real gas is given by \((P + \frac{a}{V_m^2})(V_m - b) = RT\), where P is the pressure, V_m is the molar volume, T is the temperature, R is the universal gas constant, and a and b are the van der Waals constants. The constants a and b correct the ideal gas equation for the forces between gas molecules and the volumes of the gas molecules, respectively.

Step 2: Analyze Conditions for Ideal Gas Behavior

For a real gas to behave like an ideal gas, the effects of intermolecular forces (represented by the constant a) and the volume of the molecules themselves (represented by the constant b) must be negligible.

Step 3: Determine the Required Conditions for Constants a and b

If both constants a and b are small, the corrections they apply to the ideal gas equation become insignificant, making the behavior of the real gas very close to that of an ideal gas.

Step 4: Choose the Correct Option

Therefore, a real gas will resemble an ideal gas if the constants a and b are small, which corresponds to option (a).

Key Concepts

Real Gas Behavior

The behavior of real gases often deviates from what is predicted by the ideal gas law, particularly under conditions of high pressure and low temperature. Real gases, unlike ideal gases, have intermolecular forces and their molecules occupy space.

At high pressures, the volume of the gas molecules becomes significant compared to the overall volume of gas. Furthermore, the proximity of molecules under these conditions increases the effect of intermolecular forces. At low temperatures, the attractive forces between particles are more pronounced, as the particles move slower and have more time to interact.

The van der Waals equation corrects for these differences by including two constants, known as van der Waals constants, that account for molecular volume and intermolecular forces. As pressure decreases and temperature increases, the behavior of real gases tends to align closer with the ideal gas approximation.

Ideal Gas Approximation

The ideal gas law, expressed by the equation PV = nRT, is a simplified model that describes the behavior of gases under most ordinary conditions. It assumes gases consist of point particles with no volume and no intermolecular forces.

This approximation works well when the real gas molecules are far apart and their size and interactions can be ignored. For many practical applications, particularly at standard temperature and pressure, the ideal gas law offers a sufficiently accurate description of a gas's behavior.

However, as scientists and engineers often work with gases under a wide range of conditions, the limitations of this model must be taken into account. This necessitates the use of more complex models, like the van der Waals equation, to predict gas behavior more accurately when the ideal gas law falls short.

van der Waals Constants

The van der Waals equation introduces two constants, denoted as 'a' and 'b', which are unique to each gas. These constants modify the ideal gas equation to account for real gas behavior.

The 'a' constant relates to the magnitude of the attractive forces between gas molecules. A larger 'a' value indicates stronger attractions that need to be considered when calculating pressure. On the other hand, 'b' represents the effective volume occupied by the gas molecules themselves. A larger 'b' value suggests larger molecules that take up more space.

By considering these constants in the van der Waals equation, we are able to estimate the behavior of a gas more accurately in conditions where the ideal gas law is insufficient. This highlights the significance of understanding the physical properties of the gas in question when applying the van der Waals equation to real-world problems.