Question

A real gas obeying van der Waals equation will resemble ideal gas, if the (a) constents a and \(b\) both are small (b) \(\boldsymbol{a}\) is arge and \(b\) is small (c) \(a\) is \(\mathrm{small}\) and \(b\) is large (d) constants \(a\) and \(b\) both are large.

Short answer

(a) constents a and b both are small

Step-by-step solution

Understand the van der Waals Equation

The van der Waals equation is an equation of state for a real gas that accounts for the volume occupied by gas molecules and the attractive forces between them. It is written as \( \left(P + \frac{a}{V_m^2}\right) \left(V_m - b\right) = RT \) 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 van der Waals constants dependent on the type of gas, reflecting the magnitude of intermolecular attraction forces and the volume excluded by the molecules respectively.

Conditions for a Real Gas to Resemble an Ideal Gas

For a real gas to behave like an ideal gas, the attractive forces between molecules and volume exclusion by molecules should become negligible. This implies that the constants \( a \) and \( b \) in the van der Waals equation should be very small. A small \( a \) value indicates weak intermolecular forces, analogous to none in an ideal gas, and a small \( b \) value indicates that the volume of molecules is insignificant relative to the volume of the container.

Analyse the Given Options

We need to determine which of the given options would lead to a real gas behaving as closely as possible to the ideal gas assumption. \( (a) \) Both \(a\) and \(b\) are small - this matches our requirement. \( (b) \) \(a\) is large and \(b\) is small - a large \(a\) value would mean significant intermolecular attractions, which is not ideal-like. \( (c) \) \(a\) is small and \(b\) is large - a large \(b\) means significant volume exclusion, also not ideal-like. \( (d) \) Both \(a\) and \(b\) are large - both non-ideal behaviors are significant in this case.

Key concepts

Real Gas Behavior

In contrast to ideal gases, real gases exhibit behavior that deviates from the laws of ideal gases due to two main factors: the finite size of the gas molecules and the forces of attraction and repulsion between them. Understanding real gas behavior is crucial for making accurate predictions in chemical engineering, material science, and thermodynamics.

When we apply the ideal gas law to real gases, we often observe significant discrepancies, especially under high pressure or low temperature conditions. The van der Waals equation addresses these discrepancies by introducing two correction factors. Real gases can behave more like an ideal gas under conditions where the molecules are so far apart that their size and intermolecular forces become negligible compared to the volume of their container. For instance, at very high temperatures or low pressures, the behavior of real gases tends to align more closely with ideal gas predictions.

Ideal Gas Approximation

The ideal gas approximation assumes that gas molecules are point particles with no volume and that there are no intermolecular forces between them. This simplification forms the basis of the Ideal Gas Law, expressed as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

To approach ideal gas behavior, the effects of the real gas molecules' volume and intermolecular forces must be minimized. This would typically occur when molecules are sufficiently separated so that these factors have a negligible impact on the gas's behavior. Hence, reducing the constants \( a \) and \( b \) in the van der Waals equation to small values brings a real gas closer to ideal behavior.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion between molecules. They influence the physical properties of substances, such as boiling points, melting points, and solubility. The van der Waals constants \( a \) and \( b \) in the van der Waals equation are a measure of these forces within a particular gas.

The \( a \) constant represents the magnitude of the attractive forces between gas molecules. A high value of \( a \) means that the molecules are significantly attracted to each other, which can cause deviations from ideal behavior as the gas may condense or have reduced pressure due to these attractive forces.

The \( b \) constant accounts for the volume of gas molecules themselves. A large \( b \) value indicates that the molecules have a considerable volume, affecting the gas's ability to compress and occupy space.

Understanding these forces allows us to predict how a gas will behave under different conditions, improving our ability to model processes involving gases in real-world applications.