Question

We state that the ideal gas law tends to hold best at low pressures and high temperatures. Show how the van der Waals equation simplifies to the ideal gas law under these conditions

Short answer

Under low-pressure and high-temperature conditions, the van der Waals equation \((P + a \frac{n^2}{V^2})(V - nb) = nRT\) simplifies to the ideal gas law. This is because, at low pressures, the term \(a \frac{n^2}{V^2}\) becomes negligible, and at high temperatures, the volume is much larger than the term \(nb\), allowing the equation to be simplified to \(PV = nRT\).

Step-by-step solution

Write down the van der Waals equation

The van der Waals equation is given by: \[(P + a \frac{n^2}{V^2})(V - nb) = nRT\] where - P is the pressure - V is the volume - T is the temperature - n is the number of moles - R is the gas constant - a and b are the van der Waals constants

Consider low-pressure conditions

Under low-pressure conditions, we can assume that the pressure is so small that the term \(a \frac{n^2}{V^2}\) has negligible contribution to the total pressure. Hence, the van der Waals equation can be simplified as: \[P(V - nb) = nRT\]

Consider high-temperature conditions

Under high-temperature conditions, it can be assumed that the volume is much larger than the term \(nb\). Thus, the difference between V and nb is almost equal to V itself, leading to the simplification: \[PV = nRT\]

Conclusion

Under the conditions of low pressure and high temperature, the van der Waals equation simplifies to the ideal gas law: \[PV = nRT\].

Key concepts

Van der Waals Equation

The van der Waals equation is an essential enhancement of the ideal gas law. It corrects for the observed behavior of real gases by incorporating intermolecular forces and molecular sizes. This equation is expressed as: \[(P + a \frac{n^2}{V^2})(V - nb) = nRT\] In this equation, a and b are constants that are specific to each gas.

• The term \(a \frac{n^2}{V^2}\) accounts for the attractive forces between gas particles, which reduce the pressure compared to an ideal gas scenario.
• Similarly, \(nb\) corrects for the finite volume occupied by gas particles, providing a more accurate measure of the "available" volume (\(V - nb\)).

These adjustments allow the van der Waals equation to better match the behavior of real gases, particularly at various pressures and temperatures. Understanding how these terms affect the equation helps in the transition to the ideal gas law under specific conditions.

Low Pressure

At low pressures, the assumptions leading to the ideal gas behavior become more applicable. In such conditions, the particle density inside the gas container is reduced, making interactions between molecules less frequent. This means the effect of the intermolecular forces (represented by the term \(a \frac{n^2}{V^2}\) in the van der Waals equation) becomes negligible.
When pressure is low, the van der Waals equation simplifies significantly. The attractive forces between molecules don't play a big role anymore. This allows us to disregard the correction term \(a \frac{n^2}{V^2}\). Thus, the equation becomes closer to the ideal gas law format:
\[ P(V - nb) = nRT \]
At these conditions, gas particles behave more like ideal gas particles, interacting minimally and maximizing adherence to the ideal gas law. This transformation illustrates why low pressure contributes to ideal behavior.

High Temperature

High temperature plays a significant role in making gases behave ideally. When temperatures are high, gas particles have increased kinetic energy, leading them to move rapidly and be spaced further apart. This high energy overcomes the effect of intermolecular attraction, further devaluing the importance of the \(a \frac{n^2}{V^2}\) term.
Additionally, at elevated temperatures, the volume taken by the gas molecules, expressed as \(nb\), is dwarfed by the actual volume \(V\) of the gas. Hence, \(V - nb\) approximately equals \(V\). This simplification leads us closer to the realm of ideal gas scenarios. Therefore, our equation becomes even more straightforward:
\[ PV = nRT \]
In summary, high temperatures provide conditions that support the assumptions of negligible intermolecular forces and minimal molecular volume, perfectly aligning with the ideal gas law and ensuring gases behave in an ideal manner.