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 \[ \left(P + \frac{a}{V^2}\right)(V - b) = RT \] can be simplified to the ideal gas law. This is achieved by first dividing the terms in the equation by V, resulting in \[ \left(\frac{P}{V} + \frac{a}{V^3}\right)(V - b) = RT \]. Then, we assume that the low pressure makes the term \( \frac{P}{V} \) negligible compared to \( \frac{a}{V^3} \), and high temperatures make "b" negligible compared to V. Applying these assumptions, the equation becomes \[ \frac{a}{V^2} = RT \]. Finally, under low pressure, PV is approximately equal to \( \frac{a}{V^2} \), so the equation simplifies to the ideal gas law: \[ PV = RT \].

Step-by-step solution

Divide the terms in the equation by V

We will first divide both terms inside the van der Waals equation by the volume V: \[ \left(\frac{P + \frac{a}{V^2}}{V}\right)(V - b) = RT \] This simplifies to: \[ \left(\frac{P}{V} + \frac{a}{V^3}\right)(V - b) = RT \] This rearranged equation will be helpful for our upcoming identified assumptions.

Consider low pressure and high temperature conditions

Low pressure and high temperature conditions lead to the following assumptions: 1. When pressure is low, the term \( \frac{P}{V} \) becomes negligible compared to the term \( \frac{a}{V^3} \). 2. Similarly, high temperatures cause the volume to be quite large, so "b" becomes negligible compared to V. Let's apply these assumptions to our simplified equation.

Apply the assumptions to the equation

Using the previous assumptions, our equation simplifies to: \[ \left(0 + \frac{a}{V^3}\right)(V - 0) = RT \] This simplifies further to: \[ \frac{a}{V^2} = RT \]

Compare this to the ideal gas law

From our assumption that pressure is low, we can approximate PV to be approximately equal to \( \frac{a}{V^2} \). Thus our equation under low pressure and high temperature conditions becomes: \[ PV = RT \] We have now demonstrated that, under these conditions, the van der Waals equation simplifies to the ideal gas law.