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Chapter 5: Problem 166

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

Expert verified
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

01

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
02

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: \[P(V - nb) = nRT\]
03

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: \[PV = nRT\]
04

Conclusion

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

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