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Chapter 5: Problem 158

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
At low pressures and high temperatures, the van der Waals equation \[(P+a\frac{n^2}{V^2})(V-nb)=nRT\] simplifies to the ideal gas law. When pressure is low, the term \(a\frac{n^2}{V^2}\) becomes insignificant compared to the pressure P, leading to \((P)(V-nb) \approx nRT\). Furthermore, at high temperatures, the term 'nb' becomes small compared to the volume V, allowing us to approximate the equation as \((P)(V) \approx nRT\), which is the ideal gas law.

Step by step solution

01

Write down the van der Waals equation and ideal gas law

The van der Waals equation is given by: \[(P+a\frac{n^2}{V^2})(V-nb)=nRT\] Where P is the pressure of the gas, V is the volume, n is the number of moles of the gas, a and b are constants specific to the gas, R is the ideal gas constant, and T is the temperature. The ideal gas law is given by: \[PV=nRT\]
02

Show under which conditions van der Waals equation simplifies to the ideal gas law

We want to show that if we have low pressures and high temperatures, the van der Waals equation simplifies to the ideal gas law. At low pressures, the second term in the denominator, a(n^2)/(V^2), becomes insignificant compared to the pressure P. So, we can approximate the van der Waals equation as: \[(P)(V-nb) \approx nRT\] Now, as we have high temperatures, the term 'nb' in the equation above becomes small compared to the volume V. Therefore, we can approximate the equation as: \[(P)(V) \approx nRT\] And this is the ideal gas law equation, as desired. So, under low pressures and high temperatures, the van der Waals equation simplifies to the ideal gas law.

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Most popular questions from this chapter

Cyclopropane, a gas that when mixed with oxygen is used as a general anesthetic, is composed of \(85.7 \% \mathrm{C}\) and \(14.3 \% \mathrm{H}\) by mass. If the density of cyclopropane is \(1.88 \mathrm{~g} / \mathrm{L}\) at STP, what is the molecular formula of cyclopropane?

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