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Chapter 5: Problem 146

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 \[\left( P + \frac{a}{V^2} \right) (V - b) = RT\] simplifies to the ideal gas law due to the negligible contributions of the \( \frac{a}{V^2}\) and b terms. As a result, we have \(PV \approx RT\), which is the ideal gas law.

Step by step solution

01

Write down the van der Waals equation and the ideal gas law

The van der Waals equation for one mole of a gas is given by: \[ \left( P + \frac{a}{V^2} \right) (V - b) = RT \] where P is the pressure, V is the molar volume, T is the temperature, R is the universal gas constant, and a and b are van der Waals constants specific to a particular gas. The ideal gas law for one mole of a gas is given by: \[ PV = RT \]
02

Identify low pressure impact on van der Waals equation

Under low pressure conditions, the pressure term of the van der Waals equation, P, becomes very small compared to the \( \frac{a}{V^2} \) contribution. Hence, the effect of the \( \frac{a}{V^2} \) term becomes negligible. In this situation, we can approximate the van der Waals equation as follows: \[ P \approx - \frac{a}{V^2} \]
03

Identify high temperature impact on van der Waals equation

Under high temperature conditions, the molar volume of the gas (V) increases significantly, causing the attractive forces between gas particles to become insignificant. As a result, the impact of the b term becomes negligible in the van der Waals equation. Therefore, we can approximate the van der Waals equation as follows: \[ P(V - b) \approx PV \]
04

Combine low pressure and high temperature approximations to simplify van der Waals equation

Combining the results from Steps 2 and 3, we get: \[ PV \approx RT \]
05

Compare the simplified van der Waals equation with the ideal gas law

Considering the expression that we derived in Step 4: \[ PV \approx RT \] It is now clear that the van der Waals equation simplifies to the ideal gas law under low pressure and high temperature conditions, as the simplified equation matches the ideal gas law.

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Most popular questions from this chapter

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