Question

For nearly all real gases, the quantity \(P V / R T\) decreases below the value of 1, which characterizes an ideal gas, as pressure on the gas increases. At much higher pressures, however, \(P V / R T\) increases and rises above the value of 1 . (a) Explain the initial drop in value of \(P V / R T\) below 1 and the fact that it rises above 1 for still higher pressures. (b) The effects we have just noted are smaller for gases at higher temperature. Why is this so?

Short answer

To summarize, for real gases, the value of \(PV/RT\) initially drops below 1 due to intermolecular attractive forces at low pressures. However, at higher pressures, the finite size of gas molecules becomes significant and causes the value of \(PV/RT\) to rise above 1. At higher temperatures, these deviations from ideal gas behavior become smaller, as the increase in kinetic energy of gas molecules reduces the effectiveness of intermolecular attractive forces.

Step-by-step solution

The van der Waals equation is an empirical equation that provides a more accurate description of real gases, as it accounts for two key factors that the ideal gas law does not: the finite size of gas molecules, and the intermolecular forces (attractive forces between molecules): \[\left(P + a\frac{n^2}{V^2}\right)\left(V-nb\right) = nRT \] In this equation: - \(P\) is the pressure, - \(V\) is the volume, - \(a\frac{n^2}{V^2}\) accounts for attractive forces between molecules, - \(n\) is the amount of gas (in moles), - \(nb\) accounts for the finite volume occupied by the molecules, - \(R\) is the gas constant and - \(T\) is the temperature. #Step 2: Explain the initial drop in the value of PV/RT below 1#

This initial deviation in the value of \(PV/RT\) from 1 can be attributed to the intermolecular attractive forces between the gas molecules. As the pressure increases, the gas molecules are compressed closer together, leading to a stronger interaction between the molecules, which in turn attracts them towards each other. This attractive force reduces the effective pressure that would be expected in the case of ideal gases. Therefore, at relatively low pressures, this attractive force dominates, causing the value of \(PV/RT\) to drop below 1 for real gases. #Step 3: Explain the rise in the value of PV/RT above 1 at higher pressures#

At much higher pressures, the volume occupied by the gas molecules starts to become significant compared to the total volume of the gas. The ideal gas law does not account for this finite volume occupied by real gas molecules. The volume repulsive interactions (finite volume occupied by molecules) become more prevalent at higher pressures, leading to an increase in the effective pressure and causing the value of \(PV/RT\) to rise above 1 for real gases. #Step 4: Explain why the effects are smaller for gases at higher temperatures#

At higher temperatures, the average kinetic energy of the gas molecules increases, which means that the gas molecules are moving at faster speeds and the attractive forces between them become less effective. As a result, the deviation from ideal gas behavior becomes smaller, and the value of \(PV/RT\) approaches 1. It is important to note that both attractive forces and repulsive forces are temperature dependent, but the effect of an increase in temperature has a more significant impact on the attractive forces than on the repulsive forces.