Question

The table below lists the density of substance \(\mathrm{X}\) at various temperatures and at \(101.3 \mathrm{kPa}\). The normal melting point of substance \(X\) is \(80 \mathrm{~K}\). \begin{tabular}{cc} \hline Temperature \((\mathrm{K})\) & Density \((\mathrm{mol} / \mathrm{L})\) \\ \hline 90 & 35.2 \\ 100 & 33.8 \\ 110 & 32.1 \\ 120 & 0.136 \\ 140 & 0.110 \\ 160 & 0.0893 \\ 180 & 0.0796 \\ \hline \end{tabular} (a) Over what temperature range is substance \(X\) a solid? (b) Over what temperature range is \(\mathrm{X}\) a liquid? \((\mathbf{c})\) Over what temperature range in the table is \(\mathrm{X}\) a gas? (d) Estimate the normal boiling point of \(X .\) (e) Given that \(X\) is a nonpolar molecule, suggest the kind of intermolecular forces in \(\mathrm{X}\).

Short answer

(a) Not applicable. (b) 90 K to 120 K. (c) 120 K and above. (d) Just above 110 K. (e) London dispersion forces.

Step-by-step solution

Determine the Solid Phase

The solid phase exists below the melting point, which is given as 80 K. Hence, since the table data starts from 90 K, substance X is a solid at any temperature listed below its melting point. But for the table range, there is no solid phase data because all data points are above 80 K.

Identify the Liquid Phase

The substance changes from solid to liquid at its melting point, 80 K. By analyzing the density, the liquid phase appears between densities higher than typical gaseous densities and lower than typical solid densities (90 K to around 120 K).

Determine the Gas Phase

The substance behaves as a gas at very low densities; therefore, checking the table, below 0.136 mol/L indicates gaseous state, corresponding to temperatures 120 K and above.

Estimate the Normal Boiling Point

The density drops significantly between 110 K to 120 K, indicating phase transition from liquid to gas. The point where density drastically reduces (from high to low between 120 K to 110 K) marks the boiling point, just above 110 K.

Identify Intermolecular Forces

Nonpolar molecules typically exhibit weak intermolecular forces known as London dispersion forces (van der Waals forces). Because X is nonpolar, its intermolecular forces are likely to be London dispersion forces.

Key concepts

Density and Phase Changes

When looking at a substance like X, the density provides clues about its phase at certain temperatures. Density is essentially how much of substance X is packed into a given volume.
For a solid, the molecules are tightly packed, resulting in a higher density. As the temperature heats up and approaches the melting point, the molecules begin to loosen up and transform into a liquid, decreasing the density. This trend continues as a liquid turns into a gas, where molecules are far apart, exhibiting very low density.

From the temperature range given in the table, densities at 90 K and 100 K indicate that substance X is in a solid state because the density values are significantly higher compared to gaseous substances. As temperature rises to about 120 K and beyond, the dramatic drop in density suggests a phase transition indicating a gaseous state because gases have much lower density as molecules spread out into the available space.

Intermolecular Forces

Intermolecular forces are the forces that hold molecules together. They are essential in determining a material's phase and properties.

• In nonpolar molecules like substance X, the primary forces in play are London dispersion forces.
• These forces are the weakest of the intermolecular forces and arise due to temporary shifts in electron density, creating instantaneous dipoles.
• This results in fleeting attractions between molecules.

The behavior of substance X in different phases is indicative of the strength of these forces. In the solid phase, molecules are closely packed due to stronger interactions.
However, as phase changes occur, they disperse more in response to weaker interactions, especially evident as the substance transitions into a gaseous state.

Melting and Boiling Points

Melting and boiling points are critical phase transition temperatures.
The melting point is when a solid becomes a liquid. For substance X, this is at 80 K, marking the shift from ordered, fixed-position molecules to more fluid forms that move past one another.

The boiling point is when a liquid turns into a gas, another significant phase change showing when liquid molecules escape into the air.
For substance X, the transition observed in density from 110 K to 120 K suggests its boiling point is just above 110 K. This indicates when X's molecules have enough energy to overcome intermolecular forces and spread widely, becoming a gas.
Understanding these points helps in predicting the conditions under which a substance changes states, around which physical properties dramatically shift.