Question

For each of the following pairs, choose the substance with the higher entropy per mole at a given temperature: (a) \(\operatorname{Ar}(l)\) or \(\operatorname{Ar}(g)\), (b) \(\mathrm{He}(g)\) at 3 atm pressure or \(\mathrm{He}(\mathrm{g})\) at \(1.5\) atm pressure, (c) \(1 \mathrm{~mol}\) of \(\mathrm{Ne}(g)\) in \(15.0 \mathrm{~L}\) or mol of \(\mathrm{Ne}(\mathrm{g})\) in \(1.50 \mathrm{~L}_{,}(\mathrm{d}) \mathrm{CO}_{2}(g)\) or \(\mathrm{CO}_{2}(\mathrm{~s})\)

Short answer

The substances with higher entropy per mole at a given temperature for each pair are: (a) Ar(g) (b) He(g) at 1.5 atm pressure (c) 1 mol Ne(g) in 15.0 L (d) CO2(g)

Step-by-step solution

(For Pair A)

We are given two substances, one in the liquid phase (Ar(l)) and the other in the gaseous phase (Ar(g)). Gaseous substances typically have higher entropies than liquid or solid substances due to their increased freedom of motion. Therefore, at the same temperature, Ar(g) will have a higher entropy per mole than Ar(l).

(For Pair B)

We are given Helium in gaseous form at two different pressures: 3 atm and 1.5 atm. At constant temperature, entropy decreases as pressure increases since molecules are under higher compression, and thus the freedom of motion decreases. Therefore, He(g) at 1.5 atm pressure has a higher entropy per mole than He(g) at 3 atm pressure.

(For Pair C)

We are given Neon in gaseous form in two different containers: 1 mol Ne(g) in 15.0 L and 1 mol Ne(g) in 1.50 L. Entropy of a gas will generally increase as the volume increases since the gas molecules have more space to occupy, allowing for more randomness. Therefore, 1 mol of Ne(g) in 15.0 L has a higher entropy per mole than 1 mol of Ne(g) in 1.50 L.

(For Pair D)

We are given CO2 in the gaseous form (CO2(g)) and in the solid form (CO2(s)). Similar to Pair A, gaseous substances typically have higher entropy than solid or liquid substances due to the increased freedom of motion. Therefore, CO2(g) has a higher entropy per mole than CO2(s).

Key concepts

Entropy

Entropy is a concept from thermodynamics that refers to the degree of disorder or randomness in a system. It is a measure of the number of possible microstates in a given macrostate. In simpler terms, higher entropy means more disorder or randomness in a system.
When considering entropy, remember that substances in different phases (solid, liquid, gas) have different entropy values:

• Gases typically have higher entropy than liquids.
• Liquids generally have higher entropy than solids.
• The greater the number of molecules, the higher the entropy.

Overall, entropy is driven by the number and freedom of movement of molecules. This concept is fundamental to understanding how energy is distributed in physical systems.

States of Matter

The states of matter greatly influence entropy. Matter exists in three main states: solid, liquid, and gas. The degree of entropy changes significantly between these states due to molecular arrangement and movement.
In solids, molecules are closely packed and vibrate in place, resulting in lower entropy levels. Liquids have more space between molecules, allowing them to move more freely, thus having higher entropy than solids but lower than gases.
Gases have the highest entropy because molecules are widely spaced and move independently, providing a high degree of randomness and freedom. This distinction in molecular behavior across different states explains why entropy increases as a substance moves from solid to liquid to gas.

Gas Pressure Impact on Entropy

The pressure of a gas affects its entropy due to changes in molecular freedom of motion. At lower pressures, gas molecules are less compressed and can move more freely. This increased freedom results in higher entropy.
For example, consider helium gas at different pressures. At 1.5 atm, helium molecules have more space and freedom than at 3 atm, resulting in higher entropy at the lower pressure. Generally, keeping temperature constant, decreasing pressure increases entropy.
Understanding this relationship between pressure and entropy is crucial when analyzing gases' behavior in different conditions. It highlights the importance of environmental conditions on the entropy of a gas.

Volume Effect on Entropy

The volume that a gas occupies directly impacts its entropy. As the volume increases, entropy typically increases as well. This occurs because there is more space available for molecules to move, leading to greater disorder and randomness.
Consider one mole of neon gas in volumes of 15 L and 1.5 L. The larger 15 L container allows neon molecules more room to disperse, resulting in higher entropy compared to the same number of molecules in the smaller 1.5 L volume.
This relationship is fundamental in thermodynamics, showing how expanding or compressing a gas affects its disorder. By understanding volume's effect on entropy, students can better predict and assess changes in system behavior.