Chapter 18, Problem 1PS

Which substance has the higher entropy?

1. (a) dry ice (solid CO₂) at −78 °C or CO₂(g) at 0 °C
2. (b) liquid water at 25 °C or liquid water at 50 °C
3. (c) pure alumina, Al₂O₃(s), or ruby (ruby is Al₂O₃ in which some Al³⁺ ions in the crystalline lattice are replaced with Cr³⁺ ions)
4. (d) one mole of N₂(g) at 1 bar pressure or one mole of N₂(g) at 10 bar pressure (both at 298 K)

Interpretation Introduction

Interpretation: From the given substances that is dry ice at ${\text{-78}}^{\text{o}}\text{C}$ or $C{O}_2$ at ${\text{0}}^{\text{o}}C$ the substance with higher entropy should be identified.

Concept introduction: Entropy is a measure of the randomness of the system. It is a thermodynamic quantity and an extensive property. It is represented by the symbol $\text{S}$. It can also be defined as the degree of energy dispersal. More the dispersal in energy, more value of entropy.

Answer to Problem 1PS

The substance which has the highest entropy is ${\text{CO}}_{\text{2}}\left(\text{g}\right)$ at ${\text{0 }}^{°}\text{C}$.

Explanation of Solution

Entropy increases in the order $\text{solid < liquid < gas}$.  In the gaseous state, the particles are more random. In solids, the particles have fixed positions and entropy is less. In liquids, there are constraints due to forces between the particles.

Hence, entropy is more for ${\text{CO}}_{\text{2}}\left(\text{g}\right)$ at $\text{0 }°\text{C}$ than for solid ${\text{CO}}_{\text{2}}$ at $\text{-78 }°\text{C}$ since ${\text{CO}}_{\text{2}}\left(\text{g}\right)$ is in gaseous state.

Interpretation Introduction

Interpretation: From the given substances that is liquid water at ${\text{25}}^{\text{o}}\text{C}$ or liquid water at ${\text{50}}^{\text{o}}\text{C}$ the substance with higher entropy should be identified.

Concept introduction: Entropy is a measure of the randomness of the system. It is a thermodynamic quantity and an extensive property. It is represented by the symbol $\text{S}$. It can also be defined as the degree of energy dispersal. More the dispersal in energy, more value of entropy.

Answer to Problem 1PS

The substance which has the highest entropy is liquid water at ${\text{50 }}^{°}\text{C}$.

Explanation of Solution

The entropy of a substance increase when the temperature is increased. At higher temperature, the substance is more disordered and thus the entropy is high. Also, when the temperature raised the additional energy added in the form of heat.

Therefore, liquid water has higher entropy at $\text{50 }°\text{C}$ than at $\text{25 }°\text{C}$.

Interpretation Introduction

Interpretation: From the given substances that is pure alumina and ruby the substance with higher entropy should be identified.

Concept introduction: Entropy is a measure of the randomness of the system. It is a thermodynamic quantity and an extensive property. It is represented by the symbol $\text{S}$. It can also be defined as the degree of energy dispersal. More the dispersal in energy, more value of entropy.

Answer to Problem 1PS

The substance which has the highest entropy is Ruby.

Explanation of Solution

${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ is pure alumina. The entropy of a pure crystal is zero. Ruby is ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$ doped with chromium ions. Some of the ${\text{Al}}^{\text{3+}}$ ions are replaced by ${\text{Cr}}^{\text{3+}}$ due to which the disorderness of the crystal increases and the entropy increases.

Thus, entropy is more for ruby than for pure ${\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}$.

Interpretation Introduction

Interpretation: From the given substances that is $1molof{N}_2\left(g\right)at1bar$ or $1molof{N}_2\left(g\right)at10bar$ the substance with higher entropy should be identified.

Concept introduction: Entropy is a measure of the randomness of the system. It is a thermodynamic quantity and an extensive property. It is represented by the symbol $\text{S}$. It can also be defined as the degree of energy dispersal. More the dispersal in energy, more value of entropy.

Answer to Problem 1PS

The substance which has the highest entropy is ${\text{N}}_{\text{2}}\left(\text{g}\right)$ at $\text{1 bar}$ pressure.

Explanation of Solution

The entropy of the system increase when the pressure is decreased. At high pressure, the substance is more ordered and the randomness is less. Thus, the entropy is less.

Thus, ${\text{N}}_{\text{2}}\left(\text{g}\right)$ at $\text{1 bar}$ pressure has more entropy than ${\text{N}}_{\text{2}}\left(\text{g}\right)$ at $\text{10 bar}$ pressure.