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Chapter 17: Problem 45

It is quite common for a solid to change from one structure to another at a temperature below its melting point. For example, sulfur undergoes a phase change from the rhombic crystal structure to the monoclinic crystal form at temperatures above \(95^{\circ} \mathrm{C}\). a. Predict the signs of \(\Delta H\) and \(\Delta S\) for the process \(S_{\text {rhcmbic }} \longrightarrow\) \(\mathrm{S}_{\text {monoclinic }}\) b. Which form of sulfur has the more ordered crystalline structure (has the smaller positional probability)?

Short Answer

Expert verified
a. For the phase change \(S_{\text{rhombic}} \longrightarrow S_{\text{monoclinic}}\), the signs of 𝚫H and 𝚫S are both positive, indicating an endothermic process with an increase in entropy. b. The rhombic form of sulfur has a more ordered crystalline structure (smaller positional probability) compared to the monoclinic form.

Step by step solution

01

a. Prediction of 𝚫H and 𝚫S for the phase change

In general, when a solid undergoes phase transition to another phase, the new phase is more stable at higher temperatures, due to changes in the intermolecular forces holding the solid together. Since sulfur undergoes a phase change from rhombic to monoclinic crystal form above \(95^{\circ} \mathrm{C}\), we can infer the following: 1. The enthalpy change 𝚫H is expected to be positive, as it requires energy to overcome the forces holding the rhombic structure together, showing that the rhombic structure is more stable at lower temperatures and the phase change is endothermic. 2. The entropy change 𝚫S is expected to be positive as well, since the monoclinic structure is more stable at higher temperatures, and typically, structures that are more stable at higher temperatures have higher entropy (more positional disorder). Thus, the signs of 𝚫H and 𝚫S for the phase transition are both positive.
02

b. Comparison of ordered crystalline structures

To determine which form of sulfur has a more ordered crystalline structure, we need to compare their positional probabilities. In general, a higher positional probability (higher entropy) means that the structure is more disordered. Since we established earlier that the entropy change 𝚫S is positive during the phase transition from rhombic to monoclinic, this means that the monoclinic structure has a higher entropy (more positional disorder) than the rhombic structure. Therefore, the rhombic form of sulfur would be the one with a more ordered crystalline structure (smaller positional probability) compared to the monoclinic form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Enthalpy Change (H)
Enthalpy change, denoted as H, is a measure of the total energy change in a system during a reaction or phase transition at constant pressure. When substances undergo a phase transition, such as the rhombic to monoclinic sulfur transformation, the rearrangement of atoms and molecules involves changes in energy associated with the intermolecular forces.

For the sulfur phase transition, we expect a positive H, which signifies that energy is absorbed to disrupt the stable rhombic structure. This absorption of energy is essential for any endothermic process, in which the system gains heat from the surroundings. In educational terms, you might think of H as akin to the financial cost of a process—if it’s positive, the system is 'paying' in energy to change state.
Entropy Change (S)
Entropy change (S) refers to the degree of disorder or randomness in a system. It's a fundamental concept that helps students understand the shift from organised to less-structured states.

Diving into the sulfur example, as temperature increases, the substance's atoms have more freedom to move, leading to greater disorder. A positive S, observed during the transition from rhombic to monoclinic sulfur, reflects this increase in disorder. Thus, moving to a higher entropy state suggests that the monoclinic structure has a broader range of possible positions or configurations for its constituent particles.
Crystal Structure
The crystal structure is the orderly geometric spatial arrangement of atoms in the crystalline state. The arrangement of these atoms is not random; it follows specific patterns that repeat in three-dimensional space. Educational models often use building blocks to visualize these repeating units.

For sulfur, the rhombic and monoclinic forms refer to different crystal structures, each with its unique pattern of atoms. The rhombic structure is more stable at lower temperatures, while the monoclinic takes over at temperatures above 95°C. This stability is closely tied to the specific geometric arrangements that define each structure.
Positional Probability
Positional probability is a term used to describe how likely it is to find particles in certain positions within a solid. It's analogous to the likelihood of finding someone at home—there’s a higher probability of finding them there at night than during the day.

In crystals, higher positional probability equates to a higher degree of disorder, as seen in the monoclinic structure of sulfur post-transition. Thus, a lower positional probability as in rhombic sulfur, indicates a more ordered structure where the particles have fewer places to be and tend to stay put—a bit like people staying close to their homes.

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Most popular questions from this chapter

The enthalpy of vaporization of chloroform \(\left(\mathrm{CHCl}_{3}\right)\) is \(31.4\) \(\mathrm{kJ} / \mathrm{mol}\) at its boiling point \(\left(61.7^{\circ} \mathrm{C}\right) .\) Determine \(\Delta S_{\mathrm{sys}}, \Delta S_{\mathrm{sur}}\), and \(\Delta S_{\text {univ }}\) when \(1.00 \mathrm{~mol}\) chloroform is vaporized at \(61.7^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm} .\)

Impure nickel, refined by smelting sulfide ores in a blast furnace, can be converted into metal from \(99.90 \%\) to \(99.99 \%\) purity by the Mond process. The primary reaction involved in the Mond process is $$\mathrm{Ni}(s)+4 \mathrm{CO}(g) \rightleftharpoons \mathrm{Ni}(\mathrm{CO})_{4}(g)$$ a. Without referring to Appendix 4, predict the sign of \(\Delta S^{\circ}\) for the above reaction. Explain. b. The spontaneity of the above reaction is temperature dependent. Predict the sign of \(\Delta S_{\text {sarr }}\) for this reaction. Explain. c. For \(\mathrm{Ni}(\mathrm{CO})_{4}(g), \Delta H_{\mathrm{f}}^{\circ}=-607 \mathrm{~kJ} / \mathrm{mol}\) and \(S^{\circ}=417 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\) at \(298 \mathrm{~K}\). Using these values and data in Appendix 4, calculate \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) for the above reaction. d. Calculate the temperature at which \(\Delta G^{\circ}=0(K=1)\) for the above reaction, assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature. e. The first step of the Mond process involves equilibrating impure nickel with \(\mathrm{CO}(\mathrm{g})\) and \(\mathrm{Ni}(\mathrm{CO})_{4}(g)\) at about \(50^{\circ} \mathrm{C}\). The purpose of this step is to convert as much nickel as possible into the gas phase. Calculate the equilibrium constant for the preceding reaction at \(50 .{ }^{\circ} \mathrm{C}\). f. In the second step of the Mond process, the gaseous \(\mathrm{Ni}(\mathrm{CO})_{4}\) is isolated and heated to \(227^{\circ} \mathrm{C}\). The purpose of this step is to deposit as much nickel as possible as pure solid (the reverse of the preceding reaction). Calculate the equilibrium constant for the preceding reaction at \(227^{\circ} \mathrm{C}\). g. Why is temperature increased for the second step of the Mond process? h. The Mond process relies on the volatility of \(\mathrm{Ni}(\mathrm{CO})_{4}\) for its success. Only pressures and temperatures at which \(\mathrm{Ni}(\mathrm{CO})_{4}\) is a gas are useful. A recently developed variation of the Mond process carries out the first step at higher pressures and a temperature of \(152^{\circ} \mathrm{C}\). Estimate the maximum pressure of \(\mathrm{Ni}(\mathrm{CO})_{4}(g)\) that can be attained before the gas will liquefy at \(152^{\circ} \mathrm{C}\). The boiling point for \(\mathrm{Ni}(\mathrm{CO})_{4}\) is \(42^{\circ} \mathrm{C}\) and the enthalpy of vaporization is \(29.0 \mathrm{~kJ} / \mathrm{mol}\).

Given the following data: $$\begin{array}{lr}2 \mathrm{C}_{6} \mathrm{H}_{6}(l)+15 \mathrm{O}_{2}(g) \longrightarrow 12 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l) \\ \Delta G^{\circ}=-6399 \mathrm{~kJ} \\\\\mathrm{C}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) & \Delta G^{\circ}=-394 \mathrm{~kJ} \\ \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l) & \Delta G^{\circ}=-237 \mathrm{~kJ} \end{array}$$ calculate \(\Delta G^{\circ}\) for the reaction $$6 \mathrm{C}(s)+3 \mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{6}(l)$$

Consider the reaction $$\mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{HBr}(g)$$ where \(\Delta H^{\circ}=-103.8 \mathrm{~kJ} / \mathrm{mol} .\) In a particular experiment, equal moles of \(\mathrm{H}_{2}(\mathrm{~g})\) at \(1.00 \mathrm{~atm}\) and \(\mathrm{Br}_{2}(\mathrm{~g})\) at \(1.00 \mathrm{~atm}\) were mixed in a \(1.00\) -L flask at \(25^{\circ} \mathrm{C}\) and allowed to reach equilibrium. Then the molecules of \(\mathrm{H}_{2}\) at equilibrium were counted using a very sensitive technique, and \(1.10 \times 10^{13}\) molecules were found. For this reaction, calculate the values of \(K, \Delta G^{\circ}\), and \(\Delta S^{\circ}\).

Human DNA contains almost twice as much information as is needed to code for all the substances produced in the body. Likewise, the digital data sent from Voyager II contained one redundant bit out of every two bits of information. The Hubble space telescope transmits three redundant bits for every bit of information. How is entropy related to the transmission of information? What do you think is accomplished by having so many redundant bits of information in both DNA and the space probes?
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