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Chapter 19: Problem 46

Using Appendix \(C\), compare the standard entropies at \(25^{\circ} \mathrm{C}\) for the following pairs of substances: (a) \(\mathrm{CuO}(s)\) and \(\mathrm{Cu}_{2} \mathrm{O}(\mathrm{s}) ;\) (b) \(1 \mathrm{~mol} \mathrm{~N}_{2} \mathrm{O}_{4}(g)\) and \(2 \mathrm{~mol} \mathrm{NO}_{2}(g) ;\) (c) \(\mathrm{SiO}_{2}(\mathrm{~s})\) and \(\mathrm{CO}_{2}(\mathrm{~g}) ;\) (d) \(\mathrm{CO}(\mathrm{g})\) and \(\mathrm{CO}_{2}(\mathrm{~g})\). Fur each pair, explain the difference in the entropy values.

Short Answer

Expert verified
In summary, the comparison of standard entropies at 25掳C for the given pairs are: (a) Cu鈧侽(s) has higher entropy than CuO(s) due to its more complex molecular structure. (b) 2 mol of NO鈧�(g) has higher entropy than 1 mol of N鈧侽鈧�(g) due to the higher number of molecules. (c) CO鈧�(g) has higher entropy than SiO鈧�(s) because gases have more disorder than solids. (d) CO鈧�(g) has higher entropy than CO(g) due to the presence of an extra oxygen atom, increasing molecular complexity.

Step by step solution

01

Find the standard entropies from Appendix C

Look up the standard entropies for the given pairs of substances in Appendix C at 25掳C. (a) CuO(s) and Cu鈧侽(s) (b) 1 mol N鈧侽鈧�(g) and 2 mol NO鈧�(g) (c) SiO鈧�(s) and CO鈧�(g) (d) CO(g) and CO鈧�(g)
02

Compare entropy values and identify reasons for the differences

Now, compare the standard entropy values for each pair of substances and identify the factors causing the differences. (a) CuO(s) and Cu鈧侽(s) CuO(s): \(S^{\circ} = 42.6 \frac{J}{mol \cdot K}\) Cu鈧侽(s): \(S^{\circ} = 93.8 \frac{J}{mol \cdot K}\) Cu鈧侽(s) has a higher entropy than CuO(s) because it has more atoms in its formula unit, which leads to more complex molecular structure and increased disorder in the system. (b) 1 mol N鈧侽鈧�(g) and 2 mol NO鈧�(g) N鈧侽鈧�(g): \(S^{\circ} = 304.3 \frac{J}{mol \cdot K}\) 2 mol NO鈧�(g): \(S^{\circ} = 2 \times 240.0 \frac{J}{mol \cdot K} = 480.0 \frac{J}{mol \cdot K}\) For the second pair, the difference in entropies is due to the difference in the number of molecules. Two moles of NO鈧� gas result in a higher entropy than one mole of N鈧侽鈧� gas, as the system becomes more disordered with the higher number of molecules. (c) SiO鈧�(s) and CO鈧�(g) SiO鈧�(s): \(S^{\circ} = 41.5 \frac{J}{mol \cdot K}\) CO鈧�(g): \(S^{\circ} = 213.6 \frac{J}{mol \cdot K}\) The significant difference in entropies between SiO鈧�(s) and CO鈧�(g) is due to the difference in their states. CO鈧� being a gas, has higher disorder than SiO鈧�, which is a solid. (d) CO(g) and CO鈧�(g) CO(g): \(S^{\circ} = 197.6 \frac{J}{mol \cdot K}\) CO鈧�(g): \(S^{\circ} = 213.6 \frac{J}{mol \cdot K}\) Both substances are gases, but the entropy of CO鈧�(g) is higher due to the presence of an extra oxygen atom, which adds to the molecular complexity and increases the disorder in the system.

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Key Concepts

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

Thermodynamics
Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. It describes how thermal energy is converted to and from other forms of energy and how it affects matter. The four laws of thermodynamics lay the foundation for heat transfer, energy transformations, and the relationships between properties of substances.

At the heart of thermodynamics lie concepts such as temperature, energy, and entropy. Entropy, in particular, is a central idea, representing the amount of disorder or randomness in a system. A higher entropy usually means there are more possible arrangements of a system, indicating a higher degree of uncertainty or disorder. Entropy changes give us profound insights into the direction of spontaneous processes and the feasibility of chemical reactions.
Entropy Values
In thermodynamics, the entropy value of a substance is a quantitative measure of the amount of disorder within a system. Standard entropy, denoted as 厂掳, refers to the entropy content of a substance at a standard state, often considered to be 1 bar of pressure and 25掳C for substances. These values are crucial for predicting the direction of chemical reactions and for calculating the Gibbs free energy change, which can indicate whether a process will occur spontaneously.

The values of standard entropy are affected by various factors including the complexity of the molecule (more atoms typically mean higher entropy), the phase of the matter (gases have higher entropy than liquids or solids due to their greater freedom of movement), temperature, and the number of particles, as seen in the comparison between 1 mol of N鈧侽鈧�(g) and 2 mol of NO鈧�(g), where two moles of NO鈧� gas manifest more disorder and thus, a higher entropy than one mole of N鈧侽鈧� gas.
Chemical States of Matter
Matter can exist in different chemical states, primarily solid, liquid, and gas, which significantly influence entropy. Solids have the lowest entropy because their particles are typically arranged in a fixed, orderly fashion. Liquids have higher entropy than solids because their particles can flow and move more freely, leading to more randomness. Gases have the highest entropy among the three classical states because their particles move independently and occupy the volume of their container, contributing to a substantial increase in disorder.

The differences between the chemical states of matter play a critical role in standard entropy comparisons. For instance, the comparison between SiO鈧�(s), which is a solid, and CO鈧�(g), which is a gas, clearly shows that CO鈧�, in its gaseous state, has a significantly higher entropy. This owes to the freedom of motion and the wide range of possible microstates that gas molecules can have compared to those in a rigid solid lattice.

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Most popular questions from this chapter

A certain reaction has \(\Delta H^{\circ}=-19.5 \mathrm{~kJ}\) and \(\Delta S^{\circ}=\) \(+42.7 \mathrm{~J} / \mathrm{K} .\) (a) Is the reaction exothermic or endothermic? (b) Does the reaction lead to an increase or decrease in the randomness or disorder of the system? (c) Calculate \(\Delta G^{\circ}\) for the reaction at \(298 \mathrm{~K}\). (d) Is the reaction spontaneous at \(298 \mathrm{~K}\) under standard conditions?

One way to derive Equation \(19.3\) depends on the observation that at constant \(T\) the number of ways, \(W\), of arranging \(m\) ideal-gas particles in a volume \(V\) is proportional to the volume raised to the \(m\) power: $$ W \propto V^{m} $$ Use this relationship and Boltzmann's relationship between entropy and number of arrangements (Equation 19.5) to derive the equation for the entropy change for the isothermal expansion or compression of \(n\) moles of an ideal gas.

Most liquids follow Trouton's rule, which states that the molar entropy of vaporization lies in the range of \(88 \pm 5 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\). The normal boiling points and enthalpies of vaporization of several organic liquids are as follows: $$ \begin{array}{lrl} \hline \text { Substance } & \begin{array}{l} \text { Normal Boiling } \\ \text { Point }\left({ }^{\circ} \mathrm{C}\right) \end{array} & \begin{array}{l} \Delta H_{\text {vap }} \\ \text { (kJ/mol) } \end{array} \\ \hline \text { Acetone, }\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CO} & 56.1 & 29.1 \\ \text { Dimethyl ether, }\left(\mathrm{CH}_{3}\right)_{2} \mathrm{O} & -24.8 & 21.5 \\ \text { Ethanol } \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} & 78.4 & 38.6 \\ \text { Octane, } \mathrm{C}_{8} \mathrm{H}_{18} & 125.6 & 34.4 \\ \text { Pyridine, } \mathrm{C}_{5} \mathrm{H}_{5} \mathrm{~N} & 115.3 & 35.1 \\\ \hline \end{array} $$ (a) Calculate \(\Delta \mathrm{S}_{\mathrm{vap}}\) for each of the liquids. Do all of the liquids obey Trouton's rule? (b) With reference to intermolecular forces (Section 11.2), can you explain any exceptions to the rule? (c) Would you expect water to obey Trouton's rule? By using data in Appendix \(\mathrm{B}\), check the accuracy of your conclusion. (d) Chlorobenzene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{Cl}\right)\) boils at \(131.8^{\circ} \mathrm{C}\). Use Trouton's rule to estimate \(\Delta H_{\text {vap }}\) for this substance.

(a) State the third law of thermodynamics. (b) Distinguish between translational motion, vibrational motion, and rotational motion of a molecule. (c) Illustrate these three kinds of motion with sketches for the HCl molecule.

Carbon disulfide \(\left(\mathrm{CS}_{2}\right)\) is a toxic, highly flam mable substance. The following thermodynamic data are available for \(\mathrm{CS}_{2}(l)\) and \(\mathrm{CS}_{2}(g)\) at \(298 \mathrm{~K}\) : \begin{tabular}{lrl} \hline & \(\Delta H_{f}^{\circ}(\mathbf{k J} / \mathrm{mol})\) & \(\Delta G_{f}^{0}(\mathbf{k J} / \mathrm{mol})\) \\ \hline \(\mathrm{CS}_{2}(l)\) & \(89.7\) & \(65.3\) \\ \(\mathrm{CS}_{2}(g)\) & \(117.4\) & \(67.2\) \\ \hline \end{tabular} (a) Draw the Lewis structure of the molecule. What do you predict for the bond order of the \(\mathrm{C}-\mathrm{S}\) bonds? (b) Use the VSEPR method to predict the structure of the \(\mathrm{CS}_{2}\) molecule. (c) Liquid \(\mathrm{CS}_{2}\) bums in \(\mathrm{O}_{2}\) with a blue flame, forming \(\mathrm{CO}_{2}(g)\) and \(\mathrm{SO}_{2}(g)\). Write a balanced equation for this reaction. (d) Using the data in the preceding table and in Appendix \(C\), calculate \(\Delta H^{\circ}\) and \(\Delta G^{\circ}\) for the reaction in part (c). Is the reaction exothermic? Is it spontaneous at 298 K? (e) Use the data in the preceding table to calculate \(\Delta S^{\circ}\) at \(298 \mathrm{~K}\) for the vaporization of \(\mathrm{CS}_{2}(l)\). Is the sign of \(\Delta S^{\circ}\) as you would expect for a vaporization? (f) Using data in the preceding table and your answer to part (e), estimate the boiling point of \(\mathrm{CS}_{2}(\mathrm{l})\). Do you predict that the substance will be a liquid or a gas at \(298 \mathrm{~K}\) and \(1 \mathrm{~atm}\) ?
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