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Chapter 19: Problem 47

In each of the following pairs, which compound would you expect to have the higher standard molar entropy: (a) \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) or \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) (b) \(\mathrm{CO}_{2}(g)\) or \(\mathrm{CO}(g) ?\) Explain.

Short Answer

Expert verified
In pair (a), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher standard molar entropy due to its greater number of atoms and possible arrangements in space. In pair (b), \(\mathrm{CO}_{2}(g)\) has a higher standard molar entropy for the same reason.

Step by step solution

01

Identify factors that affect molar entropy

For each pair of compounds, we will consider factors such as the complexity of the molecules (e.g., size, number of atoms) and the phase (solid, liquid, or gas) to determine which compound has a higher molar entropy.
02

Compare the complexity of molecules in pair (a)

In pair (a), we are comparing \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) and \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\). Both compounds are in the gas phase, so we don't need to consider the phase. Let's compare the complexity of these molecules. \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) contains 2 carbon atoms and 2 hydrogen atoms, for a total of 4 atoms. \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) contains 2 carbon atoms and 6 hydrogen atoms, for a total of 8 atoms. Since \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has more atoms, it has more possible arrangements in space, which means it has a higher entropy.
03

Determine the compound with higher molar entropy in pair (a)

In pair (a), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher molar entropy because it has more atoms and therefore more possible arrangements in space. So, the answer for pair (a) is \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\).
04

Compare the complexity of molecules in pair (b)

In pair (b), we are comparing \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\). Both compounds are in the gas phase, so we don't need to consider the phase. Let's compare the complexity of these molecules. \(\mathrm{CO}_{2}(g)\) contains 1 carbon atom and 2 oxygen atoms, for a total of 3 atoms. \(\mathrm{CO}(g)\) contains 1 carbon atom and 1 oxygen atom, for a total of 2 atoms. Since \(\mathrm{CO}_2(g)\) has more atoms, it has more possible arrangements in space, which means it has a higher entropy.
05

Determine the compound with higher molar entropy in pair (b)

In pair (b), \(\mathrm{CO}_{2}(g)\) has a higher molar entropy because it has more atoms and therefore more possible arrangements in space. So, the answer for pair (b) is \(\mathrm{CO}_{2}(g)\). In summary, - In pair (a), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher standard molar entropy. - In pair (b), \(\mathrm{CO}_{2}(g)\) has a higher standard molar entropy.

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Most popular questions from this chapter

(a) Express the second law of thermodynamics as a mathematical equation. (b) In a particular spontaneous process the entropy of the system decreases. What can you conclude about the sign and magnitude of \(\Delta S_{\text {surr }} ?\) (c) During a certain reversible process, the surroundings undergo an entropy change, \(\Delta S_{\text {surr }}=-78 \mathrm{~J} / \mathrm{K}\). What is the entropy change of the system for this process?

The potassium-ion concentration in blood plasma is about \(5.0 \times 10^{-3} M,\) whereas the concentration in muscle-cell fluid is much greater \((0.15 \mathrm{M})\). The plasma and intracellular fluid are separated by the cell membrane, which we assume is permeable only to \(\mathrm{K}^{+}\). (a) What is \(\Delta G\) for the transfer of \(1 \mathrm{~mol}\) of \(\mathrm{K}^{+}\) from blood plasma to the cellular fluid at body temperature \(37^{\circ} \mathrm{C} ?\) (b) What is the minimum amount of work that must be used to transfer this \(\mathrm{K}^{+} ?\)

Methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) can be made by the controlled oxidation of methane: $$ \mathrm{CH}_{4}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(g) $$ (a) Use data in Appendix C to calculate \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) for this reaction. (b) How is \(\Delta G^{\circ}\) for the reaction expected to vary with increasing temperature? (c) Calculate \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\). Under standard conditions, is the reaction spontaneous at this temperature? (d) Is there a temperature at which the reaction would be at equilibrium under standard conditions and that is low enough so that the compounds involved are likely to be stable?

Propanol \(\left(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{OH}\right)\) melts at \(-126.5^{\circ} \mathrm{C}\) and boils at \(97.4{ }^{\circ} \mathrm{C}\). Draw a qualitative sketch of how the entropy changes as propanol vapor at \(150^{\circ} \mathrm{C}\) and 1 atm is cooled to solid propanol at \(-150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

The relationship between the temperature of a reaction, its standard enthalpy change, and the equilibrium constant at that temperature can be expressed as the following linear equation: $$ \ln K=\frac{-\Delta H^{\circ}}{R T}+\text { constant } $$ (a) Explain how this equation can be used to determine \(\Delta H^{\circ}\) experimentally from the equilibrium constants at several different temperatures. (b) Derive the preceding equation using relationships given in this chapter. To what is the constant equal?
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