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Chapter 19: Problem 67

A particular constant-pressure reaction is spontaneous at \(390 \mathrm{~K}\). The enthalpy change for the reaction is \(+23.7 \mathrm{~kJ}\). What can you conclude about the sign and magnitude of \(\Delta S\) for the reaction?

Short Answer

Expert verified
Since the given reaction is spontaneous at the given temperature of 390 K, we can conclude that the entropy change, \(\Delta S\), is positive and has a magnitude greater than \(60.77 \frac{\mathrm{J}}{\mathrm{K}}\).

Step by step solution

01

Identify the given values and the Gibbs free energy equation

For this problem, we are given the following: - Temperature, \(T = 390 \mathrm{~K}\) - Enthalpy change, \(\Delta H = +23.7 \mathrm{~ kJ}\) The Gibbs free energy equation is: \(\Delta G = \Delta H - T\Delta S\)
02

Rewrite the equation to solve for entropy change, \(\Delta S\)

We know that the reaction is spontaneous, so the Gibbs free energy must be negative \(\Delta G < 0\). We can rewrite the equation to solve for \(\Delta S\): \(\Delta S = \frac{\Delta H - \Delta G}{T}\)
03

Evaluate the inequality for \(\Delta S\)

Since \(\Delta G < 0\), we have the following inequality: \(\Delta S > \frac{\Delta H}{T}\)
04

Substitute the given values and solve

Now substitute the given values of \(\Delta H\) and \(T\) into the inequality: \(\Delta S > \frac{23.7 \mathrm{~kJ}}{390 \mathrm{~K}}\) To make units consistent, we convert kJ to J: \(23.7 \mathrm{~kJ} = 23700 \mathrm{~J}\) \(\Delta S > \frac{23700 \mathrm{~J}}{390 \mathrm{~K}} \) Now, divide 23700 by 390: \(\Delta S > 60.77 \frac{\mathrm{J}}{\mathrm{K}}\) #Conclusion#Since the given reaction is spontaneous at the given temperature of 390 K, we can conclude that the entropy change, \(\Delta S\), is positive and has a magnitude greater than \(60.77 \frac{\mathrm{J}}{\mathrm{K}}\).

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Most popular questions from this chapter

For the majority of the compounds listed in Appendix \(\mathrm{C},\) the value of \(\Delta G_{f}^{\circ}\) is more positive (or less negative) than the value of \(\Delta H_{f}^{\circ} .\) (a) Explain this observation, using \(\mathrm{NH}_{3}(g), \mathrm{CCl}_{4}(l)\), and \(\mathrm{KNO}_{3}(s)\) as examples. (b) An exception to this observation is \(\mathrm{CO}(g)\). Explain the trend in the \(\Delta H_{f}^{\circ}\) and \(\Delta G_{f}^{\circ}\) values for this molecule.

(a) Using data in Appendix \(C\), estimate the temperature at which the free- energy change for the transformation from \(\mathrm{I}_{2}(s)\) to \(\mathrm{I}_{2}(g)\) is zero. What assumptions must you make in arriving at this estimate? (b) Use a reference source, such as Web Elements (www.webelements.com), to find the experimental melting and boiling points of \(\mathrm{I}_{2} .\) (c) Which of the values in part (b) is closer to the value you obtained in part (a)? Can you explain why this is so?

Trouton's rule states that for many liquids at their normal boiling points, the standard molar entropy of vaporization is about $88 \mathrm{~J} / \mathrm{mol}-\mathrm{K}$. (a) Estimate the normal boiling point of bromine, \(\mathrm{Br}_{2}\), by determining \(\Delta H_{\text {vap }}^{t}\) for \(\mathrm{Br}_{2}\) using data from Appendix C. Assume that $\Delta H_{\text {ap }}^{a}$ remains constant with temperature and that Trouton's rule holds. (b) Look hal boiling point of Br\(r_{2}\) in a chemistry handbook or at the WebElements Web site (www,webelements. com) and compare it to your calculation. What are the possible sources of error, or incorrect assumptions, in the calculation?

Consider a reaction $\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \rightleftharpoons 2 \mathrm{AB}(g),$ with atoms of A shown in red in the diagram and atoms of \(\mathrm{B}\) shown in blue. (a) If \(K_{\mathrm{c}}=1,\) which box represents the system at equilibrium? (b) If \(K_{\mathrm{c}}=1,\) which box represents the system at \(Q < K_{\mathrm{c}} ?(\mathbf{c})\) Rank the boxes in order of increasing magnitude of \(\Delta G\) for the reaction. [ Sections 19.5 and 19.7\(]\)

How does the entropy of the system change when (a) the temperature of the system increases, (b) the volume of a gas increases, \((c)\) equal volumes of ethanol and water are mixed to form a solution?
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