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Chapter 16: Problem 35

Given the values of \(\Delta H\) and \(\Delta S,\) which of the following changes will be spontaneous at constant \(T\) and \(P ?\) a. \(\Delta H=+25 \mathrm{kJ}, \Delta S=+5.0 \mathrm{J} / \mathrm{K}, T=300 . \mathrm{K}\) b. \(\Delta H=+25 \mathrm{kJ}, \Delta S=+100 . \mathrm{J} / \mathrm{K}, T=300 . \mathrm{K}\) c. \(\Delta H=-10 . \mathrm{kJ}, \Delta S=+5.0 \mathrm{J} / \mathrm{K}, T=298 \mathrm{K}\) d. \(\Delta H=-10 . \mathrm{kJ}, \Delta S=-40 . \mathrm{J} / \mathrm{K}, T=200 . \mathrm{K}\)

Short Answer

Expert verified
The spontaneous changes at constant \(T\) and \(P\) are options (c) and (d).

Step by step solution

01

Calculating Gibbs free energy change for option (a)

Calculate the Gibbs free energy change using the given values: \(\Delta G = \Delta H - T \Delta S = +25 \times 10^3 \mathrm{kJ} - 300 \mathrm{K} \times (5.0 \times 10^{-3} \mathrm{kJ/K}) = 25000 \mathrm{kJ} - 1.5 \mathrm{kJ} = 24898.5 \mathrm{kJ}\)
02

Checking for spontaneity in option (a)

Since \(\Delta G\) is positive, the process is not spontaneous.
03

Calculating Gibbs free energy change for option (b)

Calculate the Gibbs free energy change using the given values: \(\Delta G = \Delta H - T \Delta S = +25 \times 10^3 \mathrm{kJ} - 300 \mathrm{K} \times (100 \times 10^{-3} \mathrm{kJ/K}) = 25000 \mathrm{kJ} - 30 \mathrm{kJ} = 24970 \mathrm{kJ}\)
04

Checking for spontaneity in option (b)

Since \(\Delta G\) is positive, the process is not spontaneous.
05

Calculating Gibbs free energy change for option (c)

Calculate the Gibbs free energy change using the given values: \(\Delta G = \Delta H - T \Delta S = (-10 \times 10^3) \mathrm{kJ} - 298 \mathrm{K} \times (5.0 \times 10^{-3} \mathrm{kJ/K}) = -10000 \mathrm{kJ} - 1.49 \mathrm{kJ} = -10001.49 \mathrm{kJ}\)
06

Checking for spontaneity in option (c)

Since \(\Delta G\) is negative, the process is spontaneous.
07

Calculating Gibbs free energy change for option (d)

Calculate the Gibbs free energy change using the given values: \(\Delta G = \Delta H - T \Delta S = (-10 \times 10^3) \mathrm{kJ} - 200 \mathrm{K} \times (-40 \times 10^{-3} \mathrm{kJ/K}) = -10000 \mathrm{kJ} + 8 \mathrm{kJ} = -9992 \mathrm{kJ}\)
08

Checking for spontaneity in option (d)

Since \(\Delta G\) is negative, the process is spontaneous. From the analysis, the changes that will be spontaneous under the specified conditions are options (c) and (d).

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Most popular questions from this chapter

As \(\mathrm{O}_{2}(l)\) is cooled at 1 atm, it freezes at \(54.5 \mathrm{K}\) to form solid I. At a lower temperature, solid I rearranges to solid II, which has a different crystal structure. Thermal measurements show that \(\Delta H\) for the \(\mathrm{I} \rightarrow\) II phase transition is \(-743.1 \mathrm{J} / \mathrm{mol}\), and \(\Delta S\) for the same transition is \(-17.0 \mathrm{J} / \mathrm{K} \cdot\) mol. At what temperature are solids I and II in equilibrium?

The enthalpy of vaporization of ethanol is \(38.7 \mathrm{kJ} / \mathrm{mol}\) at its boiling point \(\left(78^{\circ} \mathrm{C}\right) .\) Determine \(\Delta S_{\mathrm{sys}}, \Delta S_{\text {surr }}\) and \(\Delta S_{\text {univ }}\) when 1.00 mole of ethanol is vaporized at \(78^{\circ} \mathrm{C}\) and 1.00 atm.

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 \(S_{\text {rhombic }}(s) \longrightarrow S_{\text {monoclinic }}(s).\) b. Which form of sulfur has the more ordered crystalline structure (has the smaller positional probability)?

Which of the following reactions (or processes) are expected to have a negative value for \(\Delta S^{\circ} ?\) a. \(\operatorname{SiF}_{6}(a q)+\mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{HF}(g)+\mathrm{SiF}_{4}(g)\) b. \(4 \mathrm{Al}(s)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{Al}_{2} \mathrm{O}_{3}(s)\) c. \(\mathrm{CO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{COCl}_{2}(g)\) d. \(\mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\) e. \(\mathrm{H}_{2} \mathrm{O}(s) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)\)

You remember that \(\Delta G^{\circ}\) is related to \(R T \ln (K)\) but cannot remember if it's \(R T \ln (K)\) or \(-R T \ln (K) .\) Realizing what \(\Delta G^{\circ}\) and \(K\) mean, how can you figure out the correct sign?
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