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Chapter 17: Problem 31

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
Options b, c, and d will undergo spontaneous changes at constant \(T\) and \(P\).

Step by step solution

01

Gather the given values for each option

For each option, we are given the values for \(\Delta H\), \(\Delta S\), and \(T\). We will simply plug these values into the formula for \(\Delta G\) to determine whether the reaction will be spontaneous.
02

Calculate \(\Delta G\) for option a

We have \(\Delta H = +25 \textrm{ kJ}\), \(\Delta S = +5.0 \textrm{ J/K}\), and \(T = 300 \textrm{ K}\). Converting \(\Delta H\) to J: \(\Delta H = 25 \times 1000 = 25,000 \textrm{ J}\). Now we will calculate \(\Delta G = \Delta H - T \Delta S\): \(\Delta G = 25,000 \textrm{ J} - (300 \textrm{ K})(5.0 \textrm{ J/K}) = 25,000 \textrm{ J} - 1500 \textrm{ J} = 23,500 \textrm{ J}\). Since \(\Delta G > 0\), option a is not spontaneous.
03

Calculate \(\Delta G\) for option b

For option b, we have \(\Delta H = +25 \textrm{ kJ}\), \(\Delta S = +100 \textrm{ J/K}\), and \(T = 300 \textrm{ K}\). Using the same conversion as before, we have \(\Delta H = 25,000 \textrm{ J}\). Calculating \(\Delta G = \Delta H - T \Delta S\): \(\Delta G = 25,000 \textrm{ J} - (300 \textrm{ K})(100 \textrm{ J/K}) = 25,000 \textrm{ J} - 30,000 \textrm{ J} = -5,000 \textrm{ J}\). Since \(\Delta G < 0\), option b is spontaneous.
04

Calculate \(\Delta G\) for option c

In option c, we have \(\Delta H = -10 \textrm{ kJ}\), \(\Delta S = +5.0 \textrm{ J/K}\), and \(T = 298 \textrm{ K}\). Converting \(\Delta H\) to J: \(\Delta H = -10 \times 1000 = -10,000 \textrm{ J}\). Calculating \(\Delta G = \Delta H - T \Delta S\): \(\Delta G = -10,000 \textrm{ J} - (298 \textrm{ K})(5.0 \textrm{ J/K}) = -10,000 \textrm{ J} - 1,490 \textrm{ J} = -11,490 \textrm{ J}\). Since \(\Delta G < 0\), option c is spontaneous.
05

Calculate \(\Delta G\) for option d

For option d, we have \(\Delta H = -10 \textrm{ kJ}\), \(\Delta S = -40 \textrm{ J/K}\), and \(T = 200 \textrm{ K}\). Using the same conversion as before, we have \(\Delta H = -10,000 \textrm{ J}\). Calculating \(\Delta G = \Delta H - T \Delta S\): \(\Delta G = -10,000 \textrm{ J} - (200 \textrm{ K})(-40 \textrm{ J/K}) = -10,000 \textrm{ J} + 8,000 \textrm{ J} = -2,000 \textrm{ J}\). Since \(\Delta G < 0\), option d is spontaneous.
06

Conclusion

Options b, c, and d will undergo spontaneous changes at constant \(T\) and \(P\).

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Most popular questions from this chapter

Consider the following reaction: $$\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)$$Calculate \(\Delta G\) for this reaction under the following conditions (assume an uncertainty of \(\pm 1\) in all quantities): a. \(T=298 \mathrm{~K}, P_{\mathrm{N}_{2}}=P_{\mathrm{H}_{2}}=200 \mathrm{~atm}, P_{\mathrm{NH}_{3}}=50 \mathrm{~atm}\) b. \(T=298 \mathrm{~K}, P_{\mathrm{N}_{2}}=200 \mathrm{~atm}, P_{\mathrm{H}_{2}}=600 \mathrm{~atm}, P_{\mathrm{NH}_{3}}=200 \mathrm{~atm}\)

When the environment is contaminated by a toxic or potentially toxic substance (for example, from a chemical spill or the use of insecticides), the substance tends to disperse. How is this consistent with the second law of thermodynamics? In terms of the second law, which requires the least work: cleaning the environment after it has been contaminated or trying to prevent the contamination before it occurs? Explain.

At what temperatures will the following processes be spontaneous? a. \(\Delta H=-18 \mathrm{~kJ}\) and \(\Delta S=-60 . \mathrm{J} / \mathrm{K}\) b. \(\Delta H=+18 \mathrm{~kJ}\) and \(\Delta S=+60 . \mathrm{J} / \mathrm{K}\) c. \(\Delta H=+18 \mathrm{~kJ}\) and \(\Delta S=-60 . \mathrm{J} / \mathrm{K}\) d. \(\Delta H=-18 \mathrm{~kJ}\) and \(\Delta S=+60 . \mathrm{J} / \mathrm{K}\)

For the reaction at \(298 \mathrm{~K}\), $$2 \mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{4}(g)$$ the values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are \(-58.03 \mathrm{~kJ}\) and \(-176.6 \mathrm{~J} / \mathrm{K}, \mathrm{re}-\) spectively. What is the value of \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\) ? Assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature, at what temperature is \(\Delta G^{\circ}=0 ?\) Is \(\Delta G\) negative above or below this temperature?

The synthesis of glucose directly from \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) and the synthesis of proteins directly from amino acids are both nonspontaneous processes under standard conditions. Yet it is necessary for these to occur for life to exist. In light of the second law of thermodynamics, how can life exist?
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