Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 19: Problem 55

(a) For a process that occurs at constant temperature, express the change in Gibbs free energy in terms of changes in the enthalpy and entropy of the system. (b) For a certain process that occurs at constant \(T\) and \(P,\) the value of \(\Delta G\) is positive. What can you conclude? (c) What is the relationship between \(\Delta G\) for a process and the rate at which it occurs?

Short Answer

Expert verified
In summary, (a) for a process at constant temperature, the change in Gibbs free energy \(\Delta G\) can be expressed as \(\Delta G = \Delta H - T \Delta S\). (b) A positive \(\Delta G\) value at constant \(T\) and \(P\) implies that the reaction is non-spontaneous and not energetically favorable under the given conditions. (c) There is no direct relationship between \(\Delta G\) and the rate of a process. However, \(\Delta G\) can be used to determine the equilibrium constant \(K\), which can affect the rate of the process as the reaction attains equilibrium.

Step by step solution

01

Part (a): Expressing \(\Delta G\) in terms of \(\Delta H\) and \(\Delta S\)

To find an expression for the change in Gibbs free energy, we can begin with the Gibbs free energy definition: \[G = H - TS\] Here, \(G\) is the Gibbs free energy, \(H\) is the enthalpy, \(T\) is the temperature, and \(S\) is the entropy. Now, we can take the differential of the equation: \[dG = dH - TdS - SdT\] Since the process occurs at constant temperature, the change in temperature \(dT\) is zero, therefore: \[dG = dH - TdS\] So, the change in Gibbs free energy \(\Delta G\) can be expressed as: \[\Delta G = \Delta H - T \Delta S\]
02

Part (b): Interpretation of a positive \(\Delta G\) at constant \(T\) and \(P\)

When the change in Gibbs free energy (\(\Delta G\)) is positive, the reaction is non-spontaneous under the given conditions. For a process at constant temperature (\(T\)) and pressure (\(P\)), a positive \(\Delta G\) implies that the reaction is not energetically favorable and will not proceed spontaneously in the forward direction under the given thermodynamic conditions.
03

Part (c): Relationship between \(\Delta G\) and the rate of a process

There is no direct relationship between the change in Gibbs free energy (\(\Delta G\)) and the rate at which a process occurs. \(\Delta G\) determines the thermodynamic stability and spontaneity of a process, but it does not provide any information about the reaction kinetics or how quickly the process takes place. However, the \(\Delta G\) can be used to determine the equilibrium constant \(K\). And, in turn, the rate of a process can be affected by the equilibrium constant, as the reaction attains equilibrium. The relation between \(\Delta G\) and the equilibrium constant \(K\) is given by: \[\Delta G^\circ = -RT \ln K\] In this equation, \(\Delta G^\circ\) represents the standard Gibbs free energy change, \(R\) is the gas constant, and \(T\) is the temperature. The reaction rate is dependent on the concentrations of reactants and products, as well as factors such as temperature, pressure, and catalysts - but these factors are not directly determined by \(\Delta G\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Enthalpy Change
Enthalpy change, denoted as \(\Delta H\), refers to the heat absorbed or released by a system at constant pressure during a chemical or physical process. It is a thermodynamic quantity that helps us understand the energy changes involved in a reaction. For instance, if a reaction releases heat, it is exothermic and \(\Delta H\) is negative. Conversely, if it absorbs heat, it is endothermic with a positive \(\Delta H\).

Understanding \(\Delta H\) is crucial as it helps predict energy requirements and the feasibility of industrial processes. Chemical reactions in a textbook might seem abstract, but \(\Delta H\) connects them to real-world energy considerations - from the warmth felt in an exothermic combustion reaction to the cooling effect of an endothermic reaction in a cold pack.
Entropy Change
Entropy change, symbolized as \(\Delta S\), measures the disorder or randomness within a system. It's a core concept in thermodynamics and provides insight into the direction in which a system naturally evolves. An increase in entropy (positive \(\Delta S\)) indicates a move towards disorder, while a decrease (negative \(\Delta S\)) reflects a move towards order.

Dissolving a solid into a liquid, for instance, increases entropy as the solid's structured lattice breaks down into a more disordered solution. This concept helps us understand not only chemical reactions but also the fundamental principle that the universe tends toward disorder, making \(\Delta S\) a profoundly philosophical and scientific topic.
Chemical Spontaneity
Chemical spontaneity is a term used to describe whether a chemical reaction or physical process can occur without external intervention. The 'spontaneity' of a process is governed by the Gibbs free energy change (\(\Delta G\)), which combines enthalpy and entropy changes to predict if a process will occur naturally under certain conditions.

It's important to note that 'spontaneous' doesn't mean instant; rather, it means that the reaction is energetically favorable. It's possible for a spontaneous reaction to be slow if the reaction pathway has a high activation energy. A spontaneity calculation doesn't say anything about the speed of the reaction, just its thermodynamic favorability.
Equilibrium Constant
The equilibrium constant, represented by \(K\), is a number that expresses the ratio of the concentrations of products to reactants at equilibrium for a reversible reaction. It provides a concise way to understand the position of equilibrium and thus the extent to which a reaction will proceed under a given set of conditions.

If \(K > 1\), the reaction favors the formation of products, indicating a spontaneous reaction under standard conditions. On the other hand, if \(K < 1\), the reaction favors the reactants, suggesting a non-spontaneous reaction or the need for external energy to drive it forward. It's essential for students to realize that the equilibrium constant is another way to connect the macroscopic properties of a system to the microscopic behavior of molecules in a balanced and predictable way.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use data in Appendix \(\mathrm{C}\) to calculate \(\Delta H^{\circ}, \Delta S^{\circ},\) and \(\Delta G^{\circ}\) at \(25^{\circ} \mathrm{C}\) for each of the following reactions. In each case show that \(\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}\) (a) \(2 \mathrm{Cr}(s)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CrO}_{3}(s)\) (b) \(\mathrm{BaCO}_{3}(s) \longrightarrow \mathrm{BaO}(s)+\mathrm{CO}_{2}(g)\) (c) \(2 \mathrm{P}(s)+10 \mathrm{HF}(g) \longrightarrow 2 \mathrm{PF}_{5}(g)+5 \mathrm{H}_{2}(g)\) (d) \(\mathrm{K}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{KO}_{2}(s)\)

(a) What is the meaning of the standard free-energy change, \(\Delta G^{\circ},\) as compared with \(\Delta G\) ? (b) For any process that occurs at constant temperature and pressure, what is the significance of \(\Delta G=0 ?(c)\) For a certain process, \(\Delta G\) is large and negative. Does this mean that the process necessarily occurs rapidly?

Consider the vaporization of liquid water to steam at a pressure of 1 atm. (a) Is this process endothermic or exothermic? (b) In what temperature range is it a spontaneous process? (c) In what temperature range is it a nonspontaneous process? (d) At what temperature are the two phases in equilibrium?

Indicate whether each of the following statements is true or false. If it is false, correct it. (a) The feasibility of manufacturing \(\mathrm{NH}_{3}\) from \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) depends entirely on the value of \(\Delta H\) for the process \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g) .\) (b) The re- action of \(\mathrm{Na}(s)\) with \(\mathrm{Cl}_{2}(g)\) to form \(\mathrm{NaCl}(s)\) is a spontaneous process. (c) A spontaneous process can in principle be conducted reversibly. (d) Spontaneous processes in general require that work be done to force them to proceed. (e) Spontaneous processes are those that are exothermic and that lead to a higher degree of order in the system.

Use data from Appendix \(\mathrm{C}\) to calculate the equilibrium constant, \(K,\) at \(298 \mathrm{~K}\) for each of the following reactions: (a) \(\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{Hl}(g)\) (b) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g) \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) (c) \(3 \mathrm{C}_{2} \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{6}(g)\)
See all solutions

Recommended explanations on Chemistry Textbooks

Chemistry Branches

Read Explanation

Inorganic Chemistry

Read Explanation

Making Measurements

Read Explanation

Chemical Analysis

Read Explanation

Nuclear Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free