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Chapter 17: Problem 18

Give the relation between the change in free energy for a chemical reaction and the (a) equilibrium constant. (b) maximum useful work that can be obtained. (c) free energy when reactants and products are present under standard conditions.

Short Answer

Expert verified
(a) \( \Delta G = -RT \ln K \); (b) \( \Delta G = W_{max} \); (c) \( \Delta G^0 = -RT \ln K^0 \).

Step by step solution

01

Understanding Free Energy and Equilibrium Constant

The change in free energy, \( \Delta G \), for a chemical reaction is related to the equilibrium constant, \( K \), by the equation: \[ \Delta G = -RT \ln K \] where \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. This equation shows that the free energy change is directly related to how far the reaction is from equilibrium (determined by \( K \)).
02

Relating Free Energy and Maximum Useful Work

The change in free energy, \( \Delta G \), of a process is equivalent to the maximum useful work, \( W_{max} \), that can be obtained from a process under constant temperature and pressure. Therefore, \[ \Delta G = W_{max} \] This relationship shows that the free energy change tells us the maximum amount of energy that can be harnessed to perform work.
03

Free Energy under Standard Conditions

Under standard conditions, the change in free energy, \( \Delta G^0 \), is used to describe the reaction when all reactants and products are at standard states (1 bar for gases, 1 M for solutions). \( \Delta G^0 \) is related to the equilibrium constant by: \[ \Delta G^0 = -RT \ln K ^0 \] where \( K^0 \) is the equilibrium constant under standard conditions. This indicates how favorable a reaction is under these standard settings.

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Key Concepts

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

The Role of Equilibrium Constant in Free Energy
The equilibrium constant, represented as \( K \), is crucial in understanding the direction and extent of chemical reactions. It quantifies the ratio of the concentration of the products to the reactants at equilibrium.The change in free energy, \( \Delta G \), relates to the equilibrium constant through the equation \( \Delta G = -RT \ln K \). Here:
  • \( R \) is the universal gas constant.
  • \( T \) is the temperature in Kelvin.
This equation can tell us whether a reaction is spontaneous:
  • If \( \Delta G < 0 \), the reaction proceeds in the forward direction spontaneously.
  • If \( \Delta G > 0 \), the reverse reaction is favored.
  • \( \Delta G = 0 \) indicates the reaction is at equilibrium and no net change occurs.
This relationship highlights how the value of the equilibrium constant determines where the balance of the reaction lies.
Understanding Maximum Useful Work
Free energy change \( (\Delta G) \) plays a pivotal role in determining the maximum useful work a chemical process can perform. This relationship is neatly captured in the equation \( \Delta G = W_{max} \). Here:
  • \( W_{max} \) represents the maximum work that can be done when a reaction takes place under constant temperature and pressure.
The concept of maximum useful work is vital for understanding energy efficiency in systems like batteries and engines:
  • It denotes the optimal energy output that can be harnessed from a reaction.
  • The closer the actual energy extracted from a reaction is to this theoretical maximum, the more efficient the process.
Thus, \( \Delta G \) is not just a measure of spontaneity but also of energy potential in practical applications.
Free Energy and Standard Conditions
Free energy under standard conditions, labeled as \( \Delta G^0 \), provides a baseline measure for comparing different reactions. Standard conditions mean that all reactants and products are at a specified state:
  • 1 bar pressure for gases
  • 1 M concentration for solutions
The equation \( \Delta G^0 = -RT \ln K^0 \) demonstrates how the equilibrium constant at standard conditions \( (K^0) \) correlates with free energy.This standard state indicates:
  • The inherent "driving force" of a reaction at these conditions.
  • Whether a reaction tends to form products or remains with reactants.
When \( \Delta G^0 \) is negative, the reaction is product-favoring; a positive \( \Delta G^0 \) suggests reactants are preferred. Understanding these conditions helps predict the behavior of reactions under practical, standard settings.

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Most popular questions from this chapter

When ice forms from liquid water at \(0^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure, \(\Delta G\) is zero, because the change takes place under equilibrium conditions. Explain what the signs on the enthalpy change and entropy change must be.

For Exercises 17.87 to 17.102 , assume that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not change with temperature. For each reaction, an equilibrium constant at \(298 \mathrm{~K}\) is given. Calculate \(\Delta G^{\circ}\) for each reaction. (a) \(\mathrm{Br}_{2}(\ell)+\mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{HBr}(\mathrm{g}) \quad K_{\mathrm{P}}=4.4 \times 10^{18}\) (b) \(\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \quad K_{\mathrm{P}}=3.17 \times 10^{-2}\) (c) \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g}) \quad K_{\mathrm{c}}=3.5 \times 10^{8}\)

For each process, tell whether the entropy change of the system is positive or negative. (a) A glassblower heats glass (the system) to its softening temperature. (b) A teaspoon of sugar dissolves in a cup of coffee. (The system consists of both sugar and coffee.) (c) Calcium carbonate precipitates out of water in a cave to form stalactites and stalagmites. (Consider only the calcium carbonate to be the system.)

State the first law of thermodynamics in words and in equation form. Define all symbols used in the equation.

For a process, \(w=-987 \mathrm{~J}\) and \(q=555 \mathrm{~J}\). What is \(\Delta E\) for this process?
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