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Chapter 19: Problem 53

(a) For a process that occurs at constant temperature, does the change in Gibbs free energy depend on 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. Is the process spontaneous? (c) If \(\Delta G\) for a process is large, is the rate at which it occurs fast?

Short Answer

Expert verified
(a) The change in Gibbs free energy at constant temperature depends on both the enthalpy change (\(\Delta H\)) and entropy change (\(\Delta S\)) of the system, as given by the equation \(\Delta G = \Delta H - T\Delta S\). (b) If \(\Delta G\) is positive for a process occurring at constant \(T\) and \(P\), the process is non-spontaneous. (c) A large value of \(\Delta G\) does not directly imply a fast reaction rate, as the rate depends on activation energy (\(E_a\)) and the Arrhenius equation, not the Gibbs free energy change.

Step by step solution

01

(a) Gibbs free energy dependence on enthalpy and entropy change

Using the definition of Gibbs free energy change (\(\Delta G\)) at constant temperature and pressure, we can write the equation: \[ \Delta G = \Delta H - T\Delta S, \] where \(\Delta H\) is the enthalpy change, \(T\) is the temperature, and \(\Delta S\) is the entropy change. From this equation, we can see that the change in Gibbs free energy at constant temperature depends on the changes in both the enthalpy and entropy of the system.
02

(b) Spontaneity of the process

For a process to be spontaneous, \(\Delta G\) must be negative. In this case, \(\Delta G\) is given as positive, which implies that the process is non-spontaneous.
03

(c) Relationship between \(\Delta G\) and reaction rate

The magnitude of \(\Delta G\) is related to the thermodynamic feasibility of a process, so a large value of \(\Delta G\) does not directly imply that the process occurs at a fast rate. It is important to note that the reaction rate is determined by the activation energy (\(E_a\)) and the Arrhenius equation, not the Gibbs free energy change. The large value of \(\Delta G\) could indicate that the process is thermodynamically favorable, but it does not provide information about how fast the process occurs.

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Key Concepts

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

Enthalpy Change
The concept of enthalpy change, represented by \( \Delta H \), is essential for understanding how energy is transferred during chemical reactions.
  • Enthalpy change is the heat absorbed or released at constant pressure.
  • It gives insight into whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).
The formula for Gibbs free energy \( (\Delta G) \) highlights the role of enthalpy:
\[ \Delta G = \Delta H - T\Delta S, \]where \( T \) is the temperature, and \( \Delta S \) is the entropy change.
Understanding \( \Delta H \) helps determine how much heat flows in or out during a reaction, influencing how a system's energy evolves.
Entropy Change
Entropy change, denoted by \( \Delta S \), represents the degree of disorder or randomness in a system.
  • Increasing entropy usually means the system becomes more disordered.
  • It is a crucial part of predicting reaction spontaneity.
Entropy impacts Gibbs free energy as shown in the equation:
\[ \Delta G = \Delta H - T\Delta S. \]Higher entropy change \( (\Delta S) \) reduces \( \Delta G \), making reactions potentially more spontaneous.
By understanding \( \Delta S \), we gain insight into how a reaction's direction and extent might be influenced by disorder.
Reaction Spontaneity
Understanding whether a reaction occurs spontaneously is pivotal in chemistry.
  • A reaction is spontaneous when \( \Delta G \) is negative.
  • Spontaneity doesn't imply speed; it means the reaction can occur under specified conditions.
The equation \( \Delta G = \Delta H - T\Delta S \) guides us:
- Negative \( \Delta G \): The process is spontaneous.
- Positive \( \Delta G \): The process is non-spontaneous, requiring external energy.
Evaluating spontaneity helps chemists predict whether a reaction will happen naturally.
Reaction Rate
The reaction rate is different from reaction spontaneity. It's about how fast a reaction proceeds.
  • Rate depends on factors like activation energy \( (E_a) \) and temperature.
  • Gibbs free energy \( (\Delta G) \) isn't directly linked to speed.
Even if a process is spontaneous \((\Delta G < 0)\), it could still occur slowly.
The Arrhenius equation describes how activation energy affects rate:
\[ k = A e^{-E_a/RT}, \]where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( R \) is the gas constant, and \( T \) is temperature.
Understanding reaction rate is crucial for controlling processes in industrial and laboratory settings.

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Most popular questions from this chapter

The normal boiling point of \(\mathrm{Br}_{2}(l)\) is \(58.8^{\circ} \mathrm{C},\) and its molar enthalpy of vaporization is \(\Delta H_{\mathrm{vap}}=29.6 \mathrm{kJ} / \mathrm{mol}\) (a) When \(\mathrm{Br}_{2}(l)\) boils at its normal boiling point, does its entropy increase or decrease? (b) Calculate the value of \(\Delta S\) when 1.00 mol of \(\mathrm{Br}_{2}(l)\) is vaporized at \(58.8^{\circ} \mathrm{C}\) .

Would each of the following changes increase, decrease, or have no effect on the number of microstates available to a system: (a) increase in temperature, (b) decrease in volume, (c) change of state from liquid to gas?

A system goes from state 1 to state 2 and back to state \(1 .\) (a) Is \(\Delta E\) the same in magnitude for both the forward and reverse processes? (b) Without further information, can you conclude that the amount of heat transferred to the system as it goes from state 1 to state 2 is the same or different as compared to that upon going from state 2 back to state 1\(?(\mathbf{c})\) Suppose the changes in state are reversible processes. Is the work done by the system upon going from state 1 to state 2 the same or different as compared to that upon going from state 2 back to state 1\(?\)

(a) What do you expect for the sign of \(\Delta S\) in a chemical reaction in which 2 mol of gaseous reactants are converted to 3 mol of gaseous products? (b) For which of the processes in Exercise 19.11 does the entropy of the system increase?

Using \(S^{\circ}\) values from Appendix \(\mathrm{C},\) calculate \(\Delta S^{\circ}\) values for the following reactions. In each case, account for the sign of \(\Delta S^{\circ} .\) $$ \begin{array}{l}{\text { (a) } \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g)} \\ {\text { (b) } \mathrm{N}_{2} \mathrm{O}_{4}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)} \\ {\text { (c) } \mathrm{Be}(\mathrm{OH})_{2}(s) \longrightarrow \mathrm{BeO}(s)+\mathrm{H}_{2} \mathrm{O}(g)} \\ {\text { (d) } 2 \mathrm{CH}_{3} \mathrm{OH}(g)+3 \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)}\end{array} $$
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