Question

(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

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

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: \[\Delta G = \Delta H - T \Delta S\]

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.

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 \(\Delta G\).

Key concepts

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.