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(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
The change in Gibbs free energy for a constant temperature process can be expressed as \(ΔG = ΔH - TΔS\). When ΔG is positive for a process occurring at constant T and P, it indicates a non-spontaneous process, meaning external input is required. However, ΔG does not directly give information about the rate of a process, acting only as a predictor of spontaneity. To relate ΔG to the rate of a reaction, the Arrhenius equation is used, considering activation energy and other factors.

Step by step solution

01

a. Change in Gibbs free energy

The Gibbs free energy is given by the formula: \(G = H - TS\) Here, G is the Gibbs free energy, H is the enthalpy, T is the temperature and S is the entropy. When we talk about the change in these quantities, we represent them using a delta symbol: ΔG, ΔH, and ΔS. To express the change in G in terms of changes in H and S, we look at the equation: \(ΔG = ΔH - TΔS\) This equation comes from calculating ΔG as: \(ΔG = G_{final} - G_{initial} = (H_{final} - T_{final}S_{final}) - (H_{initial} - T_{initial}S_{initial})\) The process occurs at constant temperature, so \(T_{final} = T_{initial} = T\). Therefore, we get: \(ΔG = ΔH - TΔS\)
02

b. Conclusion of positive ΔG

Given a certain process that occurs at constant T and P, we have a positive ΔG value. Looking at the equation we derived earlier: \(ΔG = ΔH - TΔS\) When ΔG is positive, we have a non-spontaneous process. In other words, the process will not occur spontaneously and requires external input, such as energy or a driving force.
03

c. Relationship between ΔG and rate

The relationship between ΔG and the rate of a process indicates whether the process is spontaneous or not. ΔG does not directly provide information about the speed or rate at which a process occurs. Instead, it dictates the direction in which the process occurs under specified conditions. However, to establish an actual connection between a positive or negative ΔG and the rate of a reaction, we would use the Arrhenius equation for the rate constant, which considers activation energy and other factors. In summary, ΔG is useful for predicting the spontaneity of a process but does not have a direct relationship to the rate of the process.

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Key Concepts

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

Enthalpy Change
Enthalpy change, denoted as \(\Delta H\), is a measure of the total heat content of a system during a process at constant pressure. In chemistry, it is an important aspect as it helps determine how much energy is absorbed or released during a chemical reaction. An endothermic reaction, where heat is absorbed, yields a positive \(\Delta H\), whereas an exothermic reaction, which releases heat, has a negative \(\Delta H\). Understanding \(\Delta H\) offers insight into the energetics of reactions, and when combined with entropy change, it can reveal whether a process is likely to be spontaneous.

For instance, when solving for Gibbs free energy, the equation \(\Delta G = \Delta H - T\Delta S\) shows that the enthalpy change is a critical factor in determining the direction and extent of a chemical reaction.
Entropy Change
Entropy change, represented by \(\Delta S\), is a key concept in thermodynamics, quantifying the change in disorder or randomness of the particles within the system. Entropy is a fundamental principle that helps us understand the second law of thermodynamics: in any natural process, the total entropy of a system and its surroundings tends to increase.

An increase in entropy (\(\Delta S > 0\)) means the system becomes more disordered, while a decrease (\(\Delta S < 0\)) indicates a shift toward order. This concept becomes crucial when assessing reaction spontaneity; a positive \(\Delta S\) favors the spontaneity of a process. In the Gibbs free energy equation \(\Delta G = \Delta H - T\Delta S\), the entropy change plays a balancing role alongside enthalpy changes to determine the feasibility of a process under constant temperature and pressure.
Thermodynamics
Thermodynamics is a branch of physics that studies heat, work, energy, and the transformations between them. It encompasses several fundamental laws, including the relationship between Gibbs free energy, enthalpy, and entropy changes, which are pivotal for understanding chemical reactions and processes.

Thermodynamic principles guide us through predicting whether a reaction will occur, designing processes and engines, and even probing the fate of the universe. By mastering these principles, students gain a powerful toolkit for solving problems across physical sciences and engineering disciplines.
Spontaneous Process
A spontaneous process is one that occurs naturally under a given set of conditions without external influence or intervention. Spontaneity is driven by the system's tendency to achieve a state of equilibrium and lower its Gibbs free energy.

Whether a process is spontaneous or not can be predicted by calculating the Gibbs free energy change (\(\Delta G\)). If \(\Delta G < 0\), the process is spontaneous; if \(\Delta G > 0\), it is non-spontaneous and requires external energy or a catalyst to proceed. This distinction is vital for students to forecast the natural direction of chemical reactions and physical changes.
Reaction Spontaneity
Reaction spontaneity involves determining the likelihood of a chemical reaction occurring without external forces. It directly relates to the Gibbs free energy change (\(\Delta G\)). A negative \(\Delta G\) signifies that a reaction is thermodynamically favorable and can occur spontaneously, usually resulting in a stable product under specified conditions.

However, even if the \(\Delta G\) for a reaction is negative, this does not imply the reaction will occur rapidly. The kinetic aspects, such as the energy barrier (activation energy), and concentration of reactants can significantly influence the rate of the reaction. Thus, students must distinguish between the thermodynamic potential and kinetic factors governing a reaction.
Chemical Equilibrium
Chemical equilibrium is the state of a reaction where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. It is a dynamic state, as reactions continue to occur, but the macroscopic properties remain constant.

In terms of Gibbs free energy, equilibrium corresponds to a state where \(\Delta G = 0\). At this point, the system's free energy is at its minimum, and no further spontaneous change is possible unless the conditions of the system change. Understanding the concept of chemical equilibrium aids students in determining the conditions required to optimize reactions in both laboratory and industrial settings.

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Most popular questions from this chapter

For a certain chemical reaction, \(\Delta H^{\circ}=-35.4 \mathrm{~kJ}\) and \(\Delta S^{\circ}=-85.5 \mathrm{~J} / \mathrm{K} .(\mathrm{a}) \mathrm{ls}\) the reaction exothermic or en- dothermic? (b) Does the reaction lead to an increase or decrease in the randomness or disorder of the system? (c) Calculate \(\Delta G^{\circ}\) for the reaction at \(298 \mathrm{~K}\). (d) Is the reaction spontaneous at \(298 \mathrm{~K}\) under standard conditions?

Acetylene gas, \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\), is used in welding. (a) Write a balanced equation for the combustion of acetylene gas to \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{I})\). (b) How much heat is produced in burning \(1 \mathrm{~mol}\) of \(\mathrm{C}_{2} \mathrm{H}_{2}\) under standard conditions if both reactants and products are brought to \(298 \mathrm{~K} ?(\mathrm{c})\) What is the maximum amount of useful work that can be accomplished under standard conditions by this reaction?

One way to derive Equation \(19.3\) depends on the observation that at constant \(T\) the number of ways, \(W\), of arranging \(m\) ideal-gas particles in a volume \(V\) is proportional to the volume raised to the \(m\) power: $$ W \propto V^{m} $$ Use this relationship and Boltzmann's relationship between entropy and number of arrangements (Equation 19.5) to derive the equation for the entropy change for the isothermal expansion or compression of \(n\) moles of an ideal gas.

The standard entropies at \(298 \mathrm{~K}\) for certain of the group \(4 \mathrm{~A}\) elements are as follows: \(\mathrm{C}(s\), diamond \()=2.43 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ; \quad \mathrm{Si}(s)=18.81 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ;\) \(\mathrm{Ge}(s)=31.09 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ; \quad\) and \(\quad \mathrm{Sn}(s)=51.18 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\) All but Sn have the diamond structure. How do you account for the trend in the \(S^{\circ}\) values?

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