Question

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

Short answer

(a) The standard free-energy change, $\mathrm{\Delta }{G}^{\circ }$, represents the change in free energy under standard conditions (1 atm pressure, 298 K temperature) from the standard state of reactants to the standard state of products. The free-energy change, $\mathrm{\Delta }G$, represents the change in free energy for a process under any conditions and is related to the standard free-energy change by $\mathrm{\Delta }G=\mathrm{\Delta }{G}^{\circ }+RT\ln Q$. (b) For a process at constant temperature and pressure, $\mathrm{\Delta }G=0$ implies that the system is at equilibrium, with the forward and reverse reactions occurring at the same rate and the concentrations of reactants and products remaining constant. (c) A large negative $\mathrm{\Delta }G$ indicates that the process is thermodynamically favorable or spontaneous, but it doesn't necessarily mean that the process occurs rapidly. The reaction rate depends on the activation energy, which can still be high for a spontaneous reaction.

Step-by-step solution

Part (a): Comparison of Standard Free-Energy Change and Free-Energy Change

In thermodynamics, the standard free-energy change, denoted as $\mathrm{\Delta }{G}^{\circ }$, is the change in free energy when a process occurs under standard conditions (1 atm pressure, 298 K temperature) and involves a change from the standard state of reactants to the standard state of products. On the other hand, the free-energy change, denoted as $\mathrm{\Delta }G$, represents the change in free energy for a process occurring under any conditions, not necessarily standard conditions. It is related to the standard free-energy change by the following equation: $$\mathrm{\Delta }G=\mathrm{\Delta }{G}^{\circ }+RT\ln Q$$ where R is the universal gas constant, T is the temperature in Kelvins, and Q is the reaction quotient, which describes the relative amounts of reactants and products present during the reaction.

Part (b): Significance of $\mathrm{\Delta }G=0$ at constant temperature and pressure

For a process that occurs at constant temperature and pressure, the significance of $\mathrm{\Delta }G=0$ lies in the fact that the system is in a state of equilibrium, and the forward and reverse reactions occur at the same rate. When $\mathrm{\Delta }G=0$, the reaction quotient, Q, is equal to the equilibrium constant, K: $$0=\mathrm{\Delta }{G}^{\circ }+RT\ln Q\Rightarrow Q=K$$ This signifies that the concentrations of reactants and products are constant, and the system is in a stable state without further change.

Part (c): Large negative $\mathrm{\Delta }G$ and reaction rate

A large negative value for $\mathrm{\Delta }G$ indicates that the process is thermodynamically favorable or spontaneous, as it results in a decrease in the system's free energy. However, it doesn't necessarily mean that the process occurs rapidly. The reaction rate depends on the activation energy, which is the energy barrier that must be overcome for the reaction to proceed. A process with a large negative $\mathrm{\Delta }G$ can still occur slowly if the activation energy is high. In summary, a large negative $\mathrm{\Delta }G$ value indicates the reaction's spontaneity, but it doesn't provide information about the reaction rate.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy in various systems. In the context of chemical reactions, it particularly addresses the energy changes that occur during the process. One key concept in thermodynamics is free energy, often represented by the symbol $G$. Free energy is a measure of the potential for reversible work when a system at a constant temperature and pressure reaches thermodynamic equilibrium with its environment.

Understanding free energy is crucial as it helps scientists and engineers predict the direction of a chemical reaction and assess the system’s ability to perform work. The change in free energy, denoted $\mathrm{\Delta }G$, helps determine whether a process is spontaneous—taking place without outside intervention—or non-spontaneous. Spontaneity in thermodynamics does not imply anything about the rate of the process, which is a common misconception; it only pertains to the energetic favorability of the process under the given conditions.

Equilibrium Constant

The equilibrium constant, represented by the capital letter $K$, is a critical concept in the study of chemical equilibrium—a state in which the forward and reverse reactions occur at the same rate, resulting in no net change in the concentration of reactants and products. The equilibrium constant is a dimensionless number defining the ratio of the concentrations of products to reactants at equilibrium, each raised to the power of their respective stoichiometric coefficients in the balanced chemical equation.

The value of $K$ is only affected by temperature, and its magnitude provides insight into the position of equilibrium. A large $K$ value suggests that, at equilibrium, products dominate over reactants, whereas a small $K$ implies a reactant-favored position. The relationship between the equilibrium constant and the standard free-energy change, $\text{Extra close brace or missing open brace}$, is described by the equation $\mathrm{\Delta }{G}^{\boxed{{\displaystyle \text{textordmasculine}}}}=-RT\ln K$ where $R$ is the gas constant and $T$ is the temperature in Kelvin. This equation illustrates how the thermodynamic favorability of a reaction and the equilibrium position are interconnected.

Reaction Quotient

The reaction quotient, denoted as $Q$, is a measure similar to the equilibrium constant but applicable at any point during a chemical reaction, not just at equilibrium. It provides a snapshot of the reaction's progress by considering the current concentrations or partial pressures of the reactants and products involved in a reaction.

The reaction quotient is calculated in the same manner as the equilibrium constant—by taking the ratio of the concentrations or pressures of products to reactants—however, these values may not necessarily be at their equilibrium levels. Comparing $Q$ with the equilibrium constant $K$ can predict the direction in which a reaction will proceed to reach equilibrium: if $Q<K$, the reaction will proceed forward to produce more products. Conversely, if $Q>K$ the reaction will shift towards the reactants. When $Q=K$, the system is at equilibrium, and no net reaction occurs.

Standard Conditions

In the study of chemical reactions and thermodynamics, standard conditions refer to a set of fixed reference conditions chosen for convenience, consistency, and comparison. Standard conditions typically include a pressure of 1 atmosphere (atm), a temperature of 298 Kelvin (K), and concentrations of 1 molar (M) for all solutions involved. It’s important to note that standard conditions are not the same as standard temperature and pressure (STP), commonly used in gas law calculations, which is set at 0°C (273.15 K) and 1 atm.

Standard free-energy changes $\mathrm{\Delta }{G}^{\boxed{{\displaystyle \text{textordmasculine}}}}$ are calculated under these conditions and help to compare the thermodynamic favorability of different reactions. When actual conditions differ from the standard, adjustments to the free-energy change can be made using the reaction quotient as seen in the equation $\mathrm{\Delta }G=\mathrm{\Delta }{G}^{\boxed{{\displaystyle \text{textordmasculine}}}}+RT\ln Q$ This accounts for the actual concentrations and pressures of reactants and products, allowing chemists to predict how far a system is from reaching equilibrium and the amount of work a system can do under non-standard conditions.