Question

Compare the values of \(\Delta G\) and \(\Delta G^{\circ}\) when: (a) \(Q<1\) (b) \(Q=1\) (c) \(Q>1\) Does the thermodynamic tendency for the reaction to occur increase or decrease as \(Q\) increases?

Short answer

(a) \(\Delta G < \Delta G^{\circ}\); (b) \(\Delta G = \Delta G^{\circ}\); (c) \(\Delta G > \Delta G^{\circ}\). As \(Q\) increases, the tendency for the reaction to occur decreases.

Step-by-step solution

Understand the Relationship

The relationship between \(\Delta G\), \(\Delta G^{\circ}\), and the reaction quotient \(Q\) is given by:\[\Delta G = \Delta G^{\circ} + RT \ln Q\]where \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. \(Q\) is the reaction quotient, which is the ratio of product concentrations to reactant concentrations at any point during the reaction.

Analyze for Q < 1

When \(Q < 1\), the natural logarithm \(\ln Q\) is negative. Thus, the term \(RT \ln Q\) is negative, and \(\Delta G\) will be less than \(\Delta G^{\circ}\). This means the reaction has a greater tendency to occur because \(\Delta G\) is more negative than \(\Delta G^{\circ}\).

Analyze for Q = 1

When \(Q = 1\), \(\ln Q = 0\), which means \(RT \ln Q\) is also zero. Therefore, \(\Delta G = \Delta G^{\circ}\). At this point, the reaction is at equilibrium if \(\Delta G^{\circ} = 0\), indicating no net driving force toward products or reactants.

Analyze for Q > 1

When \(Q > 1\), the natural logarithm \(\ln Q\) is positive. Thus, the term \(RT \ln Q\) is positive, and \(\Delta G\) will be greater than \(\Delta G^{\circ}\). This indicates that the reaction has a reduced tendency to proceed towards completion because \(\Delta G\) is less negative or more positive than \(\Delta G^{\circ}\).

Evaluate the Effect of Increasing Q

As \(Q\) increases, \(\ln Q\) becomes more positive, making \(\Delta G\) larger (less negative or more positive). This decreases the thermodynamic tendency for the reaction to occur, shifting the reaction away from completion toward the reactants, implying a lower equilibrium constant or favoring the reverse reaction.

Key concepts

Reaction Quotient

The Reaction Quotient, often denoted as \( Q \), is a measure to determine the direction in which a reaction at any given point is likely to proceed. It is calculated as the ratio of the concentrations (or pressures) of the products to the reactants at any point in the reaction, each raised to the power of their stoichiometric coefficients. This is similar to the equilibrium constant \( K \), but unlike \( K \), \( Q \) is calculated even when the system might not be at equilibrium yet. For instance, consider a simple reaction: \[ aA + bB \rightleftharpoons cC + dD \] The reaction quotient \( Q \) would be: \[ Q = \frac{{[C]^c[D]^d}}{{[A]^a[B]^b}} \]

• **If \( Q < K \):** The reaction will proceed forward, converting reactants into products.
• **If \( Q = K \):** The reaction is at equilibrium, and no net change occurs.
• **If \( Q > K \):** The reaction will proceed in reverse, converting products back to reactants.

Understanding \( Q \) helps in predicting how changes in concentration, pressure, or temperature affect the system dynamics.

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state where macroscopic properties such as pressure, temperature, and concentration remain constant over time, meaning there is no net change in the system. In this state, a reaction reaches a balance, meaning the rate of the forward reaction equals the rate of the reverse reaction. At equilibrium, the Gibbs Free Energy change \( \Delta G \) is zero, indicating no driving force for the reaction in either direction. This can be represented mathematically as: \[ \Delta G = \Delta G^{\circ} + RT \ln Q = 0 \] At equilibrium, \( Q \) becomes the equilibrium constant \( K \). Thus, \[ 0 = \Delta G^{\circ} + RT \ln K \] This is an important concept in thermodynamics as it explains why some reactions go to completion while others do not. Predicting whether a reaction mixture will reach equilibrium and in which direction it will proceed is crucial for chemical processes.

Spontaneity of Reactions

Spontaneity of a chemical reaction is determined by the Gibbs Free Energy change \( \Delta G \). A reaction is spontaneous if \( \Delta G < 0 \); it means the process can proceed without any external energy input. Conversely, if \( \Delta G > 0 \), the reaction is non-spontaneous and requires energy to proceed. The formula \( \Delta G = \Delta G^{\circ} + RT \ln Q \) helps predict spontaneity in varying conditions:

• When \( Q < 1 \), \( \Delta G \) is less than \( \Delta G^{\circ} \), often resulting in a spontaneous reaction.
• When \( Q = 1 \), \( \Delta G \) equals \( \Delta G^{\circ} \); the reaction is at a state of balance when \( \Delta G^{\circ} \) is zero.
• When \( Q > 1 \), \( \Delta G \) exceeds \( \Delta G^{\circ} \), typically non-spontaneous unless conditions favor reversibility back to equilibrium.

These principles underscore why energy changes in reactions are central to the understanding of chemical dynamics.

Equilibrium Constant

The Equilibrium Constant (\( K \)) defines the ratio of product concentrations to reactant concentrations at equilibrium, indicating the position of equilibrium in a reaction. It is a dimensionless number derived from the reaction quotient \( Q \) when the system has reached equilibrium. The formula used is: \[ K = \frac{{[C]^c[D]^d}}{{[A]^a[B]^b}} \] The value of \( K \) provides insight into the extent of a reaction:

• **If \( K > 1 \):** Products are favored; the reaction proceeds to mostly products.
• **If \( K = 1 \):** Neither reactants nor products are favored; significant amounts of both exist at equilibrium.
• **If \( K < 1 \):** Reactants are favored; the reaction forms more reactants.

The relationship between \( \Delta G^{\circ} \) and \( K \) is given by \[ \Delta G^{\circ} = -RT \ln K \] where a negative \( \Delta G^{\circ} \) signifies a reaction that favors products at equilibrium. Overall, the equilibrium constant is crucial for understanding reaction tendencies and designing chemical processes.