Problem 19 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

For a general reaction \\[ \alpha A+\beta B \rightleftharpoons \gamma C+\delta D \\] derive the relationship between the Gibbs energy change of the reaction and the reaction quotient. Use the relationship to show that \(\left.\Delta_{r} G^{\ominus}=-R T \ln K . \text { (Section } 15.3\right)\)

Step-by-Step Solution

Answer

The relationship derived is \( \Delta_{r}G^{\ominus} = -RT \ln K \).

Step 1: Understand the Concept of Gibbs Energy Change

For any chemical reaction, the change in Gibbs energy (94_{r}G) at any state is given by a function of the Gibbs energy under standard conditions (94_{r}G^{0}) and the reaction quotient (Q). This is represented by the equation:\[ 94_{r}G = 94_{r}G^{0} + RT \, ext{ln} \, Q \]where R is the gas constant and T is the temperature in Kelvin.

Step 2: Define the Reaction Quotient

The reaction quotient (Q) is a dimensionless number that gives the ratio of the concentrations of products to reactants at any point during a reaction, raised to the power of their stoichiometric coefficients. For the given reaction:\[ Q = \frac{[C]^{3} [D]^{4}}{[A]^{1} [B]^{2}} \]where [C], [D], [A], and [B] are the concentrations of the respective species.

Step 3: Relate Gibbs Energy Change to Reaction Quotient at Equilibrium

At equilibrium, 94_{r}G = 0, and the reaction quotient (Q) becomes the equilibrium constant (K). Substituting these values into the equation gives:\[ 0 = 94_{r}G^{0} + RT \, ext{ln} \, K \]

Step 4: Solve for Standard Gibbs Energy Change

Rearrange the equation to express 94_{r}G^{0} in terms of the equilibrium constant K:\[ 94_{r}G^{0} = -RT \, ext{ln} \, K \]This is the required relationship that shows how the standard Gibbs energy change is related to the equilibrium constant.

Key Concepts

Reaction Quotient

The reaction quotient, often symbolized as \( Q \), provides an insight into the state of a chemical reaction at any given point. It is calculated as the ratio of the raised concentrations of products over reactants, which reflects the progress of the reaction. Mathematically, for a reaction \( \alpha A + \beta B \rightleftharpoons \gamma C + \delta D \), the reaction quotient is given by:
\[Q = \frac{[C]^{\gamma} [D]^{\delta}}{[A]^{\alpha} [B]^{\beta}}\]Here, \([C]\), \([D]\), \([A]\), and \([B]\) are the concentrations of the substances involved. The exponents \( \gamma, \delta, \alpha, \) and \( \beta \) are the stoichiometric coefficients of the reaction. This ratio tells us whether a reaction is moving towards the products or reactants, depending on how \( Q \) compares with the equilibrium constant, \( K \).

• If \( Q < K \), the reaction will proceed forward, making more products.
• If \( Q > K \), the reaction will move in reverse, producing more reactants.
• If \( Q = K \), the system is at equilibrium and no shift occurs.

Equilibrium Constant

The equilibrium constant \( K \) is a fundamental concept in chemical reactions that signifies the point at which a reaction reaches equilibrium. At equilibrium, the forward and reverse reactions occur at the same rate, meaning the concentrations of reactants and products remain constant over time.
The value of \( K \) is determined under the same equation as \( Q \) but at equilibrium conditions:
\[K = \frac{[C]^{\gamma} [D]^{\delta}}{[A]^{\alpha} [B]^{\beta}}\]The equilibrium constant is an indicator of the extent of a reaction at equilibrium:

• A large \( K \) value (>1) implies a greater concentration of products at equilibrium.
• A small \( K \) value (<1) indicates a higher concentration of reactants.
• \( K = 1 \) signifies that neither reactants nor products are favored.

It's important to note that \( K \) values reflect the reaction's nature and depend on the temperature, but remain constant for a given temperature under defined conditions.

Standard Conditions

Standard conditions are specific conditions under which measurements such as Gibbs energy change and equilibrium constants are often made. These conditions provide a universally accepted baseline, making it easier to compare chemical reactions. Standard conditions typically refer to:

• A pressure of 1 atmosphere (atm)
• A temperature of 298.15 K (25°C)
• Concentration of 1 mol/L for solutions

Under these conditions, the change in Gibbs energy is referred to as the standard Gibbs energy change, denoted as \( \Delta_r G^\ominus \). The standard state offers a reference point to calculate how a reaction's free energy varies with changes in concentration or pressure. This is especially important when connecting the Gibbs energy change to the equilibrium constant, as they are related under standard conditions by the equation \( \Delta_r G^\ominus = -RT \ln K \). This relationship serves as an essential derived formula in thermodynamics.

Chemical Reaction

Chemical reactions involve the transformation of reactants into products through the breaking and forming of chemical bonds. They can be represented by balanced equations that show the mole ratios of reactants and products, like \( \alpha A + \beta B \rightleftharpoons \gamma C + \delta D \). Such equations illustrate the stoichiometry of the reaction, which is vital for understanding and calculating various thermodynamic properties.
At a molecular level, reactions occur due to collisions among molecules, with sufficient energy to overcome any activation energy barriers. The rate and extent of a reaction can vary based on factors such as temperature, concentration, and pressure.
Understanding the thermodynamics of reactions involves evaluating parameters like the Gibbs energy change. It predicts the spontaneity of processes:

• A negative \( \Delta_r G \) indicates a spontaneous reaction under given conditions.
• A positive \( \Delta_r G \) suggests nonspontaneity.
• \( \Delta_r G = 0 \) means the system is at equilibrium.

Overall, examining a reaction’s energy dynamics assesses how far a reaction will proceed and predicts whether it will release or require energy.