Question

What is the relationship between the free-energy change under nonstandard- state conditions, \(\Delta G\), the free-energy change under standard-state conditions, \(\Delta G^{\circ}\), and the reaction quotient, \(Q\) ?

Short answer

\( \Delta G = \Delta G^{\circ} + RT \ln Q \) relates the free-energy changes under nonstandard and standard conditions with the reaction quotient \( Q \).

Step-by-step solution

Understanding the Given Relationship

The relationship between free-energy change under nonstandard conditions, \( \Delta G \), and under standard conditions, \( \Delta G^{\circ} \), involves the reaction quotient, \( Q \). This formula connects these quantities to determine the spontaneity and direction of a reaction outside of standard conditions.

Identify the Formula

The formula that relates these quantities is given by:\[\Delta G = \Delta G^{\circ} + RT \ln Q\]where \( R \) is the universal gas constant (8.314 J/mol⋅K), and \( T \) is the temperature in Kelvin.

Explain the Formula Components

In the equation \( \Delta G = \Delta G^{\circ} + RT \ln Q \), \( \Delta G^{\circ} \) represents the free-energy change for a reaction at standard state conditions, \( Q \) is the reaction quotient that reflects the current ratio of products to reactants, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. \( RT \ln Q \) adjusts \( \Delta G^{\circ} \) for nonstandard conditions.

Implications of the Formula

This relationship implies that if \( Q = 1 \), then \( \ln Q = 0 \) and \( \Delta G = \Delta G^{\circ} \). If \( Q < 1 \), \( \ln Q \) is negative, which could make \( \Delta G \) negative, indicating a spontaneous reaction. If \( Q > 1 \), \( \ln Q \) is positive, which may lead to a positive \( \Delta G \), indicating a non-spontaneous reaction.

Key concepts

standard-state conditions

In chemistry, standard-state conditions are a reference point used to compare chemical reactions. It helps in determining properties like free energy change. Under standard-state conditions, substances are at 1 atmosphere of pressure and a specified temperature, usually 25°C (298 K). For solutions, the concentration is 1 M.Standard-state conditions are important because they provide a consistent baseline to measure and compare the thermodynamic properties of substances and reactions. Using these conditions, we can determine the standard free-energy change, \( \Delta G^{\circ} \), a crucial metric for assessing the energy efficiency and spontaneity of a process.

reaction quotient

The reaction quotient, symbolized as \( Q \), is a dimensionless number that quantifies the ratio of product to reactant concentrations at any point in a reaction. Unlike the equilibrium constant, which is only applicable at equilibrium, \( Q \) can be used at any stage of the process to understand how far a reaction has proceeded.The formula for the reaction quotient is:

• For the general reaction \( aA + bB \longrightarrow cC + dD \), \( Q \) is expressed as: \[ Q = \frac{{[C]^c[D]^d}}{{[A]^a[B]^b}} \]

Comparing \( Q \) with the equilibrium constant \( K \):

• When \( Q < K \), the reaction tends to proceed forward, favoring products.
• If \( Q > K \), it proceeds in reverse, favoring reactants.
• When \( Q = K \), the system is at equilibrium.

thermodynamics

Thermodynamics is the branch of physics concerned with heat, energy, and motion. In chemistry, it assesses how energy changes govern the transformations of matter. Key concepts include energy conservation, entropy, and the laws of thermodynamics.

• First Law (Conservation of Energy): Energy cannot be created or destroyed, only transformed.
• Second Law: Entropy, or disorder, increases in an isolated system over time, making energy conversions less efficient.
• Third Law: The entropy of a perfect crystal approaches zero at absolute zero temperature.

In chemical reactions, we use these principles to determine energy changes. The free-energy change, \( \Delta G \), is pivotal, indicating whether a reaction is spontaneous or requires external input.

spontaneity of reactions

The spontaneity of reactions is determined by the Gibbs free energy change, \( \Delta G \). This value tells us if a chemical reaction can occur on its own without needing energy from outside sources.

• A negative \( \Delta G \) indicates a spontaneous reaction, meaning it occurs naturally and releases energy, often in the form of heat.
• If \( \Delta G \) is positive, the reaction is non-spontaneous, requiring energy input to proceed.
• When \( \Delta G \) is zero, the system is at equilibrium, with no net change in reactant and product amounts.

A crucial part of calculating \( \Delta G \) for nonstandard conditions is using the equation \( \Delta G = \Delta G^{\circ} + RT \ln Q \). This takes into account standard-state conditions and the current state as reflected by the reaction quotient, \( Q \). Through this, we can assess how changes in reactant or product concentrations affect a reaction's spontaneity.