Question

The Gibbs free energy is defined as: (a) \(\mathrm{G}=\mathrm{H}-\mathrm{T} \cdot \mathrm{S}\) (b) \(\mathrm{G}=\mathrm{H}+\mathrm{T} . \mathrm{S}\) (c) \(\mathrm{G}=\mathrm{E}-\mathrm{T} . \mathrm{S}\) (d) \(\mathrm{G}=\mathrm{E}+\mathrm{T} \cdot \mathrm{S}\)

Short answer

The correct formula is (a) \( G = H - T \cdot S \).

Step-by-step solution

Understand the Formula for Gibbs Free Energy

Gibbs free energy is a thermodynamic potential that is defined for a system at constant temperature and pressure. The correct formula for Gibbs free energy is given by: \[ G = H - T \cdot S \]where \( G \) is the Gibbs free energy, \( H \) is the enthalpy, \( T \) is the temperature, and \( S \) is the entropy of the system.

Identify and Compare the Options

We need to identify which option correctly matches the standard formula for Gibbs free energy. Option (a) is:\[ \mathrm{G} = \mathrm{H} - \mathrm{T} \cdot \mathrm{S} \] This matches our known formula. Option (b) and (d) have a plus sign instead of the correct minus sign, while option (c) uses \( E \) (internal energy) instead of \( H \) (enthalpy), which is incorrect under standard conditions.

Key concepts

Thermodynamic Potential

Thermodynamic potential is a fundamental concept in thermodynamics that guides how a system can perform work or undergo transformations. Gibbs free energy is a type of thermodynamic potential that precisely measures the maximum work obtainable from a thermodynamic system at constant temperature and pressure. Understanding this potential allows scientists and engineers to predict the direction of chemical reactions and phase transitions.
Gibbs free energy can be used to determine whether a process is spontaneous. If the change in Gibbs free energy is negative, the process is spontaneous and can proceed without external energy. Conversely, a positive change implies that the process requires energy input to occur. By studying the thermodynamic potential, we can understand and control processes such as chemical reactions, industrial operations, and even biological systems.

Enthalpy

Enthalpy, represented by the symbol \( H \), is a measure of the total energy of a system. This includes the internal energy as well as the product of its pressure and volume. In simpler terms, enthalpy accounts for both the energy contained within the molecules and the energy associated with the system's volume and pressure.
Enthalpy is crucial in calculations involving heat transfer at constant pressure. In reactions and processes, it tells us how much heat is being absorbed or released. A negative change in enthalpy means that the reaction releases heat (exothermic), while a positive change indicates heat absorption (endothermic). Understanding enthalpy helps us anticipate the energy balance and heat flow in various systems, an essential component for the design and analysis of chemical processes.

Entropy

Entropy is a fundamental concept that quantifies the degree of disorder or randomness in a system. Represented by the symbol \( S \), entropy is often associated with the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time.
Entropy helps us predict the feasibility of a reaction or process. Generally, natural processes tend to move towards a state of higher entropy. In terms of Gibbs free energy, entropy plays a vital role in determining whether a reaction can happen spontaneously:

• If the entropy change is positive, it can contribute to a decrease in Gibbs free energy, promoting spontaneity.
• A negative entropy change might require compensation by other factors to maintain spontaneity.

Understanding entropy allows us to foresee how systems evolve and transform from ordered states to more disordered ones.

Temperature

Temperature is a measure of the thermal energy within a system and influences various thermodynamic properties, including Gibbs free energy. When considering Gibbs energy, the temperature is pivotal as it directly influences the entropy term in the equation \( G = H - T \cdot S \).
Temperature links the energetic and entropic components in the formula. When temperature is high, the term \( T \cdot S \) becomes more significant, meaning entropy's effect on Gibbs energy is larger. This connection explains why certain processes become more or less favorable as the temperature changes.

• At high temperatures, even small increases in entropy can drive a reaction forward.
• Conversely, at low temperatures, entropy's influence diminishes, and enthalpy becomes comparatively more influential.

Understanding how temperature impacts Gibbs free energy helps predict how changes in thermal conditions affect reaction spontaneity and equilibria in systems.