Gibbs Free Energy Example Problems

Gibbs Free Energy: Understanding and Solving Example Problems

Gibbs Free Energy (ΔG), a thermodynamic potential, predicts the spontaneity of a chemical or physical process at constant temperature and pressure. Understanding Gibbs Free Energy is crucial in various fields, from chemistry and physics to biology and engineering. This article delves into the concept of Gibbs Free Energy, provides detailed explanations, and walks you through several example problems of varying complexities, equipping you with a solid understanding of this fundamental thermodynamic principle. We'll cover various applications and address frequently asked questions, ensuring a comprehensive learning experience.

Understanding Gibbs Free Energy: A Recap

Gibbs Free Energy is defined as the maximum amount of reversible work that may be performed by a thermodynamic system at a constant temperature and pressure. It's calculated using the equation:

ΔG = ΔH - TΔS

Where:

• ΔG represents the change in Gibbs Free Energy (in Joules or Kilojoules). A negative ΔG indicates a spontaneous process (exergonic), while a positive ΔG indicates a non-spontaneous process (endergonic). A ΔG of zero suggests the system is at equilibrium.
• ΔH represents the change in enthalpy (heat content) of the system. A negative ΔH indicates an exothermic reaction (heat released), while a positive ΔH indicates an endothermic reaction (heat absorbed).
• T represents the absolute temperature (in Kelvin).
• ΔS represents the change in entropy (disorder) of the system. A positive ΔS indicates an increase in disorder, while a negative ΔS indicates a decrease in disorder.

The interplay between enthalpy and entropy determines the spontaneity of a reaction. A reaction can be spontaneous even if it's endothermic (positive ΔH) if the increase in entropy (positive ΔS) is sufficiently large to overcome the positive enthalpy change. Conversely, a reaction can be non-spontaneous even if it's exothermic (negative ΔH) if the decrease in entropy (negative ΔS) is significant enough.

Example Problems: Gibbs Free Energy Calculations

Let's work through several example problems to solidify our understanding.

Example 1: Simple Calculation

A reaction has a ΔH of -50 kJ/mol and a ΔS of +100 J/mol·K at 298 K. Calculate the Gibbs Free Energy change (ΔG) and determine if the reaction is spontaneous.

Solution:

First, ensure consistent units. Convert ΔS to kJ/mol·K: 100 J/mol·K * (1 kJ/1000 J) = 0.1 kJ/mol·K

Now, apply the Gibbs Free Energy equation:

ΔG = ΔH - TΔS ΔG = -50 kJ/mol - (298 K)(0.1 kJ/mol·K) ΔG = -50 kJ/mol - 29.8 kJ/mol ΔG = -79.8 kJ/mol

Since ΔG is negative, the reaction is spontaneous at 298 K.

Example 2: Non-Spontaneous Reaction

A reaction has a ΔH of +25 kJ/mol and a ΔS of -50 J/mol·K at 350 K. Calculate ΔG and determine spontaneity.

Solution:

Convert ΔS to kJ/mol·K: -50 J/mol·K * (1 kJ/1000 J) = -0.05 kJ/mol·K

ΔG = ΔH - TΔS ΔG = +25 kJ/mol - (350 K)(-0.05 kJ/mol·K) ΔG = +25 kJ/mol + 17.5 kJ/mol ΔG = +42.5 kJ/mol

Since ΔG is positive, the reaction is non-spontaneous at 350 K.

Example 3: Equilibrium

At what temperature will the following reaction be at equilibrium? ΔH = +10 kJ/mol and ΔS = +50 J/mol·K.

Solution:

At equilibrium, ΔG = 0. Therefore:

0 = ΔH - TΔS

Solve for T:

T = ΔH / ΔS

First convert ΔS: 50 J/mol·K * (1 kJ/1000 J) = 0.05 kJ/mol·K

T = (10 kJ/mol) / (0.05 kJ/mol·K) T = 200 K

The reaction will be at equilibrium at 200 K.

Example 4: Phase Transition

Calculate the Gibbs Free Energy change for the vaporization of water at 100°C (373 K). The enthalpy of vaporization (ΔHvap) is +40.7 kJ/mol, and the entropy of vaporization (ΔSvap) is +109 J/mol·K.

Solution:

Convert ΔSvap to kJ/mol·K: 109 J/mol·K * (1 kJ/1000 J) = 0.109 kJ/mol·K

ΔG = ΔHvap - TΔSvap ΔG = +40.7 kJ/mol - (373 K)(0.109 kJ/mol·K) ΔG = +40.7 kJ/mol - 40.7 kJ/mol ΔG = 0 kJ/mol

At the boiling point, the Gibbs Free Energy change for a phase transition is zero, indicating equilibrium between the liquid and gaseous phases.

Example 5: Reaction Quotient (Q) and Equilibrium Constant (K)

The relationship between Gibbs Free Energy, the reaction quotient (Q), and the equilibrium constant (K) is given by:

ΔG = -RTln(K/Q)

Where:

• R is the ideal gas constant (8.314 J/mol·K)
• T is the temperature in Kelvin
• K is the equilibrium constant
• Q is the reaction quotient

Let's consider a reaction: A + B <=> C. At 298 K, K = 10 and Q = 1. Calculate ΔG.

Solution:

ΔG = -RTln(K/Q) ΔG = -(8.314 J/mol·K)(298 K)ln(10/1) ΔG = -5705 J/mol or -5.7 kJ/mol

Since ΔG is negative, the reaction will proceed spontaneously towards the products to reach equilibrium.

Advanced Applications of Gibbs Free Energy

Gibbs Free Energy is not just a theoretical concept; it finds wide application in various scientific and engineering disciplines.

• Electrochemistry: Gibbs Free Energy is directly related to the cell potential (E) of an electrochemical cell: ΔG = -nFE, where n is the number of moles of electrons transferred and F is Faraday's constant. This allows us to predict the spontaneity of redox reactions.

• Chemical Equilibrium: The equilibrium constant (K) is related to the standard Gibbs Free Energy change (ΔG°) by the equation: ΔG° = -RTlnK. This allows us to calculate the equilibrium constant from thermodynamic data and vice versa.

• Biochemistry: Gibbs Free Energy is essential for understanding metabolic processes in living organisms. The free energy released during ATP hydrolysis drives many biological reactions.

• Materials Science: Gibbs Free Energy helps predict the stability of different phases of materials at various temperatures and pressures. This is crucial for understanding phase diagrams and material selection.

Frequently Asked Questions (FAQ)

Q: What are the limitations of using Gibbs Free Energy?

A: While Gibbs Free Energy is a powerful tool, it has limitations. It assumes constant temperature and pressure, which may not always be the case in real-world scenarios. It also doesn't provide information about the rate of a reaction, only its spontaneity. A reaction might be thermodynamically favorable (negative ΔG) but kinetically slow (high activation energy).

Q: How does Gibbs Free Energy differ from enthalpy and entropy?

A: Enthalpy (ΔH) measures the heat content of a system, while entropy (ΔS) measures the disorder or randomness. Gibbs Free Energy (ΔG) combines both enthalpy and entropy to predict the spontaneity of a process under constant temperature and pressure conditions. It provides a more comprehensive picture of a system's tendency to change.

Q: Can a reaction be spontaneous at one temperature but non-spontaneous at another?

A: Yes, absolutely. The spontaneity of a reaction depends on the temperature-dependent relationship between enthalpy and entropy. A reaction might be spontaneous at high temperatures (where the TΔS term dominates) but non-spontaneous at low temperatures (where the ΔH term is more influential).

Conclusion

Gibbs Free Energy is a cornerstone of thermodynamics, providing invaluable insight into the spontaneity of chemical and physical processes. By understanding the fundamental equation and its applications, we can predict whether a reaction will proceed spontaneously under given conditions. Through the examples provided, we've explored different scenarios, showcasing the practical applications and limitations of this essential thermodynamic concept. Remember that mastering Gibbs Free Energy requires practice and problem-solving. Continue to work through various examples, and you'll develop a strong understanding of this crucial tool in the world of science and engineering.