Question

(a) Write the chemical equations that correspond to $\mathrm{\Delta }{G}_i^9$ for ${\mathrm{CH}}_4(g)$ and for $\mathrm{NaCl}(s).$ (b) For these formation reactions, compare $\mathrm{\Delta }{G}_f^{\circ }$ and $\mathrm{\Delta }{H}_f$. (c) In general, under which condition is $\mathrm{\Delta }G$, more negative (less positive) than $\mathrm{\Delta }{H}_f$ ? (i) When the temperature is high, (ii) when $\mathrm{\Delta }{S}_f^{\circ }$ is positive, (iii) when the reaction is reversible.

Short answer

(a) The chemical equations for the Gibbs free energy of formation for CH₄(g) and NaCl(s) are: C(s) + 2H₂(g) → CH₄(g) Na(s) + 1/2Cl₂(g) → NaCl(s) (b) In both cases, ΔGf° is less negative (more positive) than ΔHf: For CH₄(g): ΔGf° = -50.8 kJ/mol, ΔHf = -74.8 kJ/mol For NaCl(s): ΔGf° = -384.1 kJ/mol, ΔHf = -411.0 kJ/mol (c) ΔG becomes more negative (less positive) than ΔH when the condition (ii) ΔSf° is positive.

Step-by-step solution

(a) Chemical Equations for Gibbs Free Energy of Formation

For CH₄(g) C(s) + 2H₂(g) → CH₄(g) For NaCl(s) Na(s) + 1/2Cl₂(g) → NaCl(s)

(b) Comparison of ΔGf° and ΔHf for CH₄(g) and NaCl(s)

For the formation reactions, the values of Gibbs free energy of formation (ΔGf°) and enthalpy of formation (ΔHf) can be found in standard thermodynamic tables. Let's compare the values for CH₄(g) and NaCl(s). For CH₄(g): ΔGf° = -50.8 kJ/mol ΔHf = -74.8 kJ/mol For NaCl(s): ΔGf° = -384.1 kJ/mol ΔHf = -411.0 kJ/mol In both cases, ΔGf° is less negative (more positive) than ΔHf.

(c) Condition for ΔG to be more negative (less positive) than ΔH

To analyze this situation, we need to consider the relationship between Gibbs free energy (ΔG), enthalpy (ΔH), and entropy (ΔS) at a constant temperature. The equation is given by: ΔG = ΔH - TΔS For ΔG to be more negative (less positive) than ΔH, the following condition must be true: ΔG < ΔH Substituting the equation above, we get: ΔH - TΔS < ΔH Which simplifies to: -TΔS < 0 Or: ΔS > 0 So, the condition under which ΔG becomes more negative (less positive) than ΔH is: (ii) When ΔSf° is positive.

Key concepts

Chemical Equations

Chemical equations are representations of chemical reactions where reactants on the left are transformed into products on the right. Writing these equations allows us to visualize how substances interact in a reaction. For example, the formation of methane

• Involves carbon in solid form
  combining with hydrogen gas
  to produce methane gas
  in the equation:

C(s) + 2H₂(g) → CH₄(g) Similarly, the formation of sodium chloride is presented as:

• Sodium in solid form
  reacting with gaseous chlorine
  represented as:

Na(s) + 1/2Cl₂(g) → NaCl(s) These equations help calculate the Gibbs free energy changes ( ΔG_i^o ) associated with the formation of these substances, which is crucial in predicting the spontaneity of the reactions.

Thermodynamic Tables

Thermodynamic tables are essential references for chemists. They provide critical data such as enthalpies ( ΔH ) and Gibbs free energies ( ΔG ) at standard conditions. These tables give insights into the energy changes during a reaction. By comparing values from these tables:

• For methane ( CH₄(g) ), the values from a table could show
  ΔG_f^° = -50.8 kJ/mol and ΔH_f = -74.8 kJ/mol
  
• For sodium chloride ( NaCl(s) ), they could indicate
  ΔG_f^° = -384.1 kJ/mol and ΔH_f = -411.0 kJ/mol.
  

These values show that for both substances, ΔG_f^° is less negative than ΔH_f , reflecting the amount of usable energy and the spontaneity. Using tables effectively can help determine reaction feasibility and design processes accordingly.

Entropy

Entropy ( ΔS ) is a measure of disorder or randomness in a system. In thermodynamics, it plays a pivotal role in understanding energy distribution. According to the Gibbs free energy equation: ΔG = ΔH - TΔS Where:

• ΔG : Gibbs free energy
  
• ΔH : Enthalpy
  
• T : Temperature in Kelvin
  
• ΔS : Entropy
  

Entropy, when positive, can contribute to a change in Gibbs free energy being more negative. This is because the term -TΔS becomes positive when ΔS is positive, under typical conditions. This concept helps explain why certain reactions are spontaneous at higher temperatures due to increased entropy, promoting more disordered systems.

Enthalpy

Enthalpy ( ΔH ) represents the total heat content or total energy of a system. It reflects the energy required to create a system, minus the work done by the system. In chemical reactions, changes in enthalpy indicate whether a reaction releases or absorbs heat.

• Exothermic reactions have a negative ΔH , indicating heat is released.
  
• Endothermic reactions feature a positive ΔH , where heat is absorbed.
  

Comparing enthalpy ( ΔH ) with Gibbs free energy ( ΔG ) provides insight into reaction spontaneity.

• When ΔG is more negative than ΔH , the reaction is more likely to occur spontaneously because of favorable entropy changes.
  

This makes understanding enthalpy changes crucial for developing effective and energy-efficient processes in industrial chemistry and environmental applications.