Gibbs Free Energy and Spontaneity Practice Notes

Course Overview & Intellectual Property Notice

• Course Designation: Chem 1155
• Activity Title: Class Activity 5 — Gibbs Free Energy, $\Delta G$ Practice
• Instructor: Brian Gute
• Intellectual Property Rights: This content represents the intellectual property of the instructor, Brian Gute. It may not be altered, shared for commercial purposes, or distributed in any modified or unmodified form, either during or subsequent to enrollment in the course.

Fundamental Principles of Gibbs Free Energy

• Standard Gibbs Free Energy Equation:   $\Delta G^\circ_{\text{rxn}} = \Delta H^\circ_{\text{rxn}} - T\Delta S^\circ_{\text{rxn}}$

  • $\Delta G^\circ_{\text{rxn}}$: Change in standard Gibbs free energy of the reaction
  • $\Delta H^\circ_{\text{rxn}}$: Change in standard enthalpy of the reaction
  • $\Delta S^\circ_{\text{rxn}}$: Change in standard entropy of the reaction
  • $T$: Absolute temperature expressed in Kelvin ($K$

• Criteria for Reaction Spontaneity:

  • $\Delta G < 0$ (Negative): The reaction is spontaneous in the forward direction.
  • $\Delta G > 0$ (Positive): The reaction is non-spontaneous in the forward direction (spontaneous in the reverse direction).
  • $\Delta G = 0$: The reaction is at thermodynamic equilibrium.

• Temperature Scale Requirements:

  • Temperature must always be converted to Kelvin ($K$) before performing calculations:     $T (K) = T (^\circ\text{C}) + 273.15$

• Unit Alignment between Enthalpy and Entropy:

  • Enthalpy ($\Delta H$) is typically supplied in kilojoules ($kJ$ or $kJ\,mol^{-1}$).
  • Entropy ($\Delta S$) is typically supplied in joules per Kelvin ($J\,K^{-1}$ or $J\,K^{-1}\,mol^{-1}$).
  • Prior to substitution into $\Delta G = \Delta H - T\Delta S$, entropy units must be converted to kilojoules per Kelvin ($kJ\,K^{-1}$):     $\Delta S (\text{in } kJ\,K^{-1}) = \frac{\Delta S (\text{in } J\,K^{-1})}{1000\,J\,kJ^{-1}}$

Practice 1: Determining Signs of Enthalpy, Entropy, and Reaction Spontaneity

• Reaction 1: Decomposition of Nitrous Oxide   $2N_2O(g) \rightarrow 2N_2(g) + O_2(g)$

  • Enthalpy Sign ($\Delta H^\circ_{\text{rxn}}$): Exothermic $\implies \Delta H^\circ_{\text{rxn}} < 0$ (Negative: $-$)
  • Entropy Sign ($\Delta S^\circ_{\text{rxn}}$): Positive ($+$)
  • Reasoning: $2$ moles of gaseous reactant ($2N_2O$) produce $3$ moles of gaseous products ($2N_2 + 1O_2$). An increase in the total number of moles of gas ($\Delta n_{\text{gas}} = +1$) increases molecular disorder and positional freedom.
  • Free Energy Sign ($\Delta G^\circ_{\text{rxn}}$):     $\Delta G = (-) - T(+) = (-)$
  • Spontaneity Behavior: Always spontaneous at all temperatures.

• Reaction 2: Formation of Ozone   $3O_2(g) \rightarrow 2O_3(g)$

  • Enthalpy Sign ($\Delta H^\circ_{\text{rxn}}$): Endothermic $\implies \Delta H^\circ_{\text{rxn}} > 0$ (Positive: $+$)
  • Entropy Sign ($\Delta S^\circ_{\text{rxn}}$): Negative ($-$)
  • Reasoning: $3$ moles of gaseous reactant ($3O_2$) condense into $2$ moles of gaseous product ($2O_3$). A decrease in gaseous moles ($\Delta n_{\text{gas}} = -1$) results in a reduction of overall system entropy.
  • Free Energy Sign ($\Delta G^\circ_{\text{rxn}}$):     $\Delta G = (+) - T(-) = (+) + T(+) = (+)$
  • Spontaneity Behavior: Never spontaneous at any temperature.

• Reaction 3: Freezing of Liquid Water   $H_2O(l) \rightarrow H_2O(s)$

  • Enthalpy Sign ($\Delta H^\circ_{\text{rxn}}$): Exothermic, given as $\Delta H^\circ_{\text{rxn}} = -6.01\,kJ$ (Negative: $-$)
  • Entropy Sign ($\Delta S^\circ_{\text{rxn}}$): Negative ($-$)
  • Reasoning: Transitioning from a liquid phase to an ordered solid crystal lattice structure restricts molecular movement, causing a decrease in entropy.
  • Free Energy Sign ($\Delta G^\circ_{\text{rxn}}$):     $\Delta G = (-) - T(-)$
  • Spontaneity Behavior: Temperature-dependent (Spontaneous at low temperatures, specifically below $273.15\,K$ / $0\,^\circ\text{C}$, where the negative enthalpy term dominates).

• Reaction 4: Vaporization of Liquid Water   $H_2O(l) \rightarrow H_2O(g)$

  • Enthalpy Sign ($\Delta H^\circ_{\text{rxn}}$): Endothermic, given as $\Delta H^\circ_{\text{rxn}} = +40.7\,kJ$ (Positive: $+$)
  • Entropy Sign ($\Delta S^\circ_{\text{rxn}}$): Positive ($+$)
  • Reasoning: Transitioning from a liquid phase to a disordered gas phase allows significantly greater freedom of movement, causing an increase in entropy.
  • Free Energy Sign ($\Delta G^\circ_{\text{rxn}}$):     $\Delta G = (+) - T(+)$
  • Spontaneity Behavior: Temperature-dependent (Spontaneous at high temperatures, specifically above $373.15\,K$ / $100\,^\circ\text{C}$ at standard pressure, where the negative $-T\Delta S$ term dominates).

Practice 2: General Relationship Matrix for Enthalpy, Entropy, and Spontaneity

• Matrix Case 1:

  • Enthalpy Change ($\Delta H^\circ_{\text{rxn}}$): Negative ($-$)
  • Entropy Change ($\Delta S^\circ_{\text{rxn}}$): Positive ($+$)
  • Free Energy Change ($\Delta G^\circ_{\text{rxn}}$): Always Negative ($-$)
  • Reaction Spontaneity: Spontaneous
  • Temperature Dependency: Spontaneous at all temperatures.

• Matrix Case 2:

  • Enthalpy Change ($\Delta H^\circ_{\text{rxn}}$): Positive ($+$)
  • Entropy Change ($\Delta S^\circ_{\text{rxn}}$): Negative ($-$)
  • Free Energy Change ($\Delta G^\circ_{\text{rxn}}$): Always Positive ($+$)
  • Reaction Spontaneity: Non-spontaneous
  • Temperature Dependency: Non-spontaneous at all temperatures (Never spontaneous).

• Matrix Case 3:

  • Enthalpy Change ($\Delta H^\circ_{\text{rxn}}$): Positive ($+$)
  • Entropy Change ($\Delta S^\circ_{\text{rxn}}$): Positive ($+$)
  • Free Energy Change ($\Delta G^\circ_{\text{rxn}}$): Negative at high temperatures; Positive at low temperatures
  • Reaction Spontaneity: Temperature-dependent
  • Temperature Dependency: Spontaneous at high temperatures (when $T\Delta S > \Delta H$).

• Matrix Case 4:

  • Enthalpy Change ($\Delta H^\circ_{\text{rxn}}$): Negative ($-$)
  • Entropy Change ($\Delta S^\circ_{\text{rxn}}$): Negative ($-$)
  • Free Energy Change ($\Delta G^\circ_{\text{rxn}}$): Negative at low temperatures; Positive at high temperatures
  • Reaction Spontaneity: Temperature-dependent
  • Temperature Dependency: Spontaneous at low temperatures (when $|\Delta H| > |T\Delta S|$).

Practice 3: Quantitative Analysis of Ethylene Hydrogenation

• Chemical Reaction:   $C_2H_4(g) + H_2(g) \rightarrow C_2H_6(g)$

• Thermodynamic Parameters Provided:

  • Enthalpy change: $\Delta H = -137.5\,kJ$
  • Entropy change: $\Delta S = -120.5\,J\,K^{-1}$
  • Reaction Temperature: $T = 25\,^\circ\text{C}$

• Part a: Calculation of $\Delta G^\circ$ at $25\,^\circ\text{C}$ and Spontaneity Determination

  • Convert Temperature to Kelvin:     $T = 25\,^\circ\text{C} + 273.15 = 298.15\,K$
  • Convert Entropy Units to Kilojoules per Kelvin:     $\Delta S = -120.5\,J\,K^{-1} \times \frac{1\,kJ}{1000\,J} = -0.1205\,kJ\,K^{-1}$
  • Calculate Standard Gibbs Free Energy Change ($\Delta G^\circ$):     $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$\Delta G^\circ = -137.5\,kJ - (298.15\,K \times -0.1205\,kJ\,K^{-1})$$\Delta G^\circ = -137.5\,kJ - (-35.927\,kJ)$$\Delta G^\circ = -137.5\,kJ + 35.927\,kJ = -101.573\,kJ$
  • Final Value: $\Delta G^\circ = -101.6\,kJ$
  • Spontaneity Determination: Because $\Delta G^\circ = -101.6\,kJ < 0$, the reaction is spontaneous under standard conditions at $25\,^\circ\text{C}$.

• Part b: Temperature Effect on $\Delta G$

  • Question: Does $\Delta G$ become more negative or more positive as the temperature increases?
  • Answer: More positive.
  • Mathematical Explanation: In the expression $\Delta G = \Delta H - T\Delta S$, substituting a negative entropy change ($\Delta S = -0.1205\,kJ\,K^{-1}$) turns the term $-T\Delta S$ into a positive addition ($+T \times 0.1205\,kJ\,K^{-1}$). As temperature $T$ increases, this positive quantity grows larger, driving $\Delta G$ to become progressively more positive (less negative).

Practice 4: Decomposition of Carbon Tetrachloride and Temperature Threshold Analysis

• Chemical Reaction:   $CCl_4(g) \rightarrow C(s, \text{graphite}) + 2Cl_2(g)$

• Thermodynamic Parameters Provided:

  • Enthalpy change: $\Delta H = +95.7\,kJ$
  • Entropy change: $\Delta S = +142.2\,J\,K^{-1}$

• Part a: Calculation of $\Delta G^\circ$ at $25\,^\circ\text{C}$ and Spontaneity Determination

  • Convert Temperature to Kelvin:     $T = 25\,^\circ\text{C} + 273.15 = 298.15\,K$
  • Convert Entropy Units to Kilojoules per Kelvin:     $\Delta S = +142.2\,J\,K^{-1} \times \frac{1\,kJ}{1000\,J} = +0.1422\,kJ\,K^{-1}$
  • Calculate Standard Gibbs Free Energy Change ($\Delta G^\circ$):     $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$\Delta G^\circ = +95.7\,kJ - (298.15\,K \times 0.1422\,kJ\,K^{-1})$$\Delta G^\circ = +95.7\,kJ - 42.397\,kJ = +53.303\,kJ$
  • Final Value: $\Delta G^\circ = +53.3\,kJ$
  • Spontaneity Determination: Because $\Delta G^\circ = +53.3\,kJ > 0$, the reaction is non-spontaneous at $25\,^\circ\text{C}$.

• Part b: Temperature Threshold Calculation for Spontaneity

  • Crossover Condition ($\Delta G = 0$):     $0 = \Delta H - T\Delta S \implies T = \frac{\Delta H}{\Delta S}$
  • Calculate Threshold Temperature ($T$):     $T = \frac{+95.7\,kJ}{0.1422\,kJ\,K^{-1}} = 672.996\,K$
  • Rounded Threshold Temperature: $T = 673\,K$
  • Temperature Range Determination: Since both $\Delta H$ and $\Delta S$ are positive, the reaction becomes spontaneous at temperatures greater than the calculated threshold value ($T > 673\,K$), where the entropy term $T\Delta S$ surpasses the enthalpy barrier $\Delta H$.