Thermodynamics: Spontaneous Change, Entropy & Free Energy (CHEM 120, Chapter 14)

Thermodynamics

Introduction to Thermodynamics

Thermodynamics is the study of the interrelationships among heat, work, and the energy content of a system. It provides a framework for understanding how energy is transferred and transformed during chemical reactions and physical processes.

• Thermochemistry (ΔH): Examines the heat changes in chemical reactions, distinguishing between endothermic (heat absorbed) and exothermic (heat released) processes.

• Entropy (S): Measures the degree of disorder or randomness in a system.

• Free Energy (ΔG): Determines whether a process is spontaneous or nonspontaneous.

Enthalpy, H

Definition and Measurement

Enthalpy (H) is the heat content of a system at constant pressure. The change in enthalpy, ΔH, represents the heat absorbed or released during a chemical reaction.

• Endothermic Reaction: ΔH > 0 (system absorbs heat)

• Exothermic Reaction: ΔH < 0 (system releases heat)

Direct Method for Calculating ΔHorxn

The standard enthalpy change of reaction (ΔHorxn) can be calculated using tabulated standard enthalpies of formation (ΔHof) for reactants and products at 25°C and 1 atm.

• General reaction: aA + bB → cC + dD

• Equation for direct calculation:

$\Delta H^\circ_{rxn} = [d\Delta H^\circ_f(D) + c\Delta H^\circ_f(C)] - [a\Delta H^\circ_f(A) + b\Delta H^\circ_f(B)]$

Or, more generally:

$\Delta H^\circ_{rxn} = \sum m \Delta H^\circ_f(\text{Products}) - \sum n \Delta H^\circ_f(\text{Reactants})$

• Indirect Method: Uses Hess's Law to determine ΔH for reactions by combining known enthalpy changes.

Entropy, S

Definition and Properties

Entropy (S) is a measure of the degree of disorder or randomness in a system. It quantifies the number of possible arrangements (microstates) of particles.

• Third Law of Thermodynamics: The entropy of a perfect crystal at absolute zero (0 K) is zero.

• Units: Standard entropy (So) is measured in J K-1 mol-1.

Entropy Change, ΔS

The change in entropy (ΔS) during a reaction or process is the difference in entropy between products and reactants.

• Equation:

$\Delta S_{rxn} = \sum S^\circ(\text{Products}) - \sum S^\circ(\text{Reactants})$

• Qualitative Prediction: If the number of gas molecules increases in a reaction, ΔS is usually positive (entropy increases).

• Example: For the reaction C5H12(s) + 8O2(g) → 5CO2(g) + 6H2O(g), the number of gaseous molecules increases, so ΔS > 0.

Phase Changes and Entropy

Entropy changes also occur during phase transitions (e.g., melting, vaporization). The entropy change for a phase transition is given by:

$\Delta S_{transition} = \frac{\Delta H_{transition}}{T}$

• Where ΔHtransition is the enthalpy change for the phase change and T is the temperature in Kelvin.

Entropy as a Function of Temperature

Entropy increases with temperature, as higher temperatures allow for more molecular motion and disorder.

Gibbs Free Energy, ΔG

Definition and Significance

Gibbs Free Energy (G) is the energy available to do work at constant temperature and pressure. The change in free energy (ΔG) determines the spontaneity of a process.

• ΔG < 0: Spontaneous process

• ΔG > 0: Non-spontaneous process

The absolute value of free energy cannot be measured, but changes in free energy can be calculated.

Gibbs-Helmholtz Equation

The relationship between enthalpy, entropy, and free energy is given by:

$\Delta G = \Delta H - T\Delta S$

• Where T is the temperature in Kelvin.

Standard Free Energies of Formation

Standard free energy changes (ΔGo) can be calculated using tabulated values:

$\Delta G^\circ_{rxn} = \sum \Delta G^\circ_f(\text{Products}) - \sum \Delta G^\circ_f(\text{Reactants})$

Temperature Dependence of Spontaneity

ΔH	ΔS	ΔG	Spontaneity	Example
-	+	Always -	Spontaneous at all T	2NO(g) → 2N2(g)
-	-	- at low T	Spontaneous at low T	NH3(l) → NH3(g)
+	+	- at high T	Spontaneous at high T	H2O(s) → H2O(g)
+	-	Always +	Nonspontaneous at all T	CO2(g) → C(s) + O2(g)

Gibbs Free Energy and Equilibrium

Relationship to Equilibrium Constant

At equilibrium, the change in free energy is related to the equilibrium constant (K) by:

$\Delta G^\circ = -RT \ln K$

• R = Universal Gas Constant = 8.314 J K-1 mol-1

• T = Absolute temperature (Kelvin)

• K = Equilibrium constant

Significance of ΔGo Magnitude

ΔGo (kJ/mol)	K
+200	9.1 × 10736
+100	3.0 × 10718
+50	1.7 × 1072
+10	1.8 × 102
+1.0	6.7 × 101
-1.0	15
-10	5.6 × 10-2
-50	5.8 × 10-8
-100	3.3 × 10-17
-200	1.1 × 10-87

Additional info: The table above shows how the magnitude and sign of ΔGo affect the equilibrium constant K. Large negative ΔGo values correspond to large K values (favoring products), while large positive ΔGo values correspond to small K values (favoring reactants).

Non-standard Conditions

For reactions not at standard conditions, the free energy change is given by:

$\Delta G = \Delta G^\circ + RT \ln Q$

• Q = Reaction quotient (ratio of product and reactant activities at any point)

Worked Example: Calculating ΔGo and K

Example Calculation

Given thermodynamic data, calculate ΔGo for the reaction:

COCl2(g) + 4NH3(g) → CO(NH2)2(s) + 2NH4Cl(s)

• ΔHo calculation:

$\Delta H^\circ_{rxn} = [\Delta H^\circ_f(\text{CO(NH}_2)_2) + 2\Delta H^\circ_f(\text{NH}_4\text{Cl})] - [\Delta H^\circ_f(\text{COCl}_2) + 4\Delta H^\circ_f(\text{NH}_3)]$

• ΔS calculation:

$\Delta S_{rxn} = [S^\circ(\text{CO(NH}_2)_2) + 2S^\circ(\text{NH}_4\text{Cl})] - [S^\circ(\text{COCl}_2) + 4S^\circ(\text{NH}_3)]$

• ΔG calculation at 298.2 K:

$\Delta G = \Delta H - T\Delta S$

Example: Calculating K from ΔGo

Given ΔGo = 474.4 kJ/mol for the reaction 2H2O(l) → 2H2(g) + O2(g) at 25°C, calculate the equilibrium constant K.

$\Delta G^\circ = -RT \ln K$

Rearrange to solve for K:

$K = e^{-\Delta G^\circ / RT}$

Substitute values:

$K = e^{-474400 / (8.314 \times 298.2)} = 8.3 \times 10^{-83}$

Summary Table: Key Thermodynamic Equations

Equation	Description
$\Delta H^\circ_{rxn} = \sum m \Delta H^\circ_f(\text{Products}) - \sum n \Delta H^\circ_f(\text{Reactants})$	Standard enthalpy change of reaction
$\Delta S_{rxn} = \sum S^\circ(\text{Products}) - \sum S^\circ(\text{Reactants})$	Standard entropy change of reaction
$\Delta G = \Delta H - T\Delta S$	Gibbs free energy change
$\Delta G^\circ = -RT \ln K$	Relationship between free energy and equilibrium constant
$\Delta G = \Delta G^\circ + RT \ln Q$	Free energy change under non-standard conditions

Additional info: These equations are fundamental for predicting the direction and extent of chemical reactions, as well as for calculating energy changes in chemical systems.