$$\newcommand{\xrightleftharpoons}[2]{\overset{#1}{\underset{#2}{\rightleftharpoons}}}$$ $$\newcommand{\conc}[1]{\left[\mathrm{#1}\right]}$$ $$\newcommand{\chem}[1]{\mathrm{#1}}$$

# Chapter 17 Spontaneity, Entropy, and Free Energy

Shaun Williams, PhD

### Thermodynamics vs. Kinetics

• Domain of Kinetics
• Rate of a reaction depends on the pathway from reactants to products.
• Thermodynamics tells us whether a reaction is spontaneous based only on the properties of reactants and products.

### Spontaneous Processes and Entropy

• Thermodynamics lets us predict the direction in which a process will occur but gives no information about the speed of the process.
• A spontaneous process is one that occurs without outside intervention.

### Concept Check

Consider $$2.4\,\mathrm{moles}$$ of a gas contained in a $$4.0\,\mathrm{L}$$ bulb at a constant temperature of $$32^\circ \mathrm{C}$$. This bulb is connected by a valve to an evacuated $$20.0\,\mathrm{L}$$ bulb. Assume the temperature is constant.

What should happen to the gas when you open the value?

The gas should spread evenly throughout the two bulbs.

### Entropy

• The driving force for a spontaneous process is an increase in the entropy of the universe.
• A measure of molecular randomness or disorder.
• Thermodynamic function that describes the number of arrangements that are available to a system existing in a given state.
• Nature spontaneously proceeds toward the states that have the highest probabilities of existing.

### Positional Entropy

• A gas expands into a vacuum to give a uniform distribution because the expanded state has the highest positional probability of states available to the system.
• Therefore: $$S_{solid} < S_{liquid} \ll S_{gas}$$

### Concept Check

Predict the sign of $$\Delta S$$ for each of the following:

• The evaporation of alcohol
• positive

• The freezing of water
• negative

• Compressing an ideal gas at constant temperature
• negative

• Heating an ideal gas at constant pressure
• positive

• Dissolving NaCl in water
• positive

### Second Law of Thermodynamics

• In any spontaneous process there is always an increase in the entropy of the universe.
• The entropy of the universe is increasing.
• The total energy of the universe is constant, but the entropy is increasing.
• $$S_{universe}=\Delta S_{system}+\Delta S_{surroundings}$$

### $$\Delta S_{surr}$$

• $$\Delta S_{surr} > 0$$: entropy of the universe increases
• $$\Delta S_{surr} < 0$$: process is spontaneous in opposite direction
• $$\Delta S_{surr} = 0$$: process has no tendency to occur
• The sign of $$\Delta S_{surr}$$ depends on the direction of the heat flow.
• The magnitude of $$\Delta S_{surr}$$ depends on the temperature.
• Exothermic process: $$\Delta S_{surr} = +\frac{\text{quantity of heat (J)}}{\text{temperature (K)}}$$
• Endothermic process: $$\Delta S_{surr} = -\frac{\text{quantity of heat (J)}}{\text{temperature (K)}}$$
• Heat flow (constant $$P$$) = change in enthalpy = $$\Delta H$$ $$\Delta S_{surr}=-\frac{\Delta H}{T}$$

### Interplay of $$\Delta S_{sys}$$ and $$\Delta S_{surr}$$ in Determining the Sign of $$\Delta S_{univ}$$

Signs of Entropy Changes
$$\Delta S_{sys}$$ $$\Delta S_{surr}$$ $$\Delta S_{univ}$$ Process Spontaneous?
+ + + Yes
- - - No (reaction will occur in opposite direction)
+ - ? Yes, if $$\Delta S_{sys}$$ has a larger magnitude than $$\Delta S_{surr}$$
- + ? Yes, if $$\Delta S_{surr}$$ has a larger magnitude than $$\Delta S_{sys}$$

### Free Energy ($$G$$)

$$\Delta S_{univ} = -\frac{\Delta G}{T}\,\left( \text{at constant }T\text{ and }P \right)$$

• A process (at constant $$T$$ and $$P$$ is spontaneous in the direction in which the free energy decreases.
• Negative $$\Delta G$$ means positive $$\Delta S_{univ}$$.
• $$\Delta G = \Delta H - T\Delta S$$ (at constant $$T$$ and $$P$$)

### Effect of $$\Delta H$$ and $$\Delta S$$ on Spontaneity

Case Result
$$\Delta S > 0$$, $$\Delta H <0$$ Spontaneous at all temperatures
$$\Delta S > 0$$, $$\Delta H > 0$$ Spontaneous at high temperatures (where exothermicity is relatively unimportant)
$$\Delta S < 0$$, $$\Delta H < 0$$ Spontaneous at low temperatures (where exothermicity is dominant)
$$\Delta S < 0$$, $$\Delta H > 0$$ Process not spontaneous at any temperatures (reverse process is spontaneous at all temperatures)

### Third Law of Thermodynamics

• The entropy of a perfect crystal at $$0\,\mathrm{K}$$ is zero.
• The entropy of a substance increases with temperature.

### Standard State Values

• Standard Entropy Values ($$S^\circ$$)
• Represent the increase in entropy that occurs when a substance is heated from $$0\,\mathrm{K}$$ to $$298\,\mathrm{K}$$ at 1 atm pressure. $$\Delta S^\circ_{rxn} = \sum n_pS^\circ_{products} - \sum n_r S^\circ_{reactants}$$
• Standard Free Energy Values ($$\Delta G^\circ$$)
• The change in free energy that will occur if the reactants in their standard states are converted to the products in their standard states. $$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$ $$\Delta G^\circ_{rxn} = \sum n_p G^\circ_{products} - \sum n_r G^\circ_{reactants}$$

### Concept Check

A stable diatomic molecule spontaneously forms from its atoms. Predict the sign of

• $$\Delta H^\circ$$
• negative

• $$\Delta S^\circ$$
• negative

• $$\Delta G^\circ$$
• negative

### Free Energy

• Free energy and pressure \begin{align} G &= G^\circ + RT\ln (P) \\ &\mathrm{or} \\ \Delta G &= \Delta G^\circ + RT\ln (Q) \end{align}
• Free energy and equilibrium $$\Delta G=0=\Delta G^\circ + RT\ln (K)$$ $$\Delta G^\circ = -RT \ln (K)$$

### The Meaning of $$\Delta G$$ for a Chemical Reaction

• A system can achieve the lowest possible free energy by going to equilibrium, not by going to completion.

### Qualitative Relationship Between the Change in Standard Free Energy and the Equilibrium Constant for a Given Reaction

$$\Delta G^\circ$$ $$K$$
$$\Delta G^\circ = 0$$ $$K=1$$
$$\Delta G^\circ < 0$$ $$K>1$$
$$\Delta G^\circ > 0$$ $$K<1$$

### Free Energy and Work

• Maximum possible useful work obtainable from a process at constant temperature and pressure is equal to the change in free energy. $$w_{max} = \Delta G$$
• Achieving the maximum work available from a spontaneous process can occur only via a hypothetical pathway. Any real pathway wastes energy.
• All real processes are irreversible.
• First law: You can’t win, you can only break even.
• Second law: You can’t break even.
• As we use energy, we degrade its usefulness.

/