Free Energy, Entropy, and the Second Law of Thermodynamics Study Guide

The Fundamental Driving Force of Physical and Chemical Change

• In previous study, the focus has been on chemical and physical changes through the lenses of kinetics (how fast changes occur) and equilibrium constants (predicting the extent of the change).

• We have observed specific behaviors such as acids neutralizing bases and gases expanding to fill containers.

• The central question of thermodynamics is: Why do these changes occur in the first place? What ultimately drives physical and chemical changes in matter?

• The driving force behind chemical and physical change in the universe is a quantity called entropy.

• Entropy is fundamentally related to the dispersion (spreading out) of energy.

• Nature tends toward the state in which energy is spread out to the greatest extent possible.

• Examples of processes that increase the entropy in the universe (resulting in greater energy dispersion) include:

  • The freezing of water below $0\,^{\circ}C$.

  • The dissolving of a solid into a solution.

  • The neutralization of an acid by a base.

  • The development of a person from an embryo.

• A core law of our universe is that entropy always increases.

The Second Law of Thermodynamics: Energy Dispersion and Heat Death

• Nature's Dislike for Concentrated Energy: Nature tends to move away from concentrated energy states.

  • Example: A hot cup of coffee will become cold in $30$ minutes because its concentrated thermal energy dissipates into the surrounding room.

  • Example: The concentrated energy in gasoline used to power a car dissipates into thermal energy as the car moves.

• The First Law vs. The Second Law: According to the First Law of Thermodynamics, energy cannot be created or destroyed (conservation of energy). The Second Law focuses on the dispersion or dissipation of energy from a concentrated form into a more spread-out form.

• Definition of the Second Law of Thermodynamics: The pervasive tendency for energy to spread out or dissipate when not prevented from doing so.

• Predicting Spontaneity: The Second Law determines the spontaneous direction of all processes. To know if a process will occur (such as coffee cooling or fuel burning), one must determine if the process disperses energy.

  • If energy is dispersed, the process occurs.

  • If energy is not dispersed, the process does not occur.

• Environmental Influence: The melting of ice above $0\,^{\circ}C$ disperses energy and occurs spontaneously, but below $0\,^{\circ}C$, the melting of ice does not disperse energy and therefore does not happen.

• The Fate of the Universe (Heat Death): Cosmologists use the Second Law to predict that the sun will eventually burn out as its concentrated energy disperses. Eventually, all concentrated energy in the universe will disperse until no more processes can occur. This is known as "heat death."

  • The sun is estimated to have several billion years left.

  • The universe is estimated to have approximately $10^{100}$ years before it winds down.

Spontaneous and Nonspontaneous Processes

• Definition of a Spontaneous Process: A process that occurs without ongoing outside intervention, such as the performance of work by an external force.

  • Example: A book dropping to the floor in a gravitational field.

  • Example: A ball rolling down a slope.

• Mechanical vs. Chemical Systems: Mechanical systems tend toward the lowest potential energy. Chemists seek a "chemical potential" equivalent that predicts the direction of spontaneous chemical change.

• Spontaneity vs. Speed: Spontaneity is the direction and extent to which a reaction proceeds (Thermodynamics). Speed is how fast a reaction occurs (Kinetics).

  • A reaction can be thermodynamically spontaneous but kinetically slow.

  • Example: The conversion of diamond to graphite is thermodynamically spontaneous, but the process is so slow it does not happen at a measurable rate.

• The Role of Catalysts: Catalysts increase the rate of a spontaneous process but cannot make a nonspontaneous process spontaneous. They affect only the kinetic path, not the initial and final thermodynamic states.

• Nonspontaneous Processes: A nonspontaneous process is not impossible. It can occur if:

  • It is coupled to another process that is spontaneous.

  • Energy is supplied from an external source (e.g., extracting iron metal from iron ore requires external energy, usually via another highly spontaneous reaction).

Entropy and the Limits of Enthalpy

• Enthalpy ($\Delta H$) as a Metric: Enthalpy was once considered the sole criterion for spontaneity (where exothermic reactions would be spontaneous and endothermic would not). However, many spontaneous processes are endothermic (uphill energetically).

• Examples of Endothermic Spontaneous Processes:

  • The melting of ice above $0\,^{\circ}C$.

  • The evaporation of liquid water to gaseous water.

  • The dissolution of sodium chloride ($NaCl$) in water.

• The Commonality of Spontaneity: In each of these endothermic processes, disorder or randomness increases.

  • Ice melting: Water molecules move from a highly ordered solid lattice to a disorderly liquid.

  • Evaporation: Molecules move from a somewhat disorderly liquid to a highly disorderly gas.

  • Dissolution: Ions move from an orderly crystal lattice to being randomly dispersed in solution.

The Formal Definition of Entropy ($S$)

• Informal definition: Disorder or randomness.

• Formal definition: A thermodynamic function that increases with the number of energetically equivalent ways to arrange the components of a system to achieve a particular state.

• Boltzmann’s Equation:   $S = k \ln W$

  • $k$ is the Boltzmann constant: $k = \frac{R}{N_A} = 1.38 \times 10^{-23}\,J/K$.

  • $W$ is the number of energetically equivalent ways to arrange the components of the system (unitless).

  • Units of Entropy: Joules per kelvin ($J/K$).

• Microstates and Macrostates:

  • Macrostate: Defined by conditions like Pressure ($P$), Volume ($V$), and Temperature ($T$). The energy remains constant.

  • Microstate: A specific snap-shot of the internal energy distribution among particles at a given instant. In a macrostate, energy is constantly redistributing.

  • $W$ represents the number of microstates that result in a given macrostate.

• Relationship to Energy Dispersal:

  • A state with higher entropy has a higher number of energetically equivalent ways to arrange its components, which implies greater energy dispersal.

  • Example: System A (2 particles, 1 energy level at $2\,J$) vs. System B (2 particles, 2 energy levels at $1\,J$ and $3\,J$). Both have total energy $4\,J$.

  • System A: $W = 1$ (Only one way both particles can be at $2\,J$).

  • System B: $W = 2$ (Blue at $1\,J$/Red at $3\,J$ OR Blue at $3\,J$/Red at $1\,J$).

  • System B has higher entropy because its energy is dispersed over more levels.

Entropy and the Expansion of an Ideal Gas

• Gas Expansion in a Vacuum: When a gas expands into a vacuum, external pressure ($P_{ext}$) is zero, so work ($w = -P_{ext} \Delta V$) is zero. Total energy does not change, but entropy does.

• Statistical Probability of Distribution:

  • Consider 4 gas atoms distributed between two flasks (Left and Right).

  • State A: All 4 atoms in the Left flask ($W = 1$).

  • State B: All 4 atoms in the Right flask ($W = 1$).

  • State C: 2 atoms in the Left, 2 atoms in the Right ($W = 6$).

  • There are 6 distinct microstates for State C, making it 6 times more likely than State A or B.

• The Effect of Particle Number ($n$):

  • The formula for the number of ways to arrange $r$ particles in one flask and $(n - r)$ in the other is: $\frac{n!}{(n - r)! r!}$.

  • For $10$ atoms, the equal distribution macrostate has $252$ microstates.

  • For $20$ atoms, the equal distribution macrostate has $184,756$ microstates.

  • The probability of all atoms staying in one flask remains $1$. Therefore, systems naturally move toward distribution (higher entropy).

• Entropy Change Equation:   $\Delta S = S_{final} - S_{initial}$

• Heat Flow and the Second Law: Heat travels from high temperature to low temperature because this results in greater energy randomization. The First Law would permit heat to flow from cold to hot (preserving total energy), but the Second Law prohibits it because it would involve the concentration of energy.

Questions & Discussion

• 19.1 Conceptual Connection: Iron spontaneously rusts when it comes in contact with oxygen. This is a spontaneous process driven by the second law.

• Answer Now! Problem (The Second Law): Which process is inconsistent with the Second Law of Thermodynamics?

  • (a) The spontaneous creation of energy from nothing (Violates First Law, but also thermodynamic principles).

  • (b) The spontaneous creation of matter from nothing.

  • (c) The spontaneous concentration of energy from dispersed energy (This is the specific action forbidden by the Second Law).

• Answer Now! Problem (Entropy): Consider three changes in the possible distributions of six gaseous particles within three interconnected boxes. Which change has a positive $\Delta S$?

  • This refers to transitions moving from higher concentration in fewer boxes to even distribution across more boxes.