Spontaneity, Entropy, and Free Energy

Thermodynamic Laws and the Concept of Spontaneity

The first law of thermodynamics is a fundamental statement regarding the law of conservation of energy, asserting that energy can be neither created nor destroyed. Consequently, the total energy of the universe remains constant ($E_{\text{universe}} = \text{constant}$). While the quantity of energy is conserved, its various forms can be interchanged during physical and chemical processes. For instance, dropping a book converts its initial potential energy into kinetic energy, which is subsequently transferred to the atoms in the air and floor as random thermal motion. In a chemical context, the combustion of methane ($CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2(g) + \text{energy}$) involves a lowering of potential energy stored in chemical bonds, which is released as heat. While the first law is useful for energy bookkeeping—tracking energy flow into or out of a system and identify the final form of energy—it does not explain why specific processes occur in a particular direction or identify the driving forces behind them.

The real challenge in thermodynamics is not the quantity of energy but the quality of identifying how energy becomes less useful as it is converted from one form to another. A process is defined as spontaneous if it occurs without outside intervention. Spontaneity is independent of the rate of the process; a spontaneous process may be incredibly fast or extremely slow. Thermodynamics predicts the direction of a process based on initial and final states, whereas kinetics focuses on the reaction pathway and rate. For example, thermodynamics predicts that diamond should spontaneously convert to graphite at $25\,^{\circ}C$ and $1\,atm$, but because the process is so slow, it is not observed on a human timescale.

Spontaneous Processes and the Role of Entropy

Observations of diverse physical and chemical processes reveal universal trends in spontaneity: a ball rolls down a hill but never up; steel rusts when exposed to moisture but rust does not spontaneously return to iron and oxygen; gases fill containers uniformly; and heat always flows from a hot object to a cooler one. While exothermicity often accompanies spontaneous processes, it is not the sole driving force. For example, the melting of ice above $0\,^{\circ}C$ is an endothermic process yet it occurs spontaneously. Scientists have concluded that the common characteristic of all spontaneous processes is an increase in a property called entropy ($S$). The change in the entropy of the universe serves as the measure of the driving force behind a process.

Ultimately, entropy is a measure of how energy is distributed among the energy levels within the particles of a system. Macroscopically, it reflects the natural progression from order to disorder, or from lower probability to higher probability. For example, a deck of playing cards thrown into the air is astronomically more likely to land in a disordered state than to return to its original ordered sequence because there are billions of ways to be disordered but only one way to be ordered. Nature spontaneously proceeds toward states that have the highest probabilities of existing, characterized by the greatest number of achievable microstates.

Probability, Microstates, and Positional Entropy

The probability of a particular arrangement, or state, depends on the number of microstates (configurations) that can achieve it. For a system of $N$ gas molecules in a two-bulbed container, the number of microstates for a given arrangement is calculated using the formula: $\frac{N!}{L!R!}$, where $L$ is the number of molecules in the left bulb and $R$ is the number in the right bulb. As the number of molecules increases toward macroscopic scales (on the order of $10^{23}$), the probability that the gas will be distributed nearly evenly ($50\%--50\%$) becomes overwhelming. The reverse process of all molecules migrating to one side is possible but so improbable that it is never observed. This type of randomness is termed positional probability because it relates to spatial configurations.

Positional probability increases as a substance moves from a compact solid to a liquid and finally to a gas. In a solid, molecules are fixed with few available positions; in a gas, molecules have a vast volume and many more positions available. Similarly, the mixing of two substances results in an increase in positional entropy because the molecules of each component have more available space and configurations in the mixture than they did in the separated state. For instance, adding solid sugar to water increases positional disorder as the sugar molecules disperse throughout a larger volume. Conversely, the condensation of iodine vapor into crystals represents a decrease in positional entropy as the substance moves from a large volume to a much smaller, more ordered volume.

Isothermal Expansion and the Reversibility of Ideal Gases

An isothermal process is defined as one in which the temperature of the system and surroundings remains constant ($\Delta T = 0$). For an ideal gas, the internal energy depends only on temperature, so $\Delta E = 0$, which implies that heat flow ($q$) must exactly balance work ($w$) ($q = -w$). The work performed during the expansion of a gas depends on the pathway. In a "free expansion" into a vacuum, the external pressure ($P_{\text{ex}}$) is zero, so no work is done ($w = 0$). In a one-step expansion against a constant external pressure, the work is $w = -P_{\text{ex}}\Delta V$. Increasing the number of steps in the expansion increases the amount of work the gas performs on its surroundings.

A reversible process is a hypothetical case carried out in an infinite number of steps, such that the system is always in equilibrium with its surroundings and the external pressure is always only infinitesimally different from the gas pressure ($P_{\text{ex}} \approx P_{\text{gas}}$). In this limit, the work flows out of the system is maximized. For an isothermal reversible expansion of $n$ moles of an ideal gas from volume $V_1$ to $V_2$, the work and heat are: $w_{\text{rev}} = -nRT \ln\left(\frac{V_2}{V_1}\right)$ and $q_{\text{rev}} = nRT \ln\left(\frac{V_2}{V_1}\right)$. In any real (irreversible) process, the work obtained is always less than this maximum value ($|w| < |w_{\text{rev}}|$). In a cyclic process where a gas is expanded and then compressed back to its original state, the universe only remains unchanged if both steps are reversible. In all real cyclic processes, work is converted to heat in the surroundings, increasing the disorder of the universe.

The Macroscopic and Microscopic Definitions of Entropy

Ludwig Boltzmann provided the statistical definition of entropy: $S = k_B \ln(\Omega)$, where $k_B$ is the Boltzmann constant (the gas constant per molecule, $R/N_A$) and $\Omega$ is the number of microstates. This definition shows that entropy is an additive property, while microstates are multiplicative. For the expansion of one mole of an ideal gas where each particle doubles its available positions, $\Delta S = k_B \ln(2^{N_A}) = N_A k_B \ln(2) = R \ln(2)$. More generally, for $n$ moles of gas expanding from $V_1$ to $V_2$, the entropy change is $\Delta S = nR \ln\left(\frac{V_2}{V_1}\right)$.

This statistical view connects to the macroscopic definition of entropy through heat flow. In an isothermal process, $\Delta S = \frac{q_{\text{rev}}}{T}$. This relationship allows scientists to calculate entropy changes using measurable properties like heat and temperature. For processes involving a change in temperature at constant pressure, the entropy change is calculated by $\Delta S = nC_p \ln\left(\frac{T_2}{T_1}\right)$, assuming the molar heat capacity ($C_p$) is constant. If the volume is constant, the formula uses $C_v$. For changes of state occurring at the melting point or boiling point ($T$), the process is reversible and the entropy change is $\Delta S = \frac{\Delta H}{T}$. For example, the vaporization of water at $100\,^{\circ}C$ ($373\,K$) involves $\Delta S = \frac{\Delta H_{\text{vap}}}{373\,K}$.

The Second and Third Laws of Thermodynamics

The second law of thermodynamics states that in any spontaneous process, there is always an increase in the entropy of the universe ($\Delta S_{\text{univ}} > 0$). This can be expressed as $\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}}$. If $\Delta S_{\text{univ}}$ is positive, the process is spontaneous; if negative, the reverse process is spontaneous; and if zero, the system is at equilibrium. Unlike energy, which is conserved, entropy is constantly being created in the universe.

The third law of thermodynamics establishes an absolute scale for entropy: the entropy of a perfect crystal at $0\,K$ is zero ($S = 0$). At absolute zero, all molecular motion ceases and there is only one possible microstate for a perfect regular lattice. As temperature increases, lattice vibrations and positional disorder increase, leading to higher entropy. Standard entropy values ($S^{\circ}$) at $298\,K$ and $1\,atm$ represent the entropy gained by heating a substance from $0\,K$ to $298\,K$. Generally, $S^{\circ}$ increases from solid to liquid to gas, and more complex molecules with more vibrational and rotational motions have higher standard entropies.

The Effect of Temperature on Spontaneity and Surroundings

The entropy change in the surroundings ($\Delta S_{\text{surr}}$) is driven primarily by heat flow into or out of the system. An exothermic process ($-\Delta H$) released into the surroundings increases the random motions of atoms there, thus $\Delta S_{\text{surr}}$ is positive. An endothermic process ($+\Delta H$) absorbs heat from the surroundings, decreasing their entropy ($\Delta S_{\text{surr}}$ is negative). The magnitude of this effect is temperature-dependent; the impact of heat transfer is much greater at low temperatures where existing random motion is minimal. This is defined as $\Delta S_{\text{surr}} = -\frac{\Delta H}{T}$.

Exothermicity is a major driving force for spontaneity, especially at low temperatures. In processes where $\Delta S_{\text{sys}}$ and $\Delta S_{\text{surr}}$ have opposing signs, the temperature determines which term dominates. For example, for the vaporization of water, $\Delta S_{\text{sys}}$ is positive (favorable) but $\Delta S_{\text{surr}}$ is negative (unfavorable). At temperatures above $100\,^{\circ}C$, the favorable $\Delta S_{\text{sys}}$ outweighs the unfavorable $\Delta S_{\text{surr}}$, making the process spontaneous. Below $100\,^{\circ}C$, the unfavorable $\Delta S_{\text{surr}}$ magnitude is larger, so condensation becomes spontaneous.

Free Energy and Chemical Spontaneity

Josiah Willard Gibbs defined a function called free energy ($G$) to simplify spontaneity predictions: $G = H - TS$. For a process at constant temperature and pressure, the change in free energy is $\Delta G = \Delta H - T\Delta S$. This is directly related to the entropy of the universe by $\Delta S_{\text{univ}} = -\frac{\Delta G}{T}$. Consequently, a process is spontaneous at constant $T$ and $P$ if $\Delta G$ is negative. This provides three main cases for spontaneity based on the signs of $\Delta H$ and $\Delta S$:

1. $\Delta S$ positive, $\Delta H$ negative: $\Delta G$ is always negative; spontaneous at all temperatures.
2. $\Delta S$ negative, $\Delta H$ positive: $\Delta G$ is always positive; never spontaneous.
3. $\Delta S$ and $\Delta H$ have the same sign: Spontaneity depends on temperature. If both are positive, the process is spontaneous at high temperatures (entropy driven). If both are negative, it is spontaneous at low temperatures (enthalpy driven).

Standard free energy change ($\Delta G^{\circ}$) is the change in free energy when reactants in their standard states are converted to products in their standard states. It can be calculated using $\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}$, or by using standard free energies of formation ($\Delta G^{\circ}_f$) via the summation: $\Delta G^{\circ} = \sum n_p \Delta G^{\circ}_f(\text{products}) - \sum n_r \Delta G^{\circ}_f(\text{reactants})$. Similar to Hess's Law, free energy is a state function and values for reactions can be added or manipulated according to the reaction stoichiometry. Notably, $\Delta G^{\circ}$ tells us about the equilibrium position but nothing about the speed of the reaction.

Pressure Dependence and Equilibrium

Free energy depends on the pressure of gaseous reactants and products: $G = G^{\circ} + RT \ln(P)$. For a chemical reaction, the relationship between $\Delta G$ and the reaction quotient ($Q$) is given by $\Delta G = \Delta G^{\circ} + RT \ln(Q)$. At equilibrium, the free energy of the system is at its minimum possible value, such that $\Delta G = 0$. At this point, $Q = K$ (the equilibrium constant). This leads to the fundamental equation: $\Delta G^{\circ} = -RT \ln(K)$.

This relationship allows us to predict the extent of a reaction:

• If $\Delta G^{\circ} = 0$, then $K=1$.
• If $\Delta G^{\circ} < 0$, then $K>1$ and the equilibrium lies toward the products.
• If $\Delta G^{\circ} > 0$, then $K<1$ and the equilibrium lies toward the reactants.

The temperature dependence of the equilibrium constant is described by the van't Hoff equation: $\ln(K) = -\frac{\Delta H^{\circ}}{R} \left( \frac{1}{T} \right) + \frac{\Delta S^{\circ}}{R}$. For an exothermic reaction ($\Delta H^{\circ} < 0$), a plot of $\ln(K)$ versus $1/T$ has a positive slope, meaning $K$ increases as temperature decreases. For an endothermic reaction, $K$ increases as temperature increases. This quantified relationship aligns with Le Ch"atelier's principle.

Free Energy, Work, and Adiabatic Processes

The change in free energy at constant $T$ and $P$ represents the maximum possible useful work obtainable from a process: $\Delta G = w_{\text{max useful}}$. For a spontaneous process, $\Delta G$ is the energy "free" to do work; for non-spontaneous ones, it is the minimum work required to drive the process. In any real process, some energy is always wasted as heat due to irreversibility. When energy is used, its quality is degraded as concentrated potential energy is dispersed as thermal energy ($T\Delta S$).

An adiabatic process is one in which no heat flows into or out of the system ($q = 0$), typically achieved via thermal insulation. For an adiabatic process, $\Delta E = w = nC_v\Delta T$. In an adiabatic expansion, the gas does work on the surroundings, and since no heat enters, the internal energy and temperature of the gas must decrease. For a reversible adiabatic process involving an ideal gas, the relationships are $T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}$ and $P_1 V_1^{\gamma} = P_2 V_2^{\gamma}$, where $\gamma = C_p/C_v$. Because the temperature drops during an adiabatic expansion, the final volume is significantly smaller than it would be in a comparable isothermal expansion.