Topic 9: Applications of Thermodynamics

Unit 9 ties together energy and disorder to predict whether a reaction will happen on its own. You will use entropy and Gibbs free energy to judge thermodynamic favorability, connect deltaG to the equilibrium constant, and apply the same ideas to electrochemical cells, the Nernst equation, and electrolysis.

Entropy and the Sign of deltaS

Entropy (S) measures how spread out the energy and matter of a system are among the available microstates. The more ways particles and their energy can be arranged, the higher the entropy. For a chemical change, you can usually predict the sign of deltaS by reasoning about disorder. Entropy increases (deltaS positive) when a solid melts or a liquid vaporizes, when a solid or liquid dissolves, when the number of moles of gas increases, and when temperature rises. Entropy decreases (deltaS negative) when gas moles fall, when a substance freezes or condenses, or when ions in solution combine into a precipitate. The single most reliable clue on the AP exam is the change in moles of gas: if gas moles go up, deltaS is almost always positive. Gases carry far more entropy than liquids, which carry more than solids, so phase matters more than the number of formula units alone.

Absolute Entropy and Standard Entropy Change

Unlike enthalpy, entropy has a true zero point: a perfect crystal at 0 K has zero entropy. Because of this, every substance has a positive absolute standard entropy, S degree, tabulated in J/(mol K). Note the units are joules, not kilojoules. To find the standard entropy change of a reaction, use the same products-minus-reactants pattern as Hess's law: deltaS degree = sum of (n times S degree of products) minus sum of (n times S degree of reactants), where n is the stoichiometric coefficient. A reaction that produces more gas typically gives a positive deltaS degree, confirming the qualitative reasoning from the previous section. Always include phase labels when looking up S degree values, since the gas, liquid, and solid forms of the same compound have very different entropies.

Gibbs Free Energy and Thermodynamic Favorability

Gibbs free energy (G) combines enthalpy and entropy into a single quantity that decides whether a process is thermodynamically favored at constant temperature and pressure. A reaction is thermodynamically favored (sometimes loosely called spontaneous) when deltaG is negative, unfavored when deltaG is positive, and at equilibrium when deltaG equals zero. Important caution: favorability says nothing about speed. A reaction can have a very negative deltaG yet proceed so slowly that no change is observed, because the rate depends on activation energy, not on deltaG. The standard free energy of reaction can be calculated either from standard free energies of formation (products minus reactants) or from deltaH degree and deltaS degree using the equation in the next section.

deltaG = deltaH - T deltaS (Worked Example)

The central equation of the unit is deltaG = deltaH - T deltaS, where T is the absolute temperature in kelvin. The biggest trap is units: deltaH is usually in kJ while deltaS is in J/K, so you must convert one of them before subtracting. Worked example: a reaction has deltaH = -92 kJ and deltaS = -199 J/K. Find deltaG at 298 K. First convert: deltaS = -0.199 kJ/K. Then deltaG = -92 kJ - (298 K)(-0.199 kJ/K) = -92 kJ - (-59.3 kJ) = -92 + 59.3 = -32.7 kJ. Since deltaG is negative, the reaction is thermodynamically favored at 298 K. Notice that the entropy term opposes favorability here; at a high enough temperature, -T deltaS would outweigh deltaH and deltaG would turn positive. When deltaH and deltaS have the same sign, temperature determines the outcome; when they have opposite signs, the reaction is favored (or unfavored) at all temperatures.

Relationship Between deltaG and K

Thermodynamic favorability and the equilibrium position describe the same chemistry. They are linked by deltaG degree = -RT ln K, where R = 8.314 J/(mol K), T is in kelvin, and K is the equilibrium constant. Reading this equation: when deltaG degree is negative, ln K is positive, so K is greater than 1 and products are favored at equilibrium. When deltaG degree is positive, K is less than 1 and reactants dominate. When deltaG degree equals zero, K equals 1. Under nonstandard conditions, the actual driving force is deltaG = deltaG degree + RT ln Q, where Q is the reaction quotient. A reaction proceeds in the forward direction whenever Q is less than K, which is exactly when deltaG is negative; it reaches equilibrium when Q equals K and deltaG becomes zero.

Coupled Reactions

A reaction with a positive deltaG is unfavored on its own, but it can still be driven forward by coupling it to a strongly favored reaction. Because free energy is a state function, the deltaG values of added reactions sum just as in Hess's law. If an unfavored step (positive deltaG) is paired with a favored step (large negative deltaG) that shares a common species, the overall deltaG can become negative and the combined process is favored. Living cells exploit this constantly: the hydrolysis of ATP, which has a large negative deltaG, is coupled to biosynthetic reactions that would otherwise be unfavored. In industry, the same trick lets a metal be extracted by coupling an unfavored decomposition to a favored reaction such as the oxidation of carbon.

Galvanic vs Electrolytic Cells

Electrochemistry applies thermodynamics to reactions that transfer electrons. In every cell, oxidation (loss of electrons) happens at the anode and reduction (gain of electrons) happens at the cathode; the mnemonic 'an ox, red cat' helps. A galvanic (voltaic) cell harnesses a thermodynamically favored redox reaction to produce electricity: its overall deltaG is negative and its cell potential E_cell is positive. An electrolytic cell does the opposite, using an external power source to drive an unfavored reaction: deltaG is positive and E_cell is negative, so electrical energy is supplied to force the change. The relationship deltaG = -nFE_cell makes the connection precise, where n is the moles of electrons transferred and F is Faraday's constant, 96485 C/mol. A positive E_cell guarantees a negative deltaG and a favored reaction.

Cell Potential and the Nernst Equation

Standard cell potential is found from standard reduction potentials: E_cell = E_cathode - E_anode, both taken as reduction potentials from a table. A larger (more positive) standard cell potential means a more strongly favored reaction. Standard potentials assume 1 M concentrations and 1 atm pressure; when conditions differ, the actual potential shifts according to the Nernst equation, E_cell = E_cell degree - (RT / nF) ln Q. At 298 K this is often written as E_cell = E_cell degree - (0.0592 / n) log Q. The qualitative message matches Le Chatelier: increasing reactant concentration (lowering Q) raises E_cell, while building up products (raising Q) lowers it. As a galvanic cell discharges, Q rises toward K and E_cell falls toward zero, the dead-battery condition.

Electrolysis and Faraday's Laws

Electrolysis is the use of electrical current to drive an unfavored redox reaction, such as plating a metal or decomposing water. The amount of substance produced is governed by stoichiometry of electrons. The key quantitative chain is: charge (coulombs) = current (amperes) times time (seconds); moles of electrons = charge divided by Faraday's constant (96485 C/mol); then use the half-reaction's electron coefficient to convert moles of electrons to moles, and finally grams, of product. Worked idea: passing 2.00 A for 1.00 hour delivers 2.00 times 3600 = 7200 C, which is 7200 / 96485 = 0.0746 mol of electrons. For a metal deposited by M-2+ plus 2 e- yielding M, that produces 0.0746 / 2 = 0.0373 mol of metal. Watch your time units (convert hours and minutes to seconds) and always read the charge on the ion to get the right number of electrons per atom.