Thermochemistry and Thermodynamics

Heat and Temperature

Temperature (T, units: K or °C) measures the average kinetic energy of particles in a system — it is an intensive property, independent of sample size. Heat (q, units: J) is the transfer of thermal energy between system and surroundings due to a temperature difference — it is an extensive property. The relationship between heat and temperature change is given by q = mcΔT, where m is mass, c is the specific heat capacity (J/g·K), and ΔT = T_final − T_initial. For a pure substance, the molar heat capacity C = molar mass × c. When a substance undergoes a phase change, heat is absorbed or released with NO temperature change: q = nΔH_phase_change (fusion, vaporization, sublimation). The total heat for a multi-step temperature-phase change process is the sum of q = mcΔT terms for temperature ramps plus q = nΔH terms for phase transitions. Thermochemistry classifies reactions as exothermic (q < 0, heat released to surroundings, system temperature rises if adiabatic) or endothermic (q > 0, heat absorbed from surroundings, system temperature falls). The sign convention is always from the system's perspective: exothermic means the system loses energy; endothermic means the system gains energy. The MCAT frequently tests heat transfer in calorimetry and the distinction between heat (a path function, depends on how the process occurs) and state functions like enthalpy (path-independent).

Calorimetry

Calorimetry measures the heat exchanged in a chemical or physical process. Two experimental setups dominate the MCAT. Constant-pressure calorimetry (coffee-cup calorimeter): the reaction occurs at atmospheric pressure, so the measured heat equals the enthalpy change: q_p = ΔH. The calorimeter absorbs or releases heat, calculated as q_cal = mcΔT, and by conservation of energy, q_rxn = −q_cal (assuming no heat loss to surroundings). The negative sign is critical — if the solution heats up, the reaction released that heat (exothermic, q_rxn negative). If the solution cools, the reaction absorbed heat (endothermic, q_rxn positive). Constant-volume calorimetry (bomb calorimeter): the reaction occurs in a sealed, rigid container at constant volume. Under these conditions, no PV work is done, so the measured heat equals the change in internal energy: q_v = ΔU. To find ΔH from bomb calorimetry data, use ΔH = ΔU + Δn_gas RT, where Δn_gas is the change in moles of gas. The calorimeter constant C_cal (kJ/K) is determined by calibration with a reaction of known ΔU (typically benzoic acid combustion). The heat released by the sample is q = −C_cal × ΔT. A common MCAT task: given calorimeter constant and temperature change, calculate the heat of combustion per gram or per mole. Always check whether the question asks for ΔH (per mole of reaction as written) or the heat released by a specific mass of sample.

Enthalpy

Enthalpy H is defined as H = U + PV, where U is internal energy, P is pressure, and V is volume. Enthalpy is a state function — its change depends only on the initial and final states, not the path between them. At constant pressure, the enthalpy change equals the heat transferred: ΔH = q_p. The standard enthalpy change ΔH° refers to the enthalpy change when all reactants and products are in their standard states (1 bar pressure, 1 M concentration for solutions, pure substance in its most stable form at the specified temperature — typically 298 K). The standard enthalpy of formation ΔH°_f is the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states. By definition, ΔH°_f of any element in its standard state is zero. The standard enthalpy of reaction can be calculated from formation enthalpies: ΔH°_rxn = Σ n ΔH°_f(products) − Σ n ΔH°_f(reactants), where n are the stoichiometric coefficients. This is the most commonly tested enthalpy calculation. Bond enthalpy (bond dissociation energy) provides an alternative route: ΔH°_rxn ≈ Σ (bond energies of bonds broken) − Σ (bond energies of bonds formed). Bond-breaking is endothermic (+); bond-forming is exothermic (−). The bond enthalpy method is approximate because tabulated bond energies are averages over many molecules, but it reliably gives the correct sign and approximate magnitude. The MCAT also tests standard enthalpy of combustion (ΔH°_comb), the heat released when one mole of a substance is completely burned in excess oxygen — always exothermic for fuels. Enthalpy is extensive — doubling the coefficients in a balanced equation doubles ΔH. Reversing the direction flips the sign of ΔH.

Entropy

Entropy S is a thermodynamic state function that measures the dispersal of energy and matter — loosely, the 'disorder' or 'number of accessible microstates' of a system. The second law of thermodynamics states that for any spontaneous process, the total entropy of the universe increases: ΔS_universe = ΔS_system + ΔS_surroundings > 0. A process at equilibrium has ΔS_universe = 0. The entropy change of the surroundings is related to the heat transferred: ΔS_surroundings = −ΔH_system / T (at constant T and P). An exothermic reaction (ΔH < 0) increases the entropy of the surroundings (heat is dispersed into the surroundings); an endothermic reaction (ΔH > 0) decreases it. Qualitatively, entropy increases when: gases form from liquids or solids (Δn_gas > 0), the number of moles of gas increases, a solid dissolves into ions in solution, a more complex molecule dissociates into simpler ones, or temperature increases (more accessible microstates). Entropy decreases when: gases condense, fewer gas molecules form, or highly ordered structures crystallize. The third law of thermodynamics states that a perfect crystal at absolute zero (0 K) has zero entropy — exactly one microstate. This establishes an absolute reference point for entropy values. Standard molar entropies S° (J/mol·K) are tabulated at 298 K. Key MCAT insight: gases always have much higher entropies than liquids or solids of similar molecular weight — translational entropy dominates. For a reaction, ΔS°_rxn = Σ n S°(products) − Σ n S°(reactants). The MCAT expects you to predict the sign of ΔS from the balanced equation: more gas molecules on the product side → ΔS > 0.

Gibbs Free Energy

Gibbs free energy G is the thermodynamic potential that determines spontaneity at constant temperature and pressure. It is defined as G = H − TS. At constant T, the change is ΔG = ΔH − TΔS. This is the single most important equation in MCAT thermodynamics. ΔG has units of kJ/mol. The standard free energy change ΔG° is measured under standard-state conditions (1 bar, 1 M, 298 K). ΔG° can be calculated from: (1) standard free energies of formation: ΔG°_rxn = Σ n ΔG°_f(products) − Σ n ΔG°_f(reactants); (2) enthalpy and entropy: ΔG° = ΔH° − TΔS°, where ΔH° and ΔS° are each calculated from tabulated values; (3) the equilibrium constant: ΔG° = −RT ln K, where R = 8.314 J/mol·K and T is in Kelvin. At 298 K, this simplifies to ΔG° (kJ/mol) ≈ −5.7 log₁₀ K, meaning every factor-of-10 increase in K corresponds to roughly −5.7 kJ/mol in ΔG°. This equation is tested constantly — given K, find ΔG°; given ΔG°, find K; or compare relative spontaneity of two reactions from their K values. Note that ΔG° and ΔG are different: ΔG° refers to standard-state conditions; ΔG refers to actual (nonstandard) conditions. Under nonstandard conditions: ΔG = ΔG° + RT ln Q, where Q is the reaction quotient. At equilibrium, ΔG = 0 and Q = K, recovering ΔG° = −RT ln K. The MCAT also links ΔG° to cell potential E°: ΔG° = −nFE°, where n is moles of electrons transferred and F = 96,485 C/mol (Faraday constant). This bridges thermodynamics with electrochemistry — the two domains form a single quantitative framework.

Spontaneity

A process is spontaneous (thermodynamically favored) if ΔG < 0 at the specified temperature and pressure. The interplay of ΔH and ΔS determines the temperature window of spontaneity through ΔG = ΔH − TΔS. When ΔH is negative (exothermic) and ΔS is positive, ΔG is negative at all temperatures — spontaneous always. When ΔH is negative and ΔS is negative, ΔG is negative at low T and positive at high T — spontaneous below T = ΔH/ΔS. When ΔH is positive (endothermic) and ΔS is positive, ΔG is positive at low T and negative at high T — spontaneous above T = ΔH/ΔS. When ΔH is positive and ΔS is negative, ΔG is positive at all temperatures — never spontaneous. The crossover temperature where ΔG = 0 is T = ΔH/ΔS (units must match — convert ΔH to J/mol if ΔS is in J/mol·K). Below this temperature, the ΔH term dominates; above it, the TΔS term dominates. This table is the MCAT's favorite spontaneity logic tool — expect questions that ask you to determine the temperature range for spontaneity from given signs, or to deduce the signs of ΔH and ΔS from the observation that a reaction is spontaneous only at high temperatures. Phase changes are classic examples: melting is endothermic (ΔH > 0) and increases entropy (ΔS > 0), so it is spontaneous above the melting point. Freezing is exothermic (ΔH < 0) and decreases entropy (ΔS < 0), so it is spontaneous below the freezing point. At the melting point, ΔG = 0, so T_melt = ΔH_fusion / ΔS_fusion — a direct application. The MCAT frequently tests boiling/vaporization: ΔH_vap > 0, ΔS_vap > 0, spontaneous above T_boil. Coupled reactions are a biochemical application: an unfavorable reaction (ΔG > 0) can be driven forward by coupling it to a highly favorable reaction (ΔG << 0) in the same overall process. The overall ΔG is the sum of the two ΔG values, so if ΔG_total < 0, the coupled process proceeds. ATP hydrolysis (ΔG°' ≈ −30.5 kJ/mol under cellular conditions) is the universal energy currency that drives otherwise unfavorable biosynthetic reactions.

Hess's Law

Hess's law states that the enthalpy change for an overall reaction is the sum of the enthalpy changes for a series of steps that add up to the overall reaction — enthalpy is a state function, so the path does not matter. The MCAT tests three applications. First, combining known reactions: given two or more thermochemical equations with known ΔH values, manipulate them (reverse, multiply, add) so they sum to the target reaction, and perform the same operations on their ΔH values. Reversing a reaction flips the sign of ΔH; multiplying coefficients multiplies ΔH by the same factor. Second, calculating ΔH°_rxn from standard enthalpies of formation: ΔH°_rxn = Σ n ΔH°_f(products) − Σ n ΔH°_f(reactants). This is a direct application of Hess's law using formation reactions as the intermediate steps. Third, calculating ΔH°_rxn from bond enthalpies: ΔH°_rxn ≈ Σ (BE of bonds broken) − Σ (BE of bonds formed). Bond-breaking always requires energy input (endothermic, +), and bond-forming always releases energy (exothermic, −), so the sum of bond energies of reactants minus products tracks the net enthalpy change. The bond enthalpy method works because atoms in reactants are first separated (breaking all bonds, endothermic), then reassembled into products (forming all bonds, exothermic) — the sum of these two steps gives the overall ΔH°_rxn. Hess's law also applies to ΔG and ΔS: the same additivity rules hold because G and S are state functions. The MCAT may ask you to combine multiple thermochemical equations or to identify the set of manipulations needed to reach a target equation — accuracy with coefficient multipliers and sign flips is essential.

How it works

Thermodynamics on the MCAT operates as a decision tree. Start by classifying the reaction: do you need ΔH, ΔS, ΔG, or K? For ΔH, choose your route — formation enthalpies (ΔH° = Σ n_f ΔH°_f(products) − Σ n_i ΔH°_f(reactants)), bond enthalpies (ΔH° ≈ Σ BE_broken − Σ BE_formed), calorimetry (q_rxn = −mcΔT or −C_cal ΔT), or Hess's law (combine known reactions). For ΔS, count gas molecules and phase changes qualitatively, or calculate ΔS° = Σ n S°(products) − Σ n S°(reactants) from tabulated values. For spontaneity at a given T, compute ΔG = ΔH − TΔS and check the sign. To find the crossover temperature, set ΔG = 0 to get T = ΔH/ΔS. To connect to equilibrium, use ΔG° = −RT ln K. To connect to electrochemistry, use ΔG° = −nFE°. Every variable in these equations is connected — change one, and the others shift. The MCAT rewards seeing the web of relationships rather than memorizing isolated formulas. The most common passage structure presents a novel reaction, gives ΔH° and S° values or formation data, and asks you to (1) calculate ΔG° at 298 K, (2) determine if the reaction is spontaneous at that temperature, (3) find the temperature at which spontaneity switches, (4) compute K from ΔG°, and (5) predict how increasing temperature shifts equilibrium — all from the same initial data.

How it works

Thermodynamics on the MCAT operates as a decision tree. Start by classifying the reaction: do you need ΔH, ΔS, ΔG, or K? For ΔH, choose your route — formation enthalpies (ΔH° = Σ n_f ΔH°_f(products) − Σ n_i ΔH°_f(reactants)), bond enthalpies (ΔH° ≈ Σ BE_broken − Σ BE_formed), calorimetry (q_rxn = −mcΔT or −C_cal ΔT), or Hess's law (combine known reactions). For ΔS, count gas molecules and phase changes qualitatively, or calculate ΔS° = Σ n S°(products) − Σ n S°(reactants) from tabulated values. For spontaneity at a given T, compute ΔG = ΔH − TΔS and check the sign. To find the crossover temperature, set ΔG = 0 to get T = ΔH/ΔS. To connect to equilibrium, use ΔG° = −RT ln K. To connect to electrochemistry, use ΔG° = −nFE°. Every variable in these equations is connected — change one, and the others shift. The MCAT rewards seeing the web of relationships rather than memorizing isolated formulas. The most common passage structure presents a novel reaction, gives ΔH° and S° values or formation data, and asks you to (1) calculate ΔG° at 298 K, (2) determine if the reaction is spontaneous at that temperature, (3) find the temperature at which spontaneity switches, (4) compute K from ΔG°, and (5) predict how increasing temperature shifts equilibrium — all from the same initial data.

Comparisons

• C/P (Equilibrium): ΔG° = −RT ln K. At 298 K, ΔG° (kJ/mol) ≈ −5.7 log₁₀ K. This is the quantitative bridge between thermodynamics and equilibrium — K = 10^(−ΔG°/5.7) at 298 K. When ΔG° changes (e.g., with temperature), K changes accordingly via the van't Hoff equation.
• C/P (Kinetics): Thermodynamics tells you if a reaction can happen (ΔG < 0); kinetics tells you if it will happen on a meaningful timescale. A reaction with ΔG << 0 can still be kinetically inert if the activation energy (E_a) is high. The MCAT loves contrasting thermodynamic favorability with kinetic stability.
• C/P (Electrochemistry): ΔG° = −nFE° and E° = (RT/nF) ln K. Given any one of ΔG°, K, or E°, you can compute the other two. A positive E° means ΔG° < 0 (spontaneous); a negative E° means ΔG° > 0 (nonspontaneous).
• C/P (Gases): ΔH = ΔU + Δn_gas RT. In a bomb calorimeter (constant V), you measure ΔU; to report ΔH, add the Δn_gas RT correction. In a coffee-cup calorimeter (constant P), you measure ΔH directly.
• B/B (ATP and metabolism): ATP hydrolysis (ATP → ADP + Pi) has ΔG°' ≈ −30.5 kJ/mol under cellular conditions (not standard), driving unfavorable reactions through coupled mechanisms. The overall ΔG of a coupled pathway is the sum of the individual ΔG values — ΔG is additive.
• B/B (Enzyme catalysis): Enzymes lower activation energy (kinetics) but do NOT change ΔG, ΔG°, or K — these are thermodynamic properties fixed by the reactants and products. The MCAT explicitly tests that a catalyst affects rate, not thermodynamic favorability.
• B/B (Protein folding): Protein folding is spontaneous (ΔG < 0) under physiological conditions. ΔH is typically small and negative (favorable interactions), while ΔS_system is negative (chain goes from disordered to folded) — but the hydrophobic effect releases ordered water, making ΔS_surroundings positive and ΔS_universe > 0.
• B/B (Membrane transport): Moving a solute against its concentration gradient has ΔG > 0 (nonspontaneous). Active transport couples this to ATP hydrolysis, making the overall process spontaneous.

Common confusions

• Confusing ΔG (actual conditions) with ΔG° (standard conditions). ΔG° = −RT ln K is the standard-state free energy change; ΔG = ΔG° + RT ln Q is the free energy change under any conditions. The reaction proceeds forward when ΔG < 0, not when ΔG° < 0.
• Forgetting to convert units in ΔG = ΔH − TΔS. ΔH is often given in kJ/mol while ΔS is in J/mol·K. You MUST convert one to match: either ΔH in J/mol (multiply by 1000) or TΔS in kJ/mol (divide by 1000). Failing to do this produces ΔG values off by three orders of magnitude.
• Assuming ΔH and ΔS are temperature-independent. At the MCAT level, ΔH and ΔS are treated as approximately constant over moderate temperature ranges. The crossover temperature T = ΔH/ΔS is valid only within this approximation.
• Misapplying the sign convention for bond enthalpies. ΔH°_rxn = Σ BE(bonds broken) − Σ BE(bonds formed). Bonds broken require energy (+), bonds formed release energy (−). Many students write Σ BE(reactants) − Σ BE(products) but forget that bond-breaking is endothermic — the formula already accounts for sign, so just use the tabulated positive bond energies.
• Treating ΔS_system alone as the criterion for spontaneity. The second law requires ΔS_universe > 0. An endothermic reaction (ΔS_surroundings negative) can still be spontaneous if ΔS_system is sufficiently positive to make ΔS_universe positive. ΔG = ΔH − TΔS incorporates both system and surroundings contributions in a single criterion.
• Confusing ΔH and q in bomb calorimetry. At constant volume, q_v = ΔU, not ΔH. To get ΔH, add Δn_gas RT. The MCAT often tests this distinction: 'The heat measured in a bomb calorimeter equals which thermodynamic quantity?'
• Forgetting that ΔG°_f of elements in standard states is zero — just like ΔH°_f. This is a direct parallel: both are zero for elements. Students sometimes mistakenly look up ΔG°_f values for O₂(g) or C(s, graphite).
• Mixing up the temperature effect on spontaneity. When ΔH < 0 and ΔS < 0, increasing temperature makes ΔG less negative (potentially positive) — the reaction becomes LESS favorable at high T. When ΔH > 0 and ΔS > 0, increasing temperature makes ΔG more negative (potentially spontaneous) — the reaction becomes MORE favorable at high T.
• Neglecting the difference between ΔG° and ΔG°' (biochemical standard state). ΔG°' uses pH 7 ([H⁺] = 10⁻⁷ M) and [H₂O] = 1. ATP hydrolysis ΔG°' is approximately −30.5 kJ/mol, NOT the standard ΔG° value — this is MCAT-relevant for coupled reactions in metabolism.
• Miscomputing the Δn_gas RT correction. Δn_gas = (moles of gaseous products) − (moles of gaseous reactants). Only gases count — aqueous species, liquids, and solids are excluded. At 298 K, RT ≈ 2.48 kJ/mol, so the correction is typically small but can be tested explicitly.

Quick review

• First law: ΔU = q + w (energy is conserved). At constant V, w = 0, so ΔU = q_v. At constant P, w = −PΔV, so ΔU = q_p − PΔV, giving ΔH = ΔU + PΔV = q_p.
• q = mcΔT (no phase change). m in g, c in J/g·K. For phase changes: q = nΔH_phase (constant T).
• Calorimetry: constant-P (coffee-cup) → q_p = ΔH; constant-V (bomb) → q_v = ΔU. q_rxn = −q_cal (conservation). ΔH = ΔU + Δn_gas RT.
• H = U + PV. ΔH is a state function (path-independent). ΔH°_rxn = Σ n ΔH°_f(products) − Σ n ΔH°_f(reactants).
• ΔH°_rxn ≈ Σ BE(bonds broken) − Σ BE(bonds formed). Bonds broken = endothermic (+). Bonds formed = exothermic (−).
• Second law: ΔS_universe = ΔS_system + ΔS_surroundings > 0 for spontaneous. ΔS_surroundings = −ΔH_system / T (at constant T, P).
• Entropy increases with: more gas molecules, higher T, dissolution, more particles. S°_gas >> S°_liquid > S°_solid.
• Third law: S = 0 for a perfect crystal at 0 K. Standard molar entropies S° are tabulated at 298 K.
• G = H − TS. ΔG = ΔH − TΔS (at constant T). ΔG < 0 → spontaneous; ΔG = 0 → equilibrium; ΔG > 0 → nonspontaneous.
• Spontaneity table: ΔH −, ΔS + → always spontaneous. ΔH +, ΔS − → never spontaneous. ΔH −, ΔS − → spontaneous at low T. ΔH +, ΔS + → spontaneous at high T. Crossover T = ΔH/ΔS.
• ΔG° = −RT ln K. At 298 K: ΔG° (kJ/mol) ≈ −5.7 log₁₀ K. ΔG = ΔG° + RT ln Q (nonstandard conditions).
• ΔG° = −nFE° (electrochemistry bridge). n = mol e⁻, F = 96,485 C/mol. Positive E° ⇔ negative ΔG° ⇔ spontaneous.
• Hess's law: overall ΔH = sum of ΔH of steps. Same for ΔG and ΔS (state functions). Reverse reaction → flip sign. Multiply coefficients → multiply ΔH.
• Formation values: ΔH°_f and ΔG°_f of any element in its standard state = 0. This is true for both enthalpy and free energy.
• Coupled reactions: a nonspontaneous reaction (ΔG > 0) can be driven by coupling to ATP hydrolysis or another favorable reaction. Overall ΔG = ΔG₁ + ΔG₂; process proceeds if sum < 0.
• Phase changes at transition temperature: ΔG = 0, so T_melt = ΔH_fusion / ΔS_fusion and T_boil = ΔH_vap / ΔS_vap.
• Standard conditions: 1 bar (~1 atm), 1 M (solute), pure substance in most stable form, 298 K (unless specified). Biochemical standard ΔG°' uses pH 7.