Enthalpy & Heat Capacity Big picture: heat vs enthalpy Heat q is energy in transit when there is a temperature difference. Enthalpy H is a property (a state function) of the system that makes it easy to count heat at constant pressure. In many lab and biological processes (open beakers, our bodies), pressure is nearly constant, so changes in enthalpy match the heat exchanged at constant pressure. Sign convention: heat released by the system is negative (exothermic), absorbed is positive (endothermic). Exothermic: heat leaves the system (ΔH < 0). Endothermic: heat enters the system (ΔH > 0). A property that depends only on the current state (P, V, T, composition), not on the path taken. Examples: U, H, S, G. State function Path function A quantity that depends on the process path. Examples: heat q and work w. Defining enthalpy H and connecting to heat Definition of enthalpy State function definition (valid for any state). Think U as microscopic motions (kinetic + potential inside), and PV as the work term linked to volume at pressure. 3D render: cylinder-piston with gas. Visualize internal energy U as jittering molecules, and PV as piston lifting against external pressure. Labels: U (molecular motion), PV (pressure-volume work), H = U + PV. Modern clean 3D style; no extra captions inside image. Why add PV? Many reactions are at near-constant external pressure (open beaker/atmospheric). When the system’s volume changes, it does pressure–volume work. The H = U + PV combination is designed so that at constant pressure, the heat exchanged equals the change in H. Valid only at constant external pressure. Heat at constant pressure For processes occurring at constant external pressure P. Linking H and U at constant pressure For a gas-phase reaction obeying ideal-gas behavior at constant temperature T and pressure, the change in volume relates to change in moles of gas. Using the ideal gas law, we can write a very useful NEET relation. Here n g = moles of gaseous products − moles of gaseous reactants. Gas-phase relation (ideal gas) neet-alert High-yield: For reactions with n g = 0 (or for condensed-phase processes where volume change is negligible), H U . Heat capacity: how much heat changes temperature Heat capacity tells you how much heat is needed to change temperature by 1 K. Think of it like a bucket’s size for heat: a large heat capacity needs a lot of heat to warm up a little; a small capacity warms quickly. C depends on the sample size and conditions. Definition of heat capacity Specific heat capacity (per gram) Here m is mass and c s is specific heat (J g⁻¹ K⁻¹). Molar heat capacity (per mole) Here n is moles and C m is molar heat capacity (J mol⁻¹ K⁻¹). Water’s high specific heat means it warms more slowly than sand for the same heat input. Great for climate and our bodies. Common specific heats (at ~25 C, J g⁻¹ K⁻¹) Items Substance Specific heat c s (J g⁻¹ K⁻¹) Water (liquid) 4.18 Ice 2.09 Steam 1.86 Ethanol 2.46 Copper 0.385 Aluminium 0.897 C p and C V for gases: why C p > C V At constant volume, all heat supplied raises internal energy (temperature). At constant pressure, some heat also does expansion work against the surroundings, so you need extra heat to get the same temperature rise. Hence, C p > C V . For an ideal gas, they are connected by Mayer’s relation. For 1 mol, C p - C V = R . Mayer’s relation (ideal gas) Also called heat capacity ratio; useful in adiabatic processes and sound speed in gases. Adiabatic index (heat capacity ratio) Why C p > C V : At constant pressure, the gas expands while heating; at constant volume, no expansion work. Both curves start at the same state; the constant-pressure path needs more heat for the same ΔT. Schematic for ideal gas heating: Two panels on white background. Left: heating at constant V (vertical line on P–V graph) no area (no work). Right: heating at constant P (horizontal line) with pale shaded area under curve showing expansion work. Labels: q V , q P , with q P > q V ; annotate C p > C V . Clean vector style, red arrows for heat. 2026-05-26T17:04:20.872Z gpt-image-2 Monatomic (e.g., He, Ne) 3/2 5/2 5/3 Diatomic / Linear (e.g., N2, O2, CO2 linear) 5/2 7/2 7/5 Triatomic non-linear (e.g., H2O) 4/3 Molecule type Degrees of freedom f C V (in R) C p (in R) γ = C p/C V Gas class Ideal-gas heat capacities from degrees of freedom (ignoring vibrations at ordinary T) Remember f: Mono 3, Di/Linear 5, Non-linear Tri 6 ⇒ C V = fR/2; C p = C V + R; = C p/C V . neet-alert Very tested: Apply C p - C V = R for 1 mol ideal gas. For multiple moles, use C p - C V = nR . Monatomic: C V = 3 2 R , C p = 5 2 R , = 5/3 . Diatomic (room T): C V = 5 2 R , C p = 7 2 R , = 7/5 . Calorimetry: measuring heat (ΔU vs ΔH) Calorimetry measures heat by tracking a known thermal mass’s temperature change. Two classic setups: a bomb calorimeter (strong steel vessel, constant volume) and a coffee-cup calorimeter (insulated cup, near-constant pressure). What they directly measure differs. Bomb calorimeter: sealed steel bomb in a water jacket with a stirrer and thermometer; ignition wire triggers combustion at constant volume → measures ΔU. gpt-image-2 2026-05-26T17:04:22.096Z Cutaway diagram of a bomb calorimeter: steel bomb with sample cup and ignition wire, oxygen fill valve; bomb immersed in water bath within insulated bucket; thermometer and stirrer; show heat flow to water and calorimeter body. Labels: constant V, measures ΔU. Clean vector schematic. Coffee-cup calorimeter: polystyrene cup with solution, thermometer, stirrer. Open to atmosphere (≈constant P) → measures q p = ΔH for the process in solution. Diagram of an insulated polystyrene cup with lid, thermometer, stir bar; reactants mix in solution. Arrows indicate heat flow to/from solution. Labels: constant pressure, q p = ΔH. Minimalist vector style. 2026-05-26T17:04:23.337Z gpt-image-2 Bomb calorimeter (constant V): Heat released by reaction = heat gained by water + calorimeter body. Use q = Cₐₛₜ ΔT for the calorimeter (if its heat capacity is given). For the system (reaction), q v = ΔU. Coffee-cup (constant P): Heat of reaction in solution equals negative of heat gained/lost by solution and cup: q reaction = - ( m solution c s ΔT + C cup ΔT). For processes in solution, q p = ΔH. Sign care: If the solution warms up (ΔT > 0), the reaction is exothermic ( q reaction < 0). How calculations are set up Human body thermoregulation: Water-rich tissues (high specific heat) buffer temperature swings. Metabolism of glucose is exothermic (ΔH < 0) and releases energy for cellular work. Clinically, cooling blankets or warm packs exploit heat capacities to adjust body temperature safely. clinical Quick worked ideas (NEET-style reasoning) 1) If a gas reaction has n g = -2 at 298 K and U = -100 kJ, then H = U + n g RT -100 , kJ + (-2)(8.314 10 -3 , kJ mol -1 , K -1 )(298 , K ) . The correction is about −4.95 kJ, so H -105 kJ. 2) Heating 100 g water by 10 K: q = mc s T = 100 4.18 10 = 4180 J = 4.18 kJ. Common traps (and how to avoid them) q is energy transferred during a process (path function). H is a state function. Only under constant pressure does H = q p hold. Heat (q) and enthalpy (H) are the same thing. At constant pressure, part of the heat goes into expansion work, so more heat is required for the same T . Therefore C p > C V . C p equals C V ; both just heat capacities so they must be the same. Only when n g = 0 (no net change in moles of gas) or for condensed-phase processes where P V is negligible do we get H U . In general, H = U + n g RT for ideal gases. ΔH always equals ΔU for any reaction. Specific heat capacity is the same as heat capacity. Specific heat capacity is per gram ( c s , J g⁻¹ K⁻¹); heat capacity C is for the whole object (J K⁻¹). Molar heat capacity is per mole (J mol⁻¹ K⁻¹). Glossary (quick revision) Key terms State function defined by H = U + PV. At constant pressure, heat exchanged equals ΔH ( q p ). Enthalpy (H) Heat required for a body to change its temperature by 1 K: C = q/ΔT. Heat capacity (C) Heat capacity per gram: q = m c s ΔT. Units: J g−1 K−1. Specific heat capacity ( c s ) Heat capacity per mole: q = n C m ΔT. Units: J mol−1 K−1. Molar heat capacity ( C m ) Molar heat capacity at constant pressure for gases; C p − C V = R (1 mol ideal gas). C p Molar heat capacity at constant volume for gases; lower than C p for the same gas. C V For an ideal gas: C p − C V = nR (or = R for 1 mol). Mayer’s relation Heat capacity ratio: γ = C p / C V . Adiabatic index (γ) Independent ways a molecule can store energy (translation + rotation at ordinary T; vibrations add at higher T). For ideal-gas estimates: f = 3 (monatomic), 5 (linear), 6 (non-linear). Degrees of freedom (f) Measurement of heat exchanged using temperature changes of a known thermal mass. Calorimetry Constant-volume device for combustion; measures ΔU. Bomb calorimeter Near-constant-pressure setup for solution reactions; measures q p = ΔH. Coffee-cup calorimeter Exam hack: For quick ΔH corrections from ΔU, compute n g RT at 300 K using R ≈ 8.3 J mol⁻¹ K⁻¹ → ≈ 2.5 kJ per mole of gas change (since 8.3×10⁻³ kJ K⁻¹ mol⁻¹ × 300 K ≈ 2.5 kJ mol⁻¹). tip