What is thermodynamics?
Thermodynamics is the science of energy, heat, temperature, and the way matter flows between states. Four laws, ranked in a peculiar order, organize the whole subject: the zeroth says thermal equilibrium is transitive (if A = B and B = C, then A = C), which justifies the existence of temperature as a single number; the first says energy is conserved, that ΔU = Q − W (heat in minus work out changes a system's internal energy); the second says total entropy of an isolated system never decreases — heat flows spontaneously from hot to cold, never the reverse; and the third says a system's entropy reaches a minimum at absolute zero, which can be approached but never reached. The six calculators on this panel cover the working equations: ideal-gas and combined gas laws, heat capacity, thermal expansion in three geometries, temperature unit conversion, and Carnot efficiency.
The first law is bookkeeping, the second law is direction: the first conserves energy (you cannot win); the second says even perfectly preserved energy becomes unavailable for work (you cannot break even). Together they set the rules every engine, refrigerator, and chemical reaction must obey.
Ideal gas law — PV = nRT
For a gas sufficiently dilute and hot that its molecules barely interact, pressure, volume, moles, and absolute temperature are linked by PV = nRT, with R = 0.082057 L·atm/(mol·K) (or 8.314 J/(mol·K) in SI units). The default sample (V = 22.4 L, n = 1 mol, T = 273.15 K) returns P ≈ 1.0007 atm — the textbook molar volume at STP is 22.4 L precisely because that is what 1 mole of ideal gas occupies at 1 atm and 0°C. The law is exact for an ideal gas, and approximately correct for air, oxygen, and many other real gases at room temperature and modest pressure. At high pressure or low temperature, intermolecular forces and finite molecular size matter, and the van der Waals equation (P + a/V²)(V − b) = nRT corrects with two empirical parameters. The three other gas laws (Boyle, Charles, Gay-Lussac) are special cases of PV = nRT at constant T, V, or P; the combined gas law P1V1/T1 = P2V2/T2 handles changes in all three simultaneously.
Heat transfer — Q = m·c·ΔT
The heat absorbed by a body of mass m and specific heat capacity c changing temperature by ΔT is Q = m·c·ΔT, in joules. Specific heat is the energy per kilogram per kelvin; water's famously high value of 4,186 J/(kg·K) makes it an excellent thermal buffer. The default sample (1 kg water, 4,186 J/(kg·K), ΔT = 50 K) yields Q ≈ 209,300 J (209.3 kJ) — the energy needed to warm 1 L of cold tap water to a comfortable hot. Other materials diverge sharply: aluminum is 900 J/(kg·K), copper 385, iron 450, glass 840 — a metal heats quickly and cools quickly, water warms slowly and releases heat slowly. The 4,186 number is why hot-water bottles stay warm for so long, why ocean temperatures change only slowly across seasons, and why coastal climates are milder than continental ones.
Thermal expansion — ΔL = α·L0·ΔT
Solids expand with temperature in three ways depending on geometry: linear ΔL = αL0ΔT, area ΔA = 2αA0ΔT, volume ΔV = 3αV0ΔT, where α is the linear coefficient. The default sample (1 m of steel, α = 12×10−6/K, ΔT = 100 K) yields ΔL ≈ 1.2 mm — small enough to ignore in everyday carpentry, large enough to crack a steel bridge in winter without expansion joints. Aluminum is twice as expansive (23×10−6/K), copper and brass intermediate, glass low at 9×10−6/K. Liquids typically expand more than solids (mercury 182, water behaves anomalously below 4°C), and gases obey the ideal gas law rather than any linear expansion law. The expansion gaps in railway tracks, the bimetallic strips in thermostats, and the mercury column rising in a thermometer are all α-driven.
Temperature conversion — four scales, two anchors
The four working scales are Celsius (water's freezing and boiling points at 1 atm as 0 and 100), Fahrenheit (a similar spread with different anchors, still standard in the US), Kelvin (the absolute scale, used for all physics calculations), and Rankine (the absolute Fahrenheit scale). The default sample (100°C) yields 373.15 K, 212°F, and 671.67°R. The conversion rules: K = °C + 273.15; °F = °C × 9/5 + 32; °R = °F + 459.67. Absolute zero is 0 K = −273.15°C = −459.67°F = 0°R, the temperature at which classical thermal motion stops; the third law says no finite process can actually reach it, only approach it ever more closely.
Carnot efficiency — the upper bound for any engine
The Carnot cycle, run between a hot reservoir at Th and a cold reservoir at Tc, is the most efficient possible heat engine operating between those two temperatures, with efficiency η = 1 − Tc/Th (temperatures absolute). The default sample (Th = 500 K, Tc = 300 K) returns η ≈ 40%. A real engine burning natural gas at 1500 K and exhausting to a 300 K environment has a Carnot limit of 80%; real combined-cycle power plants reach 60%. No engine can beat Carnot — a statement that follows directly from the second law and explains why perpetual motion machines are impossible. Heat pumps and refrigerators run the same physics and have a coefficient of performance COP = Tc/(Th − Tc) in heating mode or Tc/(Th − Tc) for cooling, also capped by the same reversible limit. That is why a heat pump’s COP can exceed 1: it moves heat rather than converting it, delivering 3–4 joules of warmth per joule of electricity.
Common misconceptions
- Heat and temperature are the same thing. Heat is energy in transit; temperature is the average kinetic energy of the molecules. A bathtub of warm water contains far more heat (energy) than a sparkler at 1500 °C, because mass counts. Specific heat capacity is the conversion factor that lets the same temperature change demand wildly different amounts of heat for different materials.
- Cold flows from cold to hot. Heat flows from hot to cold, always. The sensation of "cold flowing" is your hand's heat moving into the surroundings until temperatures equalize.
- Entropy means disorder. The popular gloss is misleading. Entropy is the logarithm of the number of microstates compatible with a macrostate — a precise count, not a vague measure of chaos. Gases have higher entropy than solids because their positions are more dispersed.
- An engine can be made 100% efficient by reducing friction. Friction is part of the loss, but the irreducible loss is heat dumped into the cold reservoir. Even a frictionless engine obeying Carnot's reversible cycle is capped by 1 − Tc/Th.
Related tools: Modern Physics for the statistical mechanics behind entropy, Equilibrium for the ΔG = −RT ln K bridge between thermodynamics and reaction spontaneity, and Unit Converter for switching among calories, joules, BTU, and electron-volts.
