Thermodynamics

Thermodynamics is the study of energy moving as heat and work: why heat flows from hot to cold, how much energy it takes to warm or melt something, and why no engine can turn all of its heat into work. In the 2027 event you use that physics twice. A written test checks the ideas and the calculations, and a device you build before the tournament has to soak up energy from a heat lamp and pass it to a cup of room-temperature water while you predict how warm the water will get. The same habits win both parts: think about where the energy goes, and put numbers on it quickly.

Event format

Thermodynamics is a hybrid event with the same structure in Division B and Division C.

Everything you impound has to be at room temperature, and you may not bring or use your own thermometer during Part II (or borrow the supervisor's). Your prediction comes from the data you collected at home and the graphs or tables you bring.

What the rules cover

The written test draws at least three questions from each topic area in the 2027 outline. The table paraphrases the outline and shows where each topic applies.

The rules also limit the math. Both divisions use arithmetic with ratios, exponents and absolute value, plus area and volume of common shapes. Division B algebra stays at solving one equation for one unknown; Division C may ask for multi-step derivations with several variables. Neither division may require trigonometry or calculus. Answers are expected in metric units with sensible significant figures unless a question says otherwise.

Systems, properties and temperature

Systems and surroundings

A system is whatever you choose to study: the water in the cup, the gas in a piston, a whole engine. Everything else is the surroundings, and the boundary separates them. Systems are classified by what can cross that boundary.

The cup of water in Part II is close to a closed system while it heats: energy arrives by radiation, conduction and convection, but very little water leaves (a small amount evaporates, which technically makes it open).

Intensive and extensive properties

An extensive property depends on how much material you have: mass, volume, internal energy, heat capacity, and (for Division C at State and National) entropy and enthalpy. Cut the sample in half and the property halves. An intensive property does not depend on amount: temperature, pressure, density, specific heat and molar mass. Cutting the sample in half leaves it unchanged.

A quick test: the ratio of two extensive properties is intensive. Mass ÷ volume gives density; heat capacity ÷ mass gives specific heat.

State functions and path functions

The state of a simple system at equilibrium is fixed by a few state variables: for a fixed amount of a gas, any two of $P$, $V$ and $T$ fix the third. Knowing $n$ alone is not enough; you need the amount plus two independent variables. A state function depends only on the present state, not on how the system got there: $P$, $V$, $T$, internal energy $U$, enthalpy $H$ and entropy $S$ are all state functions. So the change in a state function between two states is the same for every route, and it is zero around any complete cycle.

Heat $Q$ and work $W$ are path functions. They are energy in transit, so how much of each crosses the boundary depends on the route. On a P-V diagram, two different paths between the same two states have different areas under them (different $W$), so they need different $Q$ too, but $\Delta U = Q - W$ comes out the same. A system can be described as "having" internal energy, but never as "having" heat or work.

A system is in thermodynamic equilibrium when nothing inside it changes on its own: it is in thermal equilibrium (one uniform temperature), mechanical equilibrium (uniform pressure, no unbalanced forces) and chemical and phase equilibrium. Only equilibrium states have single, well-defined values of $P$ and $T$, which is why only equilibrium states are points on a P-V diagram.

Temperature is not heat

Temperature measures the average kinetic energy of the particles in a substance. For an ideal gas the link is exact: the average translational kinetic energy per molecule is $\tfrac{3}{2}k_B T$, with $k_B = 1.38 \times 10^{-23}$ J/K and $T$ in kelvin. Heat is energy in transit because of a temperature difference. An object does not "contain heat"; it contains internal energy, and heat is what crosses its boundary.

A swimming pool and a cup of tea can share a temperature while the pool holds vastly more internal energy. Temperature tells you which way heat will flow; how much flows depends on mass, specific heat and time. Heat always flows from the higher temperature to the lower one, even when the colder object holds more total internal energy (an iceberg holds more internal energy than a cup of hot coffee, yet heat flows from the coffee to the iceberg).

Internal energy $U$ is the total microscopic energy of a system: the kinetic energy of its particles (translation, rotation, vibration) plus the potential energy of the forces between them. Temperature tracks only the average kinetic part. That is why 1 kg of steam at 100 °C holds more internal energy than 1 kg of liquid water at 100 °C: same temperature, but the steam's molecules have been pulled apart against their attractions. For an ideal gas there are no attractions, so $U$ depends on temperature alone.

Absolute zero (0 K, −273.15 °C, −459.67 °F) is the temperature at which particle motion is at its minimum possible value. It is not quite "all motion stops": quantum mechanics leaves a small zero-point energy. Nothing can be colder, so negative kelvin readings in an answer choice are impossible (for ordinary systems), and the third law says even 0 K itself cannot be reached.

The zeroth law and thermometers

The zeroth law says that if A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A and B are in thermal equilibrium with each other. It is what makes a thermometer meaningful: the thermometer (C) comes to equilibrium with the water (A), and its reading tells you what any other object (B) at the same reading would be. The law was named "zeroth" because it was recognized as logically more basic only after the first and second laws already had their numbers. Calibrating a thermometer in ice water and boiling water, and reading a thermometer placed in a cup of tea, are the standard examples of the zeroth law at work.

Every thermometer uses a thermometric property: something that changes in a known, repeatable way with temperature. It is calibrated against fixed points such as the ice point and steam point of water or, more precisely, the triple point of water (273.16 K).

A liquid-in-glass thermometer works only because the liquid expands much more than the glass bulb holding it. Plunge one into hot water and the reading first dips slightly: the glass warms and expands before heat reaches the liquid, so the bulb's volume grows first; then the liquid catches up and the column rises. A good thermometer liquid expands uniformly (so the scale is linear), stays liquid over the range needed, and is easy to see.

The earliest instrument was Galileo's thermoscope (around 1600): air trapped in a glass bulb above a column of water, so the water level moved as the air expanded or contracted. It showed temperature changes but had no numbered scale and also responded to air pressure. Sealed liquid-in-glass thermometers with scales followed in the 1600s and 1700s. A thermometer measures temperature, not internal energy or heat, so "a thermometer measures internal energy" is false. Calorimeters, not thermometers, are the devices that measure heat.

Temperature scales and conversions

Celsius and Fahrenheit are relative scales with arbitrary zeros. Kelvin and Rankine are absolute scales that put zero at absolute zero. One kelvin is the same size as one Celsius degree, and one Rankine degree is the same size as one Fahrenheit degree.

The direct routes save steps on a test: ${}^\circ C = ({}^\circ F - 32)/1.8$; $K = ({}^\circ F + 459.67)/1.8$; ${}^\circ F = 1.8\,K - 459.67$; ${}^\circ R = 1.8\,({}^\circ C + 273.15)$.

ReferenceLandmark temperatures on four scales· 10 rows

Values at 1 atm. Body temperature is the conventional 37 °C.

10 rows

Landmark	K	°C	°F	°R
Absolute zero	0	−273.15	−459.67	0
Liquid nitrogen boils	77.4	−195.8	−320.4	139.3
Dry ice sublimes	194.65	−78.5	−109.3	350.4
Celsius and Fahrenheit agree	233.15	−40	−40	419.67
Fahrenheit zero	255.37	−17.78	0	459.67
Water freezes (ice point)	273.15	0	32	491.67
Triple point of water	273.16	0.01	32.02	491.69
Room temperature (typical)	293–298	20–25	68–77	528–537
Human body	310.15	37	98.6	558.27
Water boils	373.15	100	212	671.67

How to read it: the Kelvin and Celsius columns always differ by 273.15, and the Rankine and Fahrenheit columns by 459.67. Between the ice point and the steam point there are 100 Celsius degrees and 180 Fahrenheit degrees, which is where the factor 1.8 (= 9/5) comes from.

Some history of the scales is fair game because it explains their fixed points. Daniel Fahrenheit (1724) set 0 °F at the temperature of an ice-water-salt (brine) mixture and used roughly human body temperature as his upper reference (about 96 on his original scale); water then froze at 32 and boiled at 212. Anders Celsius (1742) originally put 100 at the freezing point and 0 at the boiling point; the scale was reversed shortly after his death and was long called "centigrade". Kelvin's absolute scale (1848) and William Rankine's Fahrenheit-sized absolute scale (1859) both start at absolute zero. On a Celsius thermometer the degree is the same size as the kelvin, which is why scientific work pairs those two. Neither scale is more "accurate"; a Fahrenheit degree is just a smaller step.

Since 2019 the kelvin has been defined by fixing the numerical value of the Boltzmann constant. Before that it was defined through the triple point of water, 273.16 K. Older tests may phrase it either way.

Heat-related units

Energy is the capacity to do work or to transfer heat. Work is energy transferred when a force acts through a distance (for a gas, when its volume changes against a pressure); heat is energy transferred because of a temperature difference. Both are measured in joules, and neither is "stored" in an object; what an object stores is internal energy.

Related quantities carry their own units: specific heat in J/(kg·K) or J/(g·°C), latent heat in J/kg, thermal conductivity in W/(m·K), and pressure in pascals (1 kPa = 1,000 Pa; 1 atm = 101,325 Pa).

ReferenceConstants for the binder· 11 rows

11 rows

Constant	Value
Gas constant $R$	8.314 J/(mol·K) = 0.08206 L·atm/(mol·K) = 1.987 cal/(mol·K) = 62.36 L·torr/(mol·K)
Boltzmann constant $k_B = R/N_A$	1.380649 × 10⁻²³ J/K (exact since 2019)
Avogadro's number $N_A$	6.022 × 10²³ mol⁻¹
Stefan-Boltzmann constant $\sigma$	5.67 × 10⁻⁸ W/(m²·K⁴)
Wien's constant $b$	2.898 × 10⁻³ m·K
Planck's constant $h$	6.626 × 10⁻³⁴ J·s
Speed of light $c$	3.00 × 10⁸ m/s
Molar volume of an ideal gas	22.4 L/mol at 0 °C and 1 atm; 24.5 L/mol at 25 °C and 1 atm
Water: $c$, $L_f$, $L_v$	4,186 J/(kg·K) (1 cal/(g·°C)); 334 kJ/kg (6.01 kJ/mol); 2,256 kJ/kg (40.7 kJ/mol)
Ice and steam: $c$	about 2,100 and 2,000 J/(kg·K)
Solar constant	about 1,361 W/m² at the top of Earth's atmosphere

The units also tell you what a quantity is. Heat capacity is J/K (or BTU/°R, BTU/°F, cal/°C); specific heat is J/(kg·K); entropy is J/K, the same units as heat capacity; molar quantities add "/mol". Emissivity is a pure ratio with no units. An answer that comes out in the wrong units signals a formula used wrongly.

Thermal expansion

Most materials expand when heated because their particles vibrate with larger amplitude and, since the forces between atoms are lopsided (easier to pull apart than to push together), the average spacing grows. Expansion is the property most thermometers use, and it is why bridges have expansion joints and power lines sag on hot days.

Why $\beta \approx 3\alpha$: a cube of side $L_0$ becomes $L_0(1 + \alpha\Delta T)$ on each side, so its volume becomes $V_0(1 + \alpha\Delta T)^3 \approx V_0(1 + 3\alpha\Delta T)$ because the squared and cubed terms in a tiny $\alpha\Delta T$ are negligible. A hole in a plate expands exactly as if it were filled with the plate material, so heating a metal lid loosens it on a glass jar.

How to use it: liquids expand roughly several to a hundred times more than solids, and gases more still. Because a container expands too, a liquid's apparent expansion is its own minus the container's, which is one reason liquid expansion is harder to measure than solid expansion. When a solid is heated its mass stays the same while its volume grows, so its density falls.

A bimetallic strip bonds two metals with different $\alpha$, such as brass and steel. On heating, the high-$\alpha$ metal lengthens more, so the strip curls with the high-$\alpha$ metal on the outside of the curve; on cooling it curls the other way. Old thermostats, dial thermometers and some circuit breakers use this.

Phases, latent heat and the ideal gas law

Phases and phase transitions

Solids hold their particles in fixed positions, so they keep their shape and volume; liquids let particles slide past one another, so they keep their volume but take the container's shape; gases let particles move freely, so they fill whatever container they are in. Plasma, a gas so hot that electrons are stripped from atoms, conducts electricity and is the most common state of visible matter in the universe (stars). A Bose-Einstein condensate, formed near absolute zero, is a fifth state seen only in labs. Which phase a substance is in depends on the competition between particle kinetic energy (set by temperature) and intermolecular attractions (and on pressure, which pushes particles together).

For almost every substance the density order is solid > liquid > gas, because particles pack most tightly in the solid. Water is the famous exception: ice is less dense than liquid water (0.917 versus 1.000 g/cm³) because hydrogen bonds hold the ice crystal in an open hexagonal framework. A gas is roughly a thousand times less dense than its liquid; that is why a gas is easy to compress and a liquid or solid is not.

Every forward transition toward more particle freedom (melting, vaporizing, subliming) is endothermic ($\Delta H > 0$) and raises entropy (the spread of energy over more possible arrangements; Division C covers it in detail); every reverse transition is exothermic and lowers the substance's entropy.

Evaporation is vaporization from the surface, at any temperature below the boiling point. Boiling is vaporization throughout the liquid, with bubbles of vapor forming inside it, and it happens when the liquid's vapor pressure equals the pressure above it. Vaporization is the umbrella term for both. Evaporation cools what is left behind, because the fastest molecules escape and the average kinetic energy of those remaining drops. That is why sweating, a wet cloth in a breeze and alcohol-based hand sanitizer all cool the skin.

Vapor pressure is the pressure exerted by a liquid's own vapor in equilibrium with it. It rises steeply with temperature and does not depend on how much liquid there is. Changing the outside pressure therefore changes the boiling point:

• At high altitude the air pressure is lower, so water boils below 100 °C (about 95 °C in Denver, about 70 °C on top of Mount Everest), and food takes longer to cook.
• A pressure cooker raises the pressure to about 2 atm absolute, so water boils near 120 °C and food cooks faster.
• A liquid boils at room temperature in a vacuum chamber, and an inverted flask of hot water starts boiling again when cold water on the outside condenses the vapor inside and lowers the pressure.

Phase changes do not always happen on schedule. A very pure, undisturbed liquid can be supercooled below its freezing point, or superheated above its boiling point, until a disturbance or seed crystal triggers the change. During any phase change at equilibrium both phases coexist, and the rate of melting equals the rate of freezing (or vaporizing equals condensing).

Sensible and latent heat

Sensible heat changes temperature without changing phase. Latent heat changes phase without changing temperature. During melting or boiling the energy goes into breaking intermolecular attractions rather than into faster particle motion, so the thermometer reading stays flat.

A heating curve plots temperature against heat added. Sloped segments are sensible heating, with slopes set by $1/(mc)$ for each phase; flat segments are phase changes, with lengths set by $mL$. Because $L_v$ for water is almost seven times $L_f$, the boiling plateau is far longer than the melting plateau.

Reading a heating curve: the x axis is heat added (or time, at a steady heating rate). On a sloped segment the particles speed up, so average kinetic energy and temperature rise; on a plateau the added energy raises the potential energy of the particles as attractions are broken, and the kinetic energy stays the same. A steeper slope means a smaller heat capacity ($\Delta T/Q = 1/(mc)$). A cooling curve is the same graph run backwards, with plateaus where the substance releases its latent heat.

In chemistry units, latent heats are molar enthalpies: for water $\Delta H_{fus} = 6.01$ kJ/mol and $\Delta H_{vap} = 40.7$ kJ/mol at 100 °C (44.0 kJ/mol at 25 °C). Convert with the molar mass: $6.01 \text{ kJ/mol} \div 18.02 \text{ g/mol} = 0.334$ kJ/g. Vaporization always takes far more energy than melting because the molecules must be separated completely rather than just loosened.

Phase diagrams

A phase diagram maps which phase is stable at each pressure and temperature. The lines are coexistence curves where two phases are in equilibrium. They meet at the triple point, where all three coexist (for water, 0.01 °C and about 612 Pa). The liquid-vapor line ends at the critical point, beyond which liquid and gas are no longer distinct (a supercritical fluid).

Read a diagram by asking two things: which region holds the given point, and which line a process crosses. Moving right at constant pressure is heating; moving up at constant temperature is compressing. Below the triple-point pressure a solid sublimates directly to vapor, which is why dry ice (CO₂) never melts at 1 atm.

The names of the parts are tested directly:

• Fusion (melting) curve: solid-liquid boundary. Vaporization curve: liquid-gas boundary, which is also the vapor-pressure curve; its intersection with the 1 atm line is the normal boiling point. Sublimation curve: solid-gas boundary below the triple point. Near the triple point the sublimation curve is steeper than the vaporization curve, because subliming takes more latent heat (fusion plus vaporization).
• Triple point: the single pressure and temperature where solid, liquid and gas coexist in equilibrium.
• Critical point: the end of the liquid-gas line. Above the critical temperature no amount of pressure can liquefy the gas; above both the critical temperature and pressure the substance is a supercritical fluid, which flows like a gas but dissolves things like a liquid (supercritical CO₂ decaffeinates coffee). At the critical point the liquid and vapor densities become equal.
• A point on a line has two phases in equilibrium; a point in a region has one.
• Gibbs phase rule: $F = C - P + 2$, where $C$ is the number of components (1 for pure water), $P$ the number of phases present and $F$ the degrees of freedom (how many of $T$ and $P$ you can change independently while keeping the same phases). In a one-phase region $F = 2$ (an area); on a two-phase line $F = 1$ (choose $T$ and $P$ is fixed); at the triple point $F = 0$ (a single point). That is why a pure substance cannot have four phases coexisting.

Why water's melting line slopes backwards: ice is less dense than liquid water, so squeezing ice favors the denser liquid, and the melting point drops as pressure rises (by about 0.0075 °C per atmosphere). For almost every other substance the solid is denser and the line slopes forward. If a question just says "a substance" and asks whether raising pressure raises or lowers its melting point, the honest answer is that it depends on whether the solid or the liquid is denser.

The same physics looks different on a pressure-volume diagram. Compress a gas slowly at a fixed temperature below its critical temperature: its pressure rises along a hyperbola-like isotherm until droplets appear, then stays perfectly flat (the vapor pressure) while gas turns to liquid, then shoots up steeply once only liquid is left, because liquids barely compress. The ends of all those flat segments trace out a dome-shaped saturation curve: its left side is the saturated-liquid line, its right side the saturated-vapor line, and inside the dome liquid and vapor coexist. The top of the dome is the critical point, where the critical isotherm has no flat part, only a horizontal inflection. Isotherms above the critical temperature never cross the dome; far above it they approach the ideal-gas hyperbolas $PV$ = constant.

The Clausius-Clapeyron relation

The slope of any coexistence line is set by the latent heat and by the volume change of the transition. This is the Clapeyron equation; it explains both water's backward melting line and how quickly vapor pressure rises with temperature.

Trouton's rule (Division C, State & National, since it uses entropy) is a handy check: for many liquids the molar entropy of vaporization at the normal boiling point is about 85 to 88 J/(mol·K), so $\Delta H_{vap} \approx 88\,T_b$ in J/mol. Water (109 J/(mol·K)) is higher because hydrogen bonding orders the liquid.

The ideal gas law

The ideal gas model treats particles as points that collide elastically and do not attract one another. Real gases behave this way at low pressure and high temperature. The ideal gas law combines Boyle's, Charles's and Gay-Lussac's laws with Avogadro's principle.

The gas laws hide a trap in the word "double". Doubling the volume of a gas at constant temperature halves its pressure (Boyle). Doubling the Celsius temperature of a rigid container of gas at 20 °C (to 40 °C) raises its pressure by only $313/293 = 1.07$ times, "less than double", because the pressure is proportional to the kelvin temperature.

Two more gas-law tools. Density form: since $n = m/M$, the ideal gas law becomes $PM = \rho RT$, so a measured density gives the molar mass (air at 1 atm and 20 °C: $\rho = 101{,}325 \times 0.029/(8.314 \times 293.15) = 1.21$ kg/m³). Standard conditions: "STP" traditionally means 0 °C and 1 atm, where one mole of ideal gas occupies 22.4 L; the current IUPAC standard is 0 °C and 1 bar, giving 22.7 L. Use the definition the question states.

Extrapolating to absolute zero. Plot the volume of any gas at constant pressure (or its pressure at constant volume) against Celsius temperature and you get a straight line. Lines for every gas, at every pressure, extend back to meet zero volume or pressure at the same point, −273.15 °C. Real gases liquefy long before, but this extrapolation is how absolute zero was first estimated and why the kelvin scale starts there.

Kinetic theory: what temperature and pressure mean

The kinetic molecular theory pictures a gas as a huge number of tiny particles in random motion, colliding elastically with each other and the walls, with no forces between collisions. Pressure is the average force per area of molecules bouncing off the walls. Temperature is proportional to the average translational kinetic energy of the molecules:

The curve is lopsided, so its three common "average" speeds come in a fixed order: most probable speed < mean speed < rms speed. The tail of fast molecules explains evaporation below the boiling point and why reactions speed up when warmed. Maxwell derived this distribution around 1860 and Boltzmann generalized it. The same $1/\sqrt{M}$ dependence gives Graham's law: the rate at which a gas effuses through a tiny hole is inversely proportional to the square root of its molar mass, $\text{rate}_1/\text{rate}_2 = \sqrt{M_2/M_1}$, so helium (4 g/mol) effuses $\sqrt{32/4} = 2.8$ times as fast as oxygen (32 g/mol).

Heat capacities of an ideal gas

Each independent way a molecule can store energy (a degree of freedom) holds on average $\tfrac{1}{2}k_BT$ (the equipartition theorem). A monatomic gas (He, Ne, Ar) has only three translational degrees of freedom. A diatomic gas (N₂, O₂, air) at room temperature adds two rotations, for five. Vibrations only switch on at high temperature.

$C_P$ is always larger than $C_V$ (by exactly $R$ per mole for an ideal gas, Mayer's relation) because at constant pressure the gas expands as it warms, so part of the heat goes into work on the surroundings instead of into internal energy. Because an ideal gas's internal energy depends only on temperature, $\Delta U = nC_V\Delta T$ for any process, not only constant-volume ones. For solids, the molar heat capacity approaches about $3R$ ≈ 25 J/(mol·K) at ordinary temperatures (the Dulong-Petit law), which is why metals with heavier atoms have smaller specific heats per gram.

Real gases

Real gases deviate from $PV = nRT$ at high pressure (molecules take up a real fraction of the volume) and at low temperature near condensation (attractions pull molecules together and lower the pressure). The ideal gas law works best at low pressure, low density and high temperature relative to the critical point. The van der Waals equation corrects both effects:

$$\left(P + \frac{a\,n^2}{V^2}\right)(V - n\,b) = nRT$$

where $a$ measures the attractions (larger for easily liquefied gases) and $b$ is the volume excluded by the molecules themselves. Below its critical temperature a gas can be liquefied by pressure alone.

Heat transfer, conductivity and heat capacity

Heat moves by three mechanisms, and tests often describe a situation in one sentence and ask which one dominates. All three matter for the Part II device.

Conduction

Conduction is energy passed through a material by particle collisions and, in metals, by free electrons. The material itself does not move. The rate through a slab follows Fourier's law.

Thermal conductivity $k$ is a material property. Approximate room-temperature values: copper about 400, aluminum about 240, glass about 1, water about 0.6, wood about 0.1 to 0.2, polystyrene foam about 0.03, and still air about 0.026 W/(m·K). Foams insulate well because they trap small pockets of air that cannot circulate.

The rate of conduction depends on the material ($k$), the area, the temperature difference and the thickness; it does not depend on the mass of the slab. Metals conduct well because free electrons carry energy quickly; nonmetals rely on lattice vibrations alone. Gases conduct poorly because their molecules are far apart, which is why trapped air is the working part of almost every insulator: foam, fiberglass, down, wool, double-pane windows, and fresh snow (snow insulates better than solid ice). Conductivity changes somewhat with temperature, so tabulated values are for a stated temperature.

ReferenceThermal conductivity of common materials· 10 rows

Approximate values near room temperature, W/(m·K). Ranges reflect different grades and sources.

10 rows

Material	$k$	Material	$k$
Diamond	900–2,300	Concrete	0.8–1.7
Silver	429	Ice (0 °C)	2.2
Copper	400	Water	0.6
Aluminum	237	Brick	0.4–0.8
Brass	about 110	Human tissue	0.2–0.5
Iron	80	Wood	0.1–0.2
Steel (carbon)	about 50	Snow (fresh)	0.05–0.25
Stainless steel	about 15	Fiberglass, wool	about 0.04
Lead	35	Polystyrene foam	about 0.03
Glass	about 1	Air (still)	0.026

Thermal resistance and R-values

Builders describe insulation by its thermal resistance, the thickness divided by the conductivity. Resistances of layers stacked in series simply add, like resistors in series, which makes multi-layer walls easy.

Convection

Convection carries energy by the bulk motion of a fluid. Warmed fluid expands, becomes less dense and rises; cooler fluid sinks to replace it. In natural (free) convection density differences alone drive the flow; in forced convection a fan, pump or wind does. Convection needs a fluid, so it cannot happen through a solid or across a vacuum, and it happens only in liquids and gases.

Examples tests use: a pot of water heated from below (circulating convection cells); a room radiator or a space heater with a fan; a forced-air furnace blowing warm air through ducts; air conditioning (cold air sinks, so vents work well high up, and heaters work well low down); a hand held above a candle burns while one beside it does not (hot gases rise); sea breezes by day (land warms faster, air rises over it and cooler sea air flows in) and land breezes at night; blood carrying heat around the body; mantle convection and magma; boiling. Wind chill is forced convection: moving air strips away the warm layer next to your skin.

The coefficient is not a material property like $k$: it depends on the fluid, its speed, the shape of the surface and the temperature difference, which is why forced flow and phase change give much larger values.

Radiation

Thermal radiation is energy carried by electromagnetic waves, mostly infrared at everyday temperatures. It needs no medium, which is how sunlight crosses space and how a heat lamp warms your hand from half a meter away. Every object above absolute zero radiates, and hotter objects radiate much more strongly. How well a surface absorbs radiation is described by its absorptivity; a good absorber is also a good emitter at the same wavelength. Matte black surfaces absorb most visible and infrared light; shiny metal surfaces reflect most of it.

When radiation strikes a surface, each part of it is absorbed, reflected or transmitted, and the three fractions add to 1: absorptivity + reflectivity + transmissivity = 1. For an opaque surface, transmissivity is 0, so a good reflector is a poor absorber. Kirchhoff's law of thermal radiation adds that, at a given wavelength and temperature, a surface's emissivity equals its absorptivity: good absorbers are good emitters, and good reflectors (shiny, low-emissivity surfaces) are poor emitters. A black matte can of hot water cools faster than a shiny one at the same temperature, and a black surface in sunlight warms faster.

Reflective "radiant barrier" insulation and emergency blankets therefore have high reflectivity and low emissivity. A vacuum flask (thermos) combines all three ideas: the vacuum between its double walls stops conduction and convection, and the silvered walls cut radiation. The greenhouse effect is a radiation story: sunlight (mostly visible and near-infrared, from a hot source) passes through glass or the atmosphere, the warmed ground re-emits long-wave infrared, and glass or greenhouse gases absorb much of that infrared and send part of it back down (a real glass greenhouse stays warm mostly because it blocks convection, so the atmospheric effect is the purer radiation example). Spacecraft and satellites, surrounded by vacuum, can shed waste heat only by radiation from radiator panels.

Tests sometimes list evaporation as a fourth cooling route; strictly it is mass transfer that carries latent heat, usually helped by convection. "Induction" is not a heat-transfer mechanism.

Heat capacity, specific heat and calorimetry

Heat capacity $C = Q/\Delta T$ is the energy an object needs per kelvin of warming; for a single material $C = mc$. A material with high specific heat warms slowly for a given energy input, which is why water is such a good heat store and why 100 mL of water in Part II needs about 419 J for every 1 °C of rise ($0.100 \text{ kg} \times 4186$ J/(kg·K)).

ReferenceSpecific heats of common materials· 10 rows

Near room temperature unless stated. Divide by 1,000 for J/(g·°C); divide J/(g·°C) by 4.186 for cal/(g·°C).

10 rows

Material	$c$, J/(kg·K)	Material	$c$, J/(kg·K)
Water (liquid)	4,186	Aluminum	about 900
Ice (near 0 °C)	about 2,100	Concrete	about 880
Steam (near 100 °C)	about 2,000	Glass	about 840
Ethylene glycol	about 2,400	Sand, granite	about 800
Ethanol	about 2,440	Iron, steel	about 450
Human body (average)	about 3,500	Copper	about 385
Wood	about 1,700	Brass	about 380
Air (constant pressure)	about 1,005	Silver	about 235
Olive or vegetable oil	about 2,000	Mercury	about 140
Paraffin wax	about 2,500	Lead, gold	about 129

Water's specific heat is unusually high because much of the heat goes into stretching and breaking its network of hydrogen bonds rather than into raising molecular speed. That is why coastal climates are mild, why water is used as a coolant and heat store, and why sand on a beach gets far hotter than the sea on the same afternoon. Metals have low specific heats per gram because their atoms are heavy: per mole, most solid elements need about the same $3R$ (Dulong-Petit), so fewer atoms per gram means less heat per gram.

Calorimetry measures heat through temperature change. In an insulated mixture, the heat lost by the warm parts equals the heat gained by the cool parts.

A calorimeter is an insulated container used to measure heat. In a coffee-cup calorimeter (constant pressure, open to the air) the measured heat equals the enthalpy change, $q_P = \Delta H$. In a bomb calorimeter the reaction happens inside a sealed steel vessel of fixed volume immersed in water, so no expansion work is done and the measured heat equals the change in internal energy, $q_V = \Delta U$. For a bomb calorimeter with a known total heat capacity $C_{\text{cal}}$, $q = C_{\text{cal}}\Delta T$, and the reaction's heat is the negative of that.

Newton's law of cooling

For modest temperature differences, an object's rate of heat loss is roughly proportional to its temperature difference from the surroundings. That makes temperature approach room temperature along an exponential curve, fast at first and then slowly.

The key fact is that the difference $T - T_{\text{env}}$ shrinks by the same fraction in every equal time interval. The constant $k$ is always positive (units of 1/time) and is larger for small objects with large surface area, good conductors and moving air. For an object whose inside stays nearly uniform in temperature, convection theory gives it directly: $\tau = 1/k = mc/(hA)$, so a large heat capacity $mc$ slows cooling and a large $hA$ speeds it. Because the difference only shrinks by a fraction each interval, the object approaches room temperature ever more slowly and, in this model, reaches it only after infinite time; in practice, after about $5\tau$ the remaining difference is under 1% of the starting one. A negative $k$ would make the difference grow, so a cooling object would get hotter than its surroundings on its own, which the second law forbids. Solve for $k$ or $t$ by taking a natural log: $k = -\dfrac{1}{t}\ln\dfrac{T - T_{\text{env}}}{T_0 - T_{\text{env}}}$. Logarithms are allowed arithmetic on a calculator; no calculus is needed.

The idea matters for Part II in reverse: as your cup warms above room temperature it starts losing heat faster, so the temperature rise per minute shrinks over the Heating Time. A device that keeps losses low holds its warming rate up for longer.

Processes, cycles, engines and the laws

The first law

The first law is energy conservation with heat included: energy cannot be created or destroyed, only transferred or converted. The change in a system's internal energy equals the heat added to it minus the work it does on its surroundings.

For a gas at constant pressure, work done by the gas is $W = P\,\Delta V$ (Pa × m³ = J; or L·atm × 101.3 = J). On a pressure-volume (P-V) graph, the work is the area under the process curve: positive when the volume grows, negative when it shrinks, zero for a vertical (constant-volume) line.

The first law forbids a perpetual motion machine of the first kind, one that puts out more energy than it takes in. An engine claimed to deliver 10 hp (7,460 W) from 6,000 W of input violates the first law; an engine that delivers work with no heat input at all does too.

The four standard processes

Here $\gamma = C_P/C_V$, about 1.67 for a monatomic gas and 1.40 for a diatomic gas such as air. On a P-V diagram an adiabat through a point is steeper than the isotherm through the same point. A bicycle pump warming as you pump and a spray can cooling as it empties are both close to adiabatic, because the change is too fast for much heat to flow. Other "iso" names: isenthalpic (constant enthalpy, as in throttling) and isentropic (constant entropy, which is a reversible adiabatic process).

Reversible, irreversible and quasi-static processes

A quasi-static process happens so slowly that the system passes through a continuous series of equilibrium states; only such a process can be drawn as a line on a P-V diagram. A reversible process is quasi-static and free of dissipation, so it could be run backwards with both the system and the surroundings returned exactly to where they started. It is an ideal limit: the total entropy of system plus surroundings stays constant.

Every real process is irreversible to some degree, and increases total entropy. The usual sources of irreversibility (textbooks group them as mechanical, thermal and chemical) are:

• friction and other dissipation (stirring, electrical resistance),
• heat flowing across a finite temperature difference,
• unrestrained (free) expansion of a gas,
• mixing of different substances, and chemical reactions,
• inelastic deformation.

Two named irreversible processes appear on tests:

• Free (Joule) expansion: a gas rushes into a vacuum inside an insulated container. $Q = 0$ and $W = 0$ (nothing is pushed), so $\Delta U = 0$ and an ideal gas's temperature is unchanged, but its entropy rises by $nR\ln(V_2/V_1)$. It is adiabatic but not isentropic.
• Throttling (Joule-Thomson expansion): a gas is forced through a valve or porous plug from high to low pressure with no heat exchange. Enthalpy is constant (isenthalpic). An ideal gas's temperature does not change, but most real gases at room temperature cool; this cooling drives refrigerator expansion valves and the liquefaction of gases (hydrogen and helium warm instead unless they are first cooled below their inversion temperatures).

Cycles and heat engines

A thermodynamic cycle returns a system to its starting state, so $\Delta U = 0$ over the whole loop and the net work equals the net heat. A heat engine runs a cycle that takes heat $Q_H$ from a hot reservoir, does work $W$, and rejects $Q_C$ to a cold reservoir.

The Carnot cycle is the ideal reversible cycle: isothermal expansion at $T_H$ (absorbing heat), adiabatic expansion down to $T_C$, isothermal compression at $T_C$ (rejecting heat), and adiabatic compression back to $T_H$. No engine between the same two reservoirs can beat it.

The Carnot result has consequences tests love. Efficiency depends only on the two reservoir temperatures, not on the working substance (air, steam, helium) or the engine's design. It rises if $T_H$ goes up or $T_C$ goes down, and lowering $T_C$ by some amount helps more than raising $T_H$ by the same amount (between 373 K and 273 K, a 10 K change gains 2.7 percentage points by lowering $T_C$ but only 1.9 by raising $T_H$). A claimed engine that beats $1 - T_C/T_H$ is impossible: one said to take 100 J from boiling water, reject heat to a 10 °C room and deliver 80 J of work claims 80% when the limit is $1 - 283/373 = 24\%$. Real engines fall well short of the limit because of friction, heat leaks and finite-speed (irreversible) heat transfer: car engines run at roughly 20–30% (diesels up to about 40%), and large power plants at about 33–45% (combined-cycle plants up to about 60%).

Named cycles

Real engines follow other cycles, and tests show their P-V diagrams. Recognize each by its shape: vertical segments are isochoric, horizontal segments are isobaric, and the steep curves are adiabats (isentropic in the ideal cycle). The Carnot cycle is the only one built from isotherms and adiabats alone; it is impractical because truly isothermal steps must run infinitely slowly.

A four-stroke engine runs intake, compression, power and exhaust strokes, two crankshaft turns per cycle. The ideal Otto efficiency depends only on the compression ratio $r = V_{\max}/V_{\min}$:

Diesel engines use higher compression ratios (about 14 to 23) because they compress only air, which cannot knock; the high ratio is what makes them more efficient than gasoline engines. At the same compression ratio, though, the ideal Otto cycle is the more efficient of the two; diesels come out ahead only because they can use much higher ratios.

On a temperature-entropy (T-S) diagram (Division C, State & National) the Carnot cycle is a rectangle: two horizontal isotherms and two vertical adiabats (constant entropy). The heat absorbed is $Q_H = T_H\Delta S$, the heat rejected is $Q_C = T_C\Delta S$, and the enclosed area $(T_H - T_C)\Delta S$ is the net work, which gives $\eta = 1 - T_C/T_H$ at a glance.

Refrigerators, air conditioners and heat pumps

Refrigerators and heat pumps are heat engines run in reverse: work $W$ moves heat $Q_C$ out of a cold space and dumps $Q_H = Q_C + W$ into a warm one. The cold space is the cold reservoir; the room (or the outdoors, for an air conditioner) is the hot reservoir. An air conditioner is a refrigerator for a house; a heat pump is the same machine used to heat the warm side.

The working fluid, the refrigerant, carries heat around a loop:

1. Compressor (driven by the work input): squeezes low-pressure vapor into hot, high-pressure vapor.
2. Condenser (coils on the back or outside): the hot vapor gives off heat $Q_H$ to the room and condenses to a liquid.
3. Expansion valve (throttling): the liquid's pressure drops suddenly, and part of it flashes to vapor, cooling it far below the cold-space temperature.
4. Evaporator (coils inside the cold space): the cold refrigerant absorbs heat $Q_C$ and evaporates, then returns to the compressor.

Home heat pumps are classed by where they get their heat: air-source heat pumps (the most common; they draw heat from outdoor air), ground-source or geothermal heat pumps (pipes buried in the ground, where the temperature stays nearly constant year-round, giving a higher COP in winter), and water-source heat pumps (a well, lake or water loop). Running the same machine backwards in summer makes it an air conditioner.

Leaving a refrigerator door open to cool a kitchen makes the kitchen warmer: the fridge dumps $Q_C + W$ into the same room it takes $Q_C$ from.

The second law

The second law has two classic statements that are logically equivalent:

• Clausius statement: heat does not flow on its own from a colder body to a hotter one. A refrigerator needs work input.
• Kelvin-Planck statement: no cyclic engine can take heat from a single reservoir and turn all of it into work. Every engine must reject some heat.

Together they explain why the Carnot limit exists and why real processes run one way. If either statement could be broken, a device could be built that breaks the other. In entropy language (Division C at State and National goes further below): the total entropy of an isolated system never decreases, and it increases in every irreversible process; an isolated system settles into equilibrium, the state of maximum entropy.

A spontaneous process is one that happens on its own once started, without continuing outside work: heat flowing from hot to cold, a gas spreading into a vacuum, ice melting in a warm room, iron rusting. The reverse processes are nonspontaneous; they can be made to happen only by doing work (a refrigerator, a pump). "Spontaneous" says nothing about speed: diamond turning to graphite is spontaneous but immeasurably slow. A spontaneous process always increases the entropy of the universe (system plus surroundings), even if the system's own entropy falls, as when water freezes below 0 °C and releases heat that raises the surroundings' entropy more.

A perpetual motion machine of the second kind would run in a cycle and turn heat from a single reservoir entirely into work (for example, a ship powered by cooling the ocean). It would not violate energy conservation, but it violates the second law. Isothermal expansion of an ideal gas turns all the absorbed heat into work in one step, which is allowed: it is not a cycle, and the gas ends up at a larger volume.

Other ideas tied to the second law: the arrow of time (we see eggs break but never unbreak); heat death of the universe, the far-future state in which everything reaches one temperature and no work can be extracted; and Maxwell's demon (see History), a thought experiment that seemed to break the second law until it was shown that the demon's information handling (above all, erasing its memory) generates at least as much entropy as its sorting removes. Biology does not escape the law: a steer must eat far more grain than the muscle it builds, because each conversion loses energy as heat.

History of thermodynamics

The 2027 outline names eight scientists. History questions come in three forms: "who did this?" (match a person to a discovery, book or quotation), "what did this person do?" (choose-all-that-apply), and "put these in order" or "which century". Learn each person with two or three contributions, a key year and a phrase that identifies them.

ReferenceThe eight named scientists· 8 rows

8 rows

Scientist (lifespan, country)	Key contributions	Key dates	Identifying phrases
Joseph Black (1728–1799, Scotland; born in Bordeaux, France)	Distinguished heat (quantity) from temperature (intensity); discovered latent heat (ice absorbs heat while melting without warming) and introduced specific heat (heat capacity); also identified carbon dioxide ("fixed air")	CO₂ 1750s; latent heat about 1761–62 (taught in Glasgow and Edinburgh lectures, published after his death)	"latent heat", "specific heat", "heat ≠ temperature"; friend and supporter of James Watt
Sadi Carnot (Nicolas Léonard Sadi Carnot, 1796–1832, France)	Reflections on the Motive Power of Fire (1824), the first theory of heat engines: the ideal reversible engine (Carnot cycle), maximum efficiency set only by the reservoir temperatures, independent of working substance; first form of the second law; first idea of reversibility	1824; died of cholera at 36	"father of thermodynamics", "motive power of fire", "perfect engine"; he still used the caloric (heat-fluid) theory
James Prescott Joule (1818–1889, England)	Showed heat and work are equivalent forms of energy: the mechanical equivalent of heat (falling weights turning a paddle wheel in insulated water); Joule's law of electrical heating ($P = I^2R$); free-expansion experiment; the Joule-Thomson effect with Kelvin; the SI unit of energy is named for him	Electrical heating 1840–41; first announced 1843 (British Association); paddle-wheel experiment 1845, refined value read in 1849 and published in 1850 (about 772 ft·lbf per BTU, about 4.16 J/cal)	Brewer from Salford near Manchester; "mechanical equivalent of heat"; led to conservation of energy (first law); helped overturn caloric theory
William Thomson, Lord Kelvin (1824–1907, born Belfast, worked in Glasgow)	Absolute temperature scale based on Carnot's theory (1848), with absolute zero near −273 °C; Kelvin statement of the second law (1851, On the Dynamical Theory of Heat); coined the term "thermodynamic(s)" (1849–1854); Joule-Thomson effect; idea of the dissipation of energy and "heat death"; transatlantic telegraph engineering	1848, 1851; made Baron Kelvin in 1892	"absolute zero", "absolute scale", "kelvin unit", defined thermodynamics as the relation of heat to forces between parts of bodies and to electrical agency
Rudolf Clausius (1822–1888, Germany)	First clear statement of the second law (1850, On the Moving Force of Heat), reconciling Carnot's engine theory with Joule's conservation of energy; introduced the quantity he later named entropy (1854; named 1865, from the Greek for "transformation"); Clausius statement (heat does not flow cold to hot unaided); kinetic theory, mean free path (1858); virial theorem (1870)	1850, 1854, 1865	"entropy", "the energy of the universe is constant; the entropy of the universe tends to a maximum"; old entropy unit the clausius = 1 cal/K
James Clerk Maxwell (1831–1879, Scotland)	Distribution of molecular speeds in a gas (1860); kinetic theory; Maxwell's demon thought experiment (1867 letter, published in his book Theory of Heat, 1871); Maxwell relations; also unified electricity and magnetism	1860, 1867, 1871	"demon", "speed distribution", Theory of Heat
Ludwig Boltzmann (1844–1906, Austria)	Statistical meaning of entropy, $S = k_B \ln W$ (1877); Maxwell-Boltzmann distribution; H-theorem and the Boltzmann equation (1872); derived the Stefan-Boltzmann $T^4$ law from theory (1884) after his teacher Josef Stefan found it experimentally (1879); equipartition	1872, 1877, 1884	"$S = k \log W$ is engraved on his tombstone in Vienna"; Boltzmann constant $k_B$ (named and introduced into the formula by Planck)
Max Planck (1858–1947, Germany)	Blackbody radiation law (1900), which required energy to come in quanta $E = hf$ and introduced Planck's constant $h$; founded quantum theory; his wording of the second law gives the Kelvin-Planck statement; wrote Boltzmann's entropy formula in its modern form	1900; Nobel Prize in Physics 1918	"quanta", "Planck's constant", solved the ultraviolet catastrophe

How the story fits together, in order:

1. 1760s, heat as a substance. Black separated the amount of heat from the temperature, discovering latent and specific heat. Like most scientists of his time he thought of heat as a fluid, later called caloric, that flowed from hot bodies to cold ones and was conserved.
2. 1824, engines. Carnot, trying to improve the steam engines that powered industry, asked what limits their efficiency. Using the caloric picture, he showed that the limit depends only on the hot and cold temperatures. His book was little noticed until others revived it.
3. 1840s, heat is energy. Joule's careful experiments showed that a fixed amount of work always produces the same amount of heat. Heat is a form of energy, not a substance, which became the first law (conservation of energy). Caloric theory was then abandoned in favor of the mechanical (kinetic) theory of heat.
4. 1848–1854, the second law. Kelvin built an absolute temperature scale on Carnot's ideas. Clausius (1850) showed that in an engine heat is not just passed along, as caloric theory said, but partly consumed, in proportion to the work produced, with the rest flowing to the cold body. He and Kelvin (1851) thus reconciled Carnot's result with Joule's: energy is conserved, and heat cannot be turned entirely into work or flow uphill on its own. The first and second laws emerged together in the mid-1800s.
5. 1850s–1880s, molecules. Clausius, Maxwell and Boltzmann explained these laws through the motion of molecules: the kinetic theory of gases, the speed distribution, and entropy as a count of microstates.
6. 1900 onward. Planck's blackbody law launched quantum physics; the third law (Walther Nernst, 1906–1912) and the zeroth law (named around 1935) completed the set.

Names outside the list still appear in older practice tests as distractors or background; you are not expected to know them in depth for 2027, but recognizing them avoids traps: Josef Stefan (the $T^4$ law, 1879), Walther Nernst (third law), William Rankine (Rankine scale and cycle), Daniel Fahrenheit and Anders Celsius (temperature scales), Benjamin Thompson, Count Rumford (cannon-boring evidence against caloric), Rudolf Diesel and Nikolaus Otto (engines), James Watt, Thomas Newcomen and Thomas Savery (steam engines), and J. Willard Gibbs (free energy, phase rule).

State & National: radiation and the third law

Blackbody radiation

A blackbody is an ideal object that absorbs all radiation falling on it and emits the maximum possible thermal radiation at every wavelength for its temperature. Its spectrum depends only on temperature. As temperature rises, the curve grows taller everywhere and its peak shifts to shorter wavelengths: a stove element glows dull red, a hotter filament glows yellow-white. Planck's 1900 formula describes this curve, and it only works if energy is exchanged in discrete quanta.

Classical physics (the Rayleigh-Jeans law) predicted that a blackbody should emit ever more energy at ever shorter wavelengths, an infinite total: the ultraviolet catastrophe. Planck fixed it in 1900 by assuming that an oscillator of frequency $f$ can only gain or lose energy in whole packets, $E = hf$ ($h = 6.626 \times 10^{-34}$ J·s). At short wavelengths these packets are so large that thermal energy (about $k_BT$) rarely supplies one, so emission falls off, matching experiment. That assumption was the start of quantum theory.

Key properties of blackbody radiation: the spectrum is continuous; it depends only on temperature, not on the material or shape; a blackbody is both the best possible absorber and the best possible emitter at every wavelength; and a small hole in a hollow box (a cavity) is the best real approximation. Stars, glowing metal and incandescent filaments are close to blackbodies. The most nearly perfect blackbody spectrum ever measured is the cosmic microwave background, the leftover radiation of the early universe, at 2.725 K (peak near 1 mm, in the microwave band). Real objects that emit a fixed fraction $\varepsilon < 1$ of the blackbody amount at every wavelength are called gray bodies.

Color tracks temperature: as an object heats it glows dull red (about 800 K), then orange, yellow and white; the hottest stars look blue-white. A body's color and peak wavelength tell you its temperature; its total output tells you its temperature and size.

Radiant exitance and the Stefan-Boltzmann law

Radiant exitance $M$ is the radiant power emitted per unit area of a surface, in W/m². For a blackbody it depends only on temperature, through the Stefan-Boltzmann law. Real surfaces emit a fraction $\varepsilon$ (the emissivity, from 0 to 1) of the blackbody value.

Exitance and irradiance share units but point in opposite directions: exitance leaves a surface, irradiance arrives at one. The solar constant is an irradiance.

Luminosity, distance and comparing stars

Total power from a sphere is its luminosity, $L = 4\pi R^2\sigma T^4$, so $L \propto R^2T^4$. That power spreads over ever larger spheres, so the irradiance at distance $d$ follows the inverse-square law: $S = L/(4\pi d^2)$.

Radiative balance and planetary temperature

A planet (or any object in sunlight) settles at the temperature where the radiation it absorbs equals the radiation it emits. Sunlight of intensity $S$ (the solar constant, about 1,361 W/m² at Earth) falls on the planet's cross-section, a disk of area $\pi R^2$. A fraction $a$, the albedo, is reflected (about 0.30 for Earth). The planet emits from its whole surface, $4\pi R^2$.

If a surface absorbs a flux $F$ from all around and re-emits it, $\sigma T^4 = F$: 1,000 W/m² gives $T = (1000/5.67 \times 10^{-8})^{1/4} = 364$ K. Because the absorbed sunlight falls as $1/d^2$ and $T \propto (\text{absorbed})^{1/4}$, a planet's equilibrium temperature scales as $1/\sqrt{d}$: with Earth's albedo, a planet at Mars's distance (1.52 AU) would sit at $255/\sqrt{1.52} = 207$ K.

The third law

The third law says that the entropy of a perfect crystal of a pure substance approaches a constant, taken as zero, as its temperature approaches absolute zero. In Boltzmann's terms, a perfect crystal at 0 K has exactly one possible arrangement, $W = 1$, so $S = k_B \ln 1 = 0$. That gives entropy an absolute zero point, which is why tables list absolute ("standard molar") entropies $S°$ rather than just changes, and why every substance above 0 K has a positive $S°$.

A practical consequence is that absolute zero cannot be reached in a finite number of steps: each cooling step removes a smaller and smaller amount of entropy, so the approach is endless (laboratories have reached billionths of a kelvin, never zero). A Carnot engine would reach 100% efficiency only with a 0 K cold reservoir, and the third law says that reservoir cannot exist; but the law that forbids a 100%-efficient engine is the second law (Kelvin-Planck). The law grew out of Walther Nernst's heat theorem (1906–1912) and is stated in Planck's form as "S → 0 for a perfect crystal".

Real crystals often keep some residual entropy near 0 K because their molecules are frozen into slightly disordered arrangements (carbon monoxide molecules pointing CO or OC at random, the hydrogen positions in ice). Glasses and other non-crystalline solids are not covered by the "S = 0" statement. In Division B at State and National, expect the statement and its consequences rather than entropy calculations.

Division C, State & National: entropy and enthalpy

Entropy

Entropy $S$ is a state function that measures how spread out energy is, or statistically, how many microscopic arrangements (microstates, $W$) fit a given macroscopic state. It is often described as "disorder", which is a useful shorthand but not a definition. The second law in entropy language: the total entropy of an isolated system never decreases, and it increases in any irreversible process; it stays constant only in an idealized reversible process. Entropy is extensive, measured in J/K (or J/(mol·K) per mole).

Because the rules exclude calculus, expect entropy questions at fixed temperature (phase changes, reservoirs), sign reasoning (does entropy rise or fall?), and the statistical picture. Formulas for changing temperature or volume are given as results to apply:

Predicting the sign of ΔS. Entropy increases when:

• a solid melts, a liquid vaporizes, or a solid sublimes ($S_{\text{gas}} \gg S_{\text{liquid}} > S_{\text{solid}}$);
• a reaction produces more moles of gas than it consumes (for example $\text{CaCO}_3(s) \to \text{CaO}(s) + \text{CO}_2(g)$);
• a molecule breaks into smaller pieces, a solid dissolves, or two gases or liquids mix;
• a gas expands into a larger volume, or any substance is heated;
• larger and more complex molecules, and planets with atmospheres, have more ways to store energy and higher entropy.

It decreases for the reverse: freezing, condensing, deposition, fewer gas moles (for example $\text{SnO}_2(s) + 2\text{H}_2(g) \to \text{Sn}(s) + 2\text{H}_2\text{O}(l)$), crystallizing from solution. Count moles of gas first, because gas dominates entropy. When the gas moles are equal on both sides the gas term roughly cancels and the other phases decide: burning solid glucose, $\text{C}_6\text{H}_{12}\text{O}_6(s) + 6\text{O}_2(g) \to 6\text{CO}_2(g) + 6\text{H}_2\text{O}(g)$, has 6 mol of gas in and 12 out, so $\Delta S > 0$; with liquid water as the product (6 in, 6 out) the solid breaking into many small molecules still makes $\Delta S$ positive (about +260 J/K per mole of glucose from standard-entropy tables). A system's entropy can fall, as when water freezes or sugar crystallizes as its solution evaporates, but only if the surroundings gain more, so $\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0$ for every spontaneous process. At constant temperature and pressure, $\Delta S_{\text{surroundings}} = -\Delta H_{\text{system}}/T$, so an exothermic process raises the surroundings' entropy.

Standard molar entropies $S°$ are absolute values (thanks to the third law), so reaction entropies come from a table the same way reaction enthalpies do:

At a phase change in equilibrium, $\Delta S = \Delta H/T$, so the transition temperature is $T = \Delta H/\Delta S$. Example: ethanol has $\Delta H_{vap} = 38.6$ kJ/mol and $\Delta S_{vap} = 109.8$ J/(mol·K), so it boils at $38{,}600/109.8 = 351.5$ K = 78.4 °C.

Enthalpy

Enthalpy is defined as $H = U + PV$: the internal energy plus the energy needed to make room for the system at its pressure. It is an extensive state function, measured in joules. At constant pressure, the heat absorbed equals the enthalpy change, $Q_P = \Delta H$, which is why heats of reaction, fusion and vaporization are tabulated as enthalpy changes. A negative $\Delta H$ is exothermic (heat released to the surroundings, which warm); a positive $\Delta H$ is endothermic (heat absorbed from the surroundings, which cool). Enthalpy is a state function, so Hess's law lets you add the $\Delta H$ values of steps to get the overall change. The name is usually credited to Heike Kamerlingh Onnes (around 1909) and came into wide use in the 1920s.

For reactions involving gases, the same idea gives $\Delta H = \Delta U + \Delta n_{\text{gas}}RT$, where $\Delta n_{\text{gas}}$ is moles of gaseous products minus gaseous reactants. When the moles of gas fall (as in $2\text{H}_2(g) + \text{O}_2(g) \to 2\text{H}_2\text{O}(l)$, $\Delta n = -3$), $\Delta U$ is larger (less negative) than $\Delta H$. For melting and freezing the volume barely changes, so $\Delta H \approx \Delta U$.

Standard enthalpy of formation, $\Delta H_f°$, is the enthalpy change to form one mole of a compound from its elements in their standard states (the stable form at 1 bar, usually 25 °C). By definition $\Delta H_f° = 0$ for an element in its standard state, such as O₂(g), N₂(g), C(graphite) or Br₂(l), but not for H(g) atoms, O₃ or diamond.

Endothermic examples: melting, boiling, sublimation (dry ice: positive $\Delta H$), dissolving ammonium nitrate (cold packs), photosynthesis. Exothermic examples: combustion, freezing, condensation, neutralization, dissolving magnesium sulfate or calcium chloride (the solution warms), hand warmers (iron oxidizing). In an endothermic reaction the products have higher potential energy than the reactants; in an exothermic one, lower. An exothermic process need not release its energy only as heat: a burning candle also gives off light, a battery's reaction delivers electrical work, and an explosion releases sound and mechanical work.

Internal energy, enthalpy and entropy are all state functions; heat and work are not, because their values depend on the path between two states.

The heat-collecting device (Part II)

What the device must do

Your device collects energy from a supervisor's heat lamp and delivers it to 100 mL of room-temperature water in a clear 9 oz plastic party cup. The rules treat this as a heating problem, not an insulation problem: you are scored on how much the water warms and on how well you predict its final temperature.

Construction limits

• Materials. Almost anything is allowed. The exceptions are hazardous materials such as asbestos, mineral wool and fiberglass; thermometers and probes; and commercially made thermoses, coolers or vacuum-sealed containers. A device with a hazardous material may not be set up or tested at all, and the team takes only the written test.
• No energy sources. No electrical parts, battery heaters, chemical reactions or similar. A device found to use one scores zero for both TS and PS, and the supervisor may ask you to take the device apart after testing to check.
• Size. With the cup of water in place, the whole setup must fit inside a 30.0 cm cube.
• The cup. You must be able to put the supervisor's cup into the device and take it out. Device parts may reach inside the cup, but nothing may touch the water and nothing may be added to it.
• Temperature. No part may be noticeably warmer or colder than the room at the start, so pre-heated or chilled parts are out.
• Questions. Be ready to explain how the device was designed, built and operates.

The test setup

The supervisor uses a clamp lamp with a metal reflector and a 100-125 W heat-lamp bulb that is not tinted and not an LED, halogen or ceramic bulb, hung pointing down over the table. Your device sits centered beneath it, and the lamp stand's feet stay at least 15 cm from the bulb's center. The bulb height and Heating Time are the same for every team in a tournament but change between tournaments:

Because the device may be at most 30 cm tall, there is always at least 10 cm between the top of a 30 cm device and the bottom of the bulb.

The testing sequence

1. The supervisor announces bulb height, Heating Time, room temperature, water-bath temperature and the bulb's specifications (wattage and dimensions).
2. Device Setup Time, 3 minutes. Set up or adjust the device, fix any construction problems, and start your prediction. The clock keeps running through any re-checks, and after this only inserting or removing the cup is allowed.
3. The supervisor pours 100 mL from the bath into your cup, measures its starting temperature to 0.1 °C, and tells you.
4. Water Setup Time, up to 1 minute. Put the cup in the device without otherwise handling the cup or water.
5. The lamp turns on and the Heating Time starts. Touching the device or the lamp setup during heating zeroes both TS and PS.
6. Lock in your prediction, in °C, before the Heating Time ends.
7. When the lamp goes off, lift the cup fully out and set it on the table apart from the device; the supervisor reads the final temperature to 0.1 °C, aiming to finish within a minute.
8. Review your recorded data with the supervisor before leaving. If you plan to appeal anything about Part II, leave your device in the competition area.

The physics of a good collector

Start with an energy budget. Warming 100 mL of water by $\Delta T$ needs $Q = 0.100 \times 4186 \times \Delta T \approx 419\,\Delta T$ joules.

Most of the lamp's 100-125 W never reaches the cup. A heat lamp radiates over a wide cone, much of it misses the cup's small mouth (about 7.5 cm across), and some of what arrives is reflected or warms device parts instead of water. A good design works on four levers:

1. Capture more radiation. Reflective surfaces (for example aluminum foil on a cone or funnel) can redirect rays that would miss the cup back toward it. The wider the opening that catches light within the 30 cm limit, the more power is intercepted.
2. Absorb it where it counts. Water absorbs infrared strongly, and the clear cup lets much radiation through to it. Dark, matte surfaces absorb well; a dark sleeve hugging the outside of the cup turns radiation into heat right at the cup wall. Keep absorbers thin and light: every part that warms stores energy that never reaches the water ($Q = mc\Delta T$ again).
3. Stop the heat from leaving. Warm air above the water rises away by convection; a clear cover that transmits the lamp's radiation but blocks air flow traps it (the greenhouse idea). Foam under and around the lower cup cuts conduction into the table.
4. Get closer, carefully. Radiation spreads out with distance, so intensity falls roughly with the square of the distance from a small source. A lamp in a reflector is not a point source, so treat the inverse-square law as a rough guide and calibrate. Plastics and paper near a 125 W bulb can soften or scorch; build with heat-tolerant materials near the top and never aim a lens or mirror's focus onto the plastic cup.

Calibrating your prediction

The Prediction Score rewards a careful testing log more than a clever design. Build the lamp setup at home with the same bulb types and a 9 oz clear cup, then:

• Run every combination you might face: heights 40, 45, 50 and 55 cm for Regionals and States (and finer steps up to 65 cm for Nationals), and Heating Times of 10 to 15 minutes.
• Record room temperature, starting water temperature, height, time, bulb wattage and final temperature for each run. Repeat each condition at least three times and note the spread.
• Plot temperature gain against height for each Heating Time, and against time for each height. Bring these graphs and tables to the event; the rules allow them.
• Note how the gain changes when the starting water is cooler or warmer than the room. Water that starts below room temperature also gains heat from the air.

During the event, find the announced height and time in your table, adjust for the announced wattage and start temperature, and write final = start + expected gain.

Scoring

All scoring uses °C, and the highest final score wins.

Penalties and special cases:

Ties are broken by best TS, then best Exam Score, then preselected test questions.

Two features of the formulas shape strategy. First, PE divides your miss by the final temperature in °C, so with water near 30 °C a 1 °C miss costs only about 3% of PE; most teams' PE values are close together, and the gap between teams comes mostly from TS and ES. Second, TS is relative to the best device in the room, so a strong collector that heats the water a few degrees more than the field earns a large share of the 30 points.

Strategy

Splitting the work

Decide roles before the tournament. One partner owns the device: setup, cup insertion and removal, and the prediction from the calibration tables. The other starts the written test immediately and keeps going. If the supervisor allows test work during the Heating Time, both of you can return to the test once the prediction is locked in. On the test, one partner can take calculations (gas laws, engines, calorimetry, Stefan-Boltzmann) while the other takes definitions, identification and history; swap to check any multi-step calculation.

Building the binder

Notes are unlimited, so organize for speed rather than volume:

• A formula page with units for every equation in this guide, and the constants $R$, $k_B$, $\sigma$, Wien's constant, and water's $c$, $L_f$ and $L_v$.
• Conversion tables for temperature and energy units.
• The history table with one-line contributions.
• A process table (what is constant, what is zero, the P-V shape) and labeled diagrams of the Carnot, Otto and Diesel cycles.
• A labeled phase diagram for water and one for CO₂.
• The Condensed facts at the end of this guide as a front-of-binder summary, followed by the reference tables for specific heats, conductivities, expansion coefficients and latent heats.
• Your device calibration graphs and tables, tabbed by height and Heating Time, kept separate from the test notes so the device partner can find them in seconds.

Time management

Bank quick definition and identification questions first. For calculations, write down the given values converted to kelvin, joules, kilograms and pascals before choosing a formula; most lost points are unit slips. Leave long multi-part derivations (Division C) for the end.

Common pitfalls

How tournaments test this

The public archive holds 1,279 Thermodynamics questions from two earlier seasons (labeled 2017-18 and 2018-19): 934 tagged Division C, 315 Division B and 30 tagged for both. The practice bank holds 4,533 questions, mostly from 2017 to 2019: about 400 exported from these archive tests, about 600 from a 2025 import of tests the archive also holds, and about 3,500 from older tournament tests the archive does not have. Those tests were written for earlier rules, so they cover the core topics (temperature, heat transfer, laws, engines) well but rarely touch radiant exitance, blackbody radiation or the 2027 history list, and some ask about topics outside the 2027 outline.

• Format. About 59% of archived questions are multiple choice, 28% free response, 9% numeric, and the rest true/false or select-all.
• Topic tags (overlapping): laws about 43%, heat transfer about 31%, temperature about 23%, engines about 9%, efficiency about 6%.
• What the questions are about (our classification of the in-scope bank questions, by main topic): temperature scales and conversions 12%; gas laws and kinetic theory 7%; specific heat 7%; the four processes and P-V work 6%; history 6%; first law 6%; conduction and insulation 6%; entropy 5%; latent heat and phase changes 5%; Carnot efficiency 5%; units 4.5%; calorimetry 4%; enthalpy 4%; second law 3%; named engine cycles 2%; systems and properties 2%; and 1–2% each for states of matter, radiation, the zeroth and third laws, Stefan-Boltzmann, thermal expansion, phase diagrams, convection, refrigerators and heat pumps, Newton's law of cooling, blackbody radiation and thermometers. About 10% of the bank is outside the 2027 outline, mostly history of people not on the list (Rumford, Fahrenheit, Nernst, Gibbs, Savery) and Gibbs free energy.
• Tasks. About 12% of archived questions ask for a calculation, typically one formula with a short setup; others are identification ("which of the following"), definitions and short explanations.
• Recurring terms. Heat, temperature, energy, entropy, Carnot, efficiency, Kelvin and adiabatic lead the prompts; specific heat, ideal gas, heat capacity, internal energy, heat engine and Carnot cycle are the most common two-word phrases.
• Most common correct answers. Convection, Joule, radiation, conduction, adiabatic, Clausius and specific heat.

The archive reuses a handful of setups: a Carnot engine between two reservoirs, a phase change of water, two masses mixed in a calorimeter, a gas in a piston, and a "which mechanism" heat-transfer scenario. Master those and the 2027 additions (systems and properties, the history list, and the State/National radiation, third-law, entropy and enthalpy topics), and you cover nearly everything a 2027 test can ask.

Test yourself

These questions come from official keys in the public archive and fall inside the 2027 outline; each links to its full test. We re-solved each one.

Test yourselfHeat transfer, specific heat and phase changes

0 / 0 correct

1. 1

   You walk outside on a hot summer day, and immediately you feel the heat from the sun's rays on your face. Which of the following is the type of heat transfer you just experienced?

   Question 1 options

   AConductionBConvectionCRadiationDIR

   From: University of Texas at Austin Invitational, 2018-19, Division C

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Test yourselfTemperature, units and the ideal gas law

0 / 0 correct

1. 1

   The absolute temperature of an ideal gas is directly proportional to which of the following quantities?

   Question 1 options

   Athe mass of its moleculesBthe average speed of its moleculesCthe average kinetic energy of its moleculesDthe average momentum of its moleculesEIt is proportional to all of the above quantities.

   From: Tiger Invitational, 2017-18, Division C

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Test yourselfProcesses, engines and efficiency

0 / 0 correct

1. 1

   A bomb calorimeter tests samples under ____ conditions.

   Question 1 options

   AIsobaricBAdiabaticCIsolatedDIsochoric

   From: University of Texas at Austin Invitational, 2018-19, Division C

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Test yourselfThe laws and their history

0 / 0 correct

1. 1

   Which scientist is responsible for the concept of the infamous “perfect” engine?

   Question 1 options

   AJohn EricssonBWilliam RankineCGeorge BraytonDNicolas Carnot

   From: University of Georgia Invitational, 2018-19, Division C

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Test yourselfKinetic theory, gas behavior and expansion

0 / 0 correct

1. 1

   A sample of an ideal gas is heated and its Kelvin temperature doubles. If the root-mean-square speed of its molecules was originally v, what is the new root-mean-square speed?

   Question 1 options

   A4vB2vCv√2Dv/√2Ev/4

   From: Cypress Falls Invitational 2017-18, Division C

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Test yourselfConduction, convection and radiation

0 / 0 correct

1. 1

   Which of the following is not a factor in determining the rate of thermal conductivity of a material?

   Question 1 options

   AThe temperature difference across the materialBThe thickness of the materialCThe area of the materialDThe mass of the material

   From: Forsyth Invitational, Division C

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Test yourselfProcesses, cycles and refrigerators

0 / 0 correct

1. 1

   In a system undergoing adiabatic compression, what are the values of internal energy and heat if work done on the system is 500 J?

   Question 1 options

   AInternal energy = +500 J, Heat = 0 JBInternal energy = 0 J, Heat = -500 JCInternal energy = 0 J, Heat = +500 JDInternal energy = -500 J, Heat = 0 J

   From: Ladera Vista Invitational 2018, Division B

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Test yourselfHistory, entropy and enthalpy

0 / 0 correct

1. 1

   James Joule worked in what occupation?

   Question 1 options

   AAccountantBBrewmasterCFarmerDTeacherEWine Merchant

   From: Solon Invitational 2018, Division C

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Condensed facts

References and further reading

• Science Olympiad, 2027 rules manuals (Division B and C, Thermodynamics) and the event scoring spreadsheet: https://www.soinc.org
• OpenStax, University Physics Volume 2, chapters on temperature and heat, kinetic theory, the first law, and the second law and entropy (free online): https://openstax.org/details/books/university-physics-volume-2
• OpenStax, Chemistry 2e (thermochemistry, enthalpy, phase diagrams, gas laws): https://openstax.org/details/books/chemistry-2e
• NIST, constants and SI definitions (Boltzmann constant, Stefan-Boltzmann constant, kelvin): https://physics.nist.gov/cuu/Constants/
• NIST Chemistry WebBook, thermophysical properties of water: https://webbook.nist.gov/chemistry/fluid/
• HyperPhysics (Georgia State University), heat and thermodynamics pages, including blackbody radiation and Stefan-Boltzmann: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thermcon.html
• SciOly Wiki, Thermodynamics (event summary from past seasons): https://scioly.org/wiki/index.php/Thermodynamics
• Practice more: the Scio.ly test archive has public Thermodynamics tests from earlier seasons.