Heat & Thermodynamics

Touch a cup of tea and a glass of cold water: your hand says 'hot' and 'cold', but a hand is a poor judge, because it only tells you how a body feels compared with your skin. Physics needs a number. Temperature is the degree of hotness or coldness of a body, and in the kinetic picture it measures the average kinetic energy of the particles of the body. Heat is different: it is the energy that flows from a body at higher temperature to a body at lower temperature because of the temperature difference. Heat is energy (SI unit joule), while temperature is only a measure of hotness (SI unit kelvin). Two bodies at the same temperature exchange no heat.

A bucket of water at 40 °C and a cup of water at 40 °C have the same temperature, but the bucket holds much more heat, because it has far more particles, each carrying energy. Think of a lake and a cup of tea: the tea is hotter, yet the lake contains more heat. Temperature tells how hot; heat tells how much energy.

Three scales are used. Celsius (Anders Celsius, Sweden, 1742) takes the freezing point of water as 0 °C and the boiling point as 100 °C, with 100 equal divisions. Fahrenheit (Daniel Gabriel Fahrenheit, 1724) takes them as 32 °F and 212 °F, with 180 divisions. Kelvin (Lord Kelvin, 1848) is the SI scale: ice point 273.15 K and steam point 373.15 K, again 100 divisions, so a rise of 1 °C is the same as a rise of 1 K. Absolute zero is 0 K = −273.15 °C = −459.67 °F, the lowest possible temperature, at which the particles have the minimum possible energy. The Kelvin scale has no negative values, and the degree sign is not written with it (we write 300 K, not 300 °K).

On a summer afternoon the beach sand burns your feet while the sea water beside it feels merely warm. Both received the same sunshine. The difference lies in a property of matter called specific heat capacity (c): the amount of heat needed to raise the temperature of 1 kg of a substance by 1 K (or 1 °C). The formula is Q = m c ΔT, where Q is the heat absorbed or released, m the mass and ΔT the change in temperature. The SI unit of c is J kg⁻¹ K⁻¹. A substance with a small c heats up and cools down quickly (sand, metals); one with a large c heats up and cools down slowly (water).

Heat capacity (thermal capacity) of a whole body is C = m c, the heat needed to raise the temperature of the entire body by 1 K; its unit is J K⁻¹. Specific heat capacity is per kilogram and belongs to the material; heat capacity belongs to the particular object. Water has the highest specific heat among common substances, about 4186 J kg⁻¹ K⁻¹. Water-equivalent of a body is the mass of water that would need the same heat for the same rise in temperature.

Calorimetry is the measurement of heat. Its basis is the principle of calorimetry (method of mixtures): when two bodies at different temperatures are mixed in an insulated container, heat lost by the hot body = heat gained by the cold body (this is just conservation of energy), until both reach a common final temperature. For equal amounts of the same liquid at T₁ and T₂ the final temperature is simply the average, (T₁ + T₂) / 2. For unequal masses of the same liquid: T = (m₁T₁ + m₂T₂) / (m₁ + m₂). The instrument is a calorimeter, a metal vessel (copper or aluminium) with a stirrer, placed inside an insulating jacket so that no heat leaks out.

Why do railway tracks have small gaps between rails, and why do telephone wires sag in summer? The answer begins inside matter. The particles of a body are always jiggling about their mean positions. When the body is heated, they take in energy, so their average kinetic energy rises and they vibrate with larger amplitude. Because the forces between particles are not perfectly symmetric, the average distance between them grows, and the whole body expands. That is thermal expansion. On cooling, the reverse happens. So heating makes the particles of any substance move faster, and nearly every substance expands when heated.

A hole in a plate expands on heating exactly as if the hole were made of the same metal: the hole becomes bigger, not smaller. Reinforced concrete works because steel and concrete have almost the same coefficient of expansion, so they expand together and the bond does not break. A glass tumbler cracks when hot water is poured in because the inside expands before the outside; Pyrex (borosilicate glass) has a very low expansion and survives. Invar (an iron-nickel alloy) hardly expands at all and is used in clock pendulums and measuring tapes. A pendulum clock runs slow in summer because the pendulum lengthens and its time period increases.

Water breaks the general rule. Most liquids keep contracting as they cool, but water behaves differently between 0 °C and 4 °C. If you heat water from 0 °C, it contracts until 4 °C, and only after that does it expand normally. So water has its minimum volume and therefore maximum density at 4 °C (1000 kg/m³, or 1 g/cm³ for practical purposes). Below 4 °C the density falls again, and at 0 °C ice forms with a density of about 917 kg/m³, which is why ice floats on water with roughly nine-tenths of it submerged. This behaviour is called the anomalous expansion of water.

Put a pan of ice on a stove with a thermometer in it. The temperature rises from below zero up to 0 °C, then something strange happens: the ice keeps melting, the stove keeps supplying heat, yet the thermometer stays at 0 °C until the last bit of ice is gone. The same happens at 100 °C when water boils. The heat is not raising the temperature; it is being used to loosen the bonds between particles and change the state. This hidden heat is called latent heat ('latent' means hidden). Latent heat is the heat absorbed or released by a substance during a change of state at constant temperature: Q = m L, with the unit J kg⁻¹.

Why are the two latent heats so different? Melting only loosens the bonds between particles (they still stay close), while vaporisation must pull the particles completely apart against attraction and also push back the atmosphere to make room for the vapour. That is why latent heat of vaporisation (about 2260 kJ/kg) is nearly seven times the latent heat of fusion (334 kJ/kg). A direct result: steam at 100 °C causes a worse burn than boiling water at 100 °C, since each kilogram of steam carries an extra 2260 kJ of latent heat which it gives up on condensing on your skin.

Leave a metal spoon in a hot cup of tea: the handle soon becomes hot. Sit near a fireplace: your face warms even though the air between is not much heated. Boil water in a pan: the water at the bottom rises and the cooler water at the top sinks. Heat reaches from one place to another in exactly three ways: conduction, convection and radiation. In conduction heat passes through matter without the particles moving from their places; in convection the heated fluid itself moves and carries the heat; in radiation heat travels as electromagnetic waves and needs no medium at all.

Conduction. When one end of a rod is heated, its particles vibrate more strongly and pass the energy on to their neighbours. In metals the free electrons also carry energy quickly, so metals are good conductors (silver is the best, followed by copper, then gold and aluminium). Wood, plastic, glass, air, wool and water are poor conductors (insulators of heat); gases are the poorest. The rate of conduction depends on the material's thermal conductivity, the cross-section, the temperature difference and inversely on the length. That is why the bottom of a cooking vessel is made of copper (a good conductor) and its handle of wood or plastic (a bad conductor). A steel spoon feels colder than a wooden one at the same temperature because it conducts heat away from the hand faster.

Convection. When a fluid is heated at the bottom, the heated part expands, becomes less dense and rises; cooler, denser fluid sinks to take its place. This circulation is a convection current. It explains boiling water, the sea breeze (by day, land is hotter than sea, warm air rises over land and cool air flows in from the sea; at night the land breeze flows from land to sea), ventilators near the roof of a room (hot air leaves, fresh air comes in), the freezer compartment kept at the top of a refrigerator (cold air sinks and cools the food below), the winds and ocean currents, and even the slow movement of molten rock in the Earth's mantle. Convection cannot take place in solids or in a vacuum.

Radiation. Every body above absolute zero emits thermal radiation, mostly infrared. It travels at the speed of light, needs no medium, and is how the Sun's energy reaches us through empty space. Dark, rough surfaces are good absorbers and also good emitters; bright, shiny, white surfaces are poor absorbers and good reflectors. A black body absorbs all the radiation falling on it and, when hot, is the best emitter. Hence we wear light colours in summer and dark colours in winter, and a black pan cooks faster than a shiny one. The Stefan-Boltzmann law: the energy radiated per second per unit area by a black body is proportional to the fourth power of its absolute temperature (E ∝ T⁴); doubling T makes the radiation 16 times. Wien's displacement law: the wavelength of maximum emission is inversely proportional to the absolute temperature (λₘ T = constant), so hotter bodies glow with shorter wavelengths: a heated iron rod turns from dull red to orange to white, and hot stars look blue-white while cool stars look red.

Newton's law of cooling (Isaac Newton, about 1701): the rate at which a body loses heat is directly proportional to the difference between the temperature of the body and the temperature of its surroundings, provided the difference is small: −dT/dt ∝ (T − T₀). So a hot cup of tea cools fast at first and more and more slowly later; its temperature falls in an exponential curve towards room temperature and, in theory, never goes below it. A body hotter than its surroundings by 40 °C loses heat twice as fast as one hotter by 20 °C.

Thermodynamics studies how heat, work and energy are related. It begins by choosing the part of the universe we are studying, the system, and calling the rest the surroundings. A system is open if both matter and energy can cross its boundary (a pan of boiling water without a lid), closed if only energy can (a sealed steel cylinder of gas that can still exchange heat), and isolated if neither can (an ideal thermos flask). The condition of a system is described by state variables: pressure P, volume V, temperature T and internal energy U. They depend only on the present state, not on how it was reached. Heat (Q) and work (W) are not state variables; they depend on the path.

Zeroth law: if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other. This is what makes a thermometer meaningful and gives the idea of temperature (the law was named by R. H. Fowler around 1931 because it logically comes before the first two). First law: it is the law of conservation of energy for heat and work: ΔQ = ΔU + W. The heat given to a system either raises its internal energy or is used by the system to do work. Take Q positive when heat is absorbed, W positive when the system does work, and ΔU positive when internal energy rises. For an ideal gas, internal energy depends only on temperature. The first law shows that when a body absorbs heat but does no work, all of it goes into internal energy: the temperature rises or the state changes.

Why does a cycle tyre that bursts suddenly feel cold at the nozzle? The air escapes so fast that it has no time to take heat from the surroundings (Q = 0); it expands against the atmosphere and does work, so by ΔU = −W its internal energy and therefore temperature fall. That is an adiabatic process. Likewise, pumping air into a tyre quickly makes the pump warm. In an adiabatic expansion where the gas does 200 J of work, its internal energy decreases by 200 J. In an isothermal expansion (temperature constant) the internal energy of an ideal gas does not change at all.

Second law. The first law would allow heat to flow from a cold body to a hot one, but we never see it. The second law explains the direction. Kelvin-Planck statement: no heat engine can convert all the heat it takes in into work; some must be rejected to a colder body (so no engine is 100 per cent efficient). Clausius statement: heat cannot by itself flow from a colder body to a hotter body; it needs external work (a refrigerator). Both statements are equivalent. In terms of entropy (the measure of disorder, unit J/K, named by Clausius), the entropy of an isolated system never decreases; natural processes increase entropy, which gives time its 'arrow'. Third law: the entropy of a perfect crystal is zero at absolute zero, and absolute zero can be approached but never reached in a finite number of steps.

Heat engines. A heat engine takes heat from a hot source, converts part of it to work and rejects the rest to a cold sink. Its efficiency is η = W / Q₁ = 1 − Q₂ / Q₁. The Carnot engine (Sadi Carnot, French, 1824) is an ideal, reversible engine working between two temperatures; its efficiency is η = 1 − T₂ / T₁ (temperatures in kelvin). No engine working between the same two temperatures can be more efficient. To raise the efficiency, raise the source temperature or lower the sink temperature; the efficiency of a Carnot engine depends only on these two temperatures, not on the working substance. Example: an engine taking in 800 J and rejecting 500 J does 300 J of work, so η = 300 / 800 = 37.5 per cent. A Carnot engine between 500 K and 250 K has η = 1 − 250/500 = 50 per cent, so from 1000 J of heat it can give at most 500 J of work. Real examples: steam engines (external combustion, improved by James Watt), petrol engines (Otto cycle, spark ignition), diesel engines (Diesel cycle, compression ignition, higher compression ratio and efficiency than petrol engines).