Heat — Laboratorium

Thermal contact: energy flows from hot to cold

Block A — temperature90 °C

Block B — temperature10 °C

Block B — mass1 kg

Temperature A: 90.0 °C
Temperature B: 10.0 °C
Heat A → B: 0.0 kJ
Predicted final temp.: 50.0 °C

Each dot is an atom vibrating around its place in the crystal lattice. The more vibration energy, the brighter and warmer its color. Where the blocks meet, atoms “jostle” each other and pass energy along — in both directions, but more of it flows from hot to cold. Once the temperatures are equal the exchange goes on, but on balance nothing flows anymore. We assume the blocks are made of copper (c ≈ 385 J/(kg·K)) and A has a mass of 1 kg. Model: random exchange of energy between neighbors.

Temperature, heat, internal energy

Temperature tells you how fast, on average, the molecules move (vibrate, spin, dart around). Internal energy is the total energy of all the molecules in a body. Heat is energy that flows between bodies because of a temperature difference — a body doesn’t “contain heat”, only internal energy.

Ē_k = 32·k·Taverage translational kinetic energy of a gas molecule; k = 1.380649·10⁻²³ J/K (Boltzmann constant), T in kelvins

Scale converter

Celsius (°C)

Kelvin (K)

Fahrenheit (°F)

20 °C = 293.15 K = 68 °F.

Absolute zero (0 K = −273.15 °C = −459.67 °F) is the state of lowest possible energy. It can’t be reached — labs get down to billionths of a kelvin, but never to zero (the third law of thermodynamics). Even then particles don’t freeze completely — quantum vibrations remain.

Latent heat: ice → water → steam

Heater power2,000 W

Temperature: −20.0 °C
State: ice
Heating time: 0 s

While melting and boiling, the heater keeps supplying energy, yet the thermometer stands still: the energy goes into breaking the bonds between molecules (latent heat). Melting 1 kg of ice absorbs about 334 kJ, and boiling away 1 kg of water a whopping 2,257 kJ — more than heating it from 0 to 100 °C (419 kJ). More in States of matter. The model ignores heat losses to the surroundings.

Who heats up fastest? Specific heat

Power of each heater1,000 W

Heating time0.0 s

Each material: 1 kg, starting at 20 °C, a race to +10 °C. No heat losses; the animation runs 2× faster, the time shown is real.

Q = m · c · ΔT

Specific heat c tells you how much energy it takes to warm 1 kg of a substance by 1 K. Water has one of the highest specific heats among ordinary substances (about 4.19 kJ/(kg·K)) — that’s why the sea warms up slowly in summer and gives off heat for a long time in winter. Coastal climates are milder: cooler summers and warmer winters than far inland. For the same reason water is great at carrying heat in radiators.

A house in winter: where does the heat go?

Wall insulation5 cm

Roof insulation10 cm

Floor insulation0 cm

Outdoor temperature−10 °C

Indoor temperature20 °C

Assumed heat price (example)0.35 PLN/kWh

Insulation material

Windows

Ventilation

Thermal bridges (balcony, lintels, ring beam — uninsulated)

Loss now: 7,660 W
Heat loss coefficient H: 255 W/K
Per year (rough): 23,300 kWh
Cost / year at this price: 8,160 PLN

Losses relative to the starting house: 100%.

Model: a single-story house of about 100 m² (walls 94 m², windows 18 m², top-floor ceiling under the roof 110 m², floor on the ground 100 m², volume 260 m³). Heat-loss power is calculated with Q = U·A·ΔT for each part of the envelope plus ventilation. Annual energy comes from degree-days (for 20 °C in Poland usually about 3,500–4,200 K·d depending on the region; we assume 3,800). Gains from the sun, people and appliances are ignored — real consumption is usually lower. The price (in Polish złoty, PLN) is just an example assumption; set your own.

Three ways heat escapes

Conduction

Faster molecules nudge their slower neighbors. Metals conduct superbly, air very poorly — which is why insulators (polystyrene, wool, down) are mostly trapped air. Fourier’s law:

q = λ·ΔTd · P = U·A·ΔTλ — thermal conductivity of the material (W/(m·K)), d — thickness; U — heat transfer coefficient (U-value) of the wall or window (W/(m²·K))

Convection

Warm air is less dense, so it rises, while cool air sinks — a circulating current forms by a radiator. Convection needs a moving fluid: there is none in a vacuum.

Radiation

Every body glows in the infrared — the warmer it is, the brighter. Radiation passes even through a vacuum (that’s how the Sun warms the Earth).

P = ε·σ·A·T⁴σ = 5.67·10⁻⁸ W/(m²·K⁴), ε — emissivity (0–1), T in kelvins

Thermos

A thermos blocks all three routes: the vacuum between its walls doesn’t conduct and allows no convection, and the silvered walls reflect infrared. Only the neck and the stopper remain.

Atmosphere: the greenhouse effect and radiation Gas: molecules in motion

Mini-experiment: what keeps tea hot the longest?

The tea starts at 85 °C in a room at 20 °C; we show after how many minutes it drops below 60 °C. Cooling follows Newton’s law: T(t) = T_room + (T₀ − T_room)·e^−t/τ. The τ values are rough — in a real mug a lot depends on the amount of tea, evaporation and wall thickness. A lid limits evaporation, which carries away a great deal of heat.

The refrigeration cycle: heat “uphill”

Outdoor air−2 °C

Heating flow temperature35 °C

Device quality (% of Carnot)45%

Carnot COP (limit): 9.34
COP (simplified): 4.20
1 kWh of electricity gives heat: 4.20 kWh
Of which taken from the air: 3.20 kWh

1 · CompressorSucks in cold refrigerant vapor and compresses it. Pressure and temperature rise — the vapor gets hotter than the water in the heating system. This is where electrical energy goes in.
2 · CondenserThe hot vapor gives off heat (to the heating water or the kitchen air) and condenses into a liquid. Condensing releases a lot of latent heat.
3 · Expansion valveThe high-pressure liquid squeezes through a narrow opening. The pressure drops sharply, part of the liquid evaporates and the refrigerant cools down a lot.
4 · EvaporatorThe refrigerant is colder than the outdoor air (or the inside of the fridge), so heat flows into it by itself. The liquid boils and turns into vapor — and the cycle closes.

The same cycle works in a fridge, an air conditioner and a heat pump — the only difference is which side we consider “useful”. Here we calculate COP simply: as a fraction (40–55%) of the Carnot limit computed from the air and flow temperatures. Real devices have their own characteristics; the pressures (refrigerant R290 — propane, in the fridge R600a — isobutane) are approximate.

COP vs. weather and flow temperature

The smaller the temperature difference between the “source” and the “sink”, the less work it takes to pump heat. That’s why underfloor heating (flow at about 30–35 °C) lets a heat pump work much more efficiently than old, small radiators needing 55–65 °C. On a freezing day COP drops, but usually stays above 1 — still more heat than from an electric heater.

SCOP — the average over a Polish winter

SCOP (seasonal): 4.05
Season hours: 5,530 h
Season avg. temp.: 3.1 °C

Bars: an approximate distribution of heating-season hours in central Poland. Most hours fall at temperatures around zero, while hard frosts are short. SCOP is the ratio of heat delivered over the whole season to the electricity used. Simplifications: constant flow temperature (in practice the controller lowers it on warmer days, which raises SCOP), no defrosting and no backup heater.

Comparison: how much heat from 1 kWh of purchased energy

CO₂ footprint per 1 kWh of heat

Grid carbon intensity0.65 kg/kWh

A gas boiler turns about 90–98% of the gas energy into heat (a condensing one the most), an electric heater 100% of the electricity, and a heat pump typically delivers 300–400% over a season — because it takes most of the heat from its surroundings. How much CO₂ that emits depends on how the electricity was generated: from coal, gas, nuclear, or wind and sun (see Power plant). We assume about 0.20 kg CO₂ per kWh of gas and 0.34 kg per kWh of coal; for Poland the average carbon intensity of electricity in recent years has been about 0.6–0.7 kg/kWh and is falling. This is a physical comparison, not a cost one — fuel and electricity prices change, and subsidies (e.g. Poland’s “Clean Air” program) have their own rules.

Where does freezing air get its heat?

−10 °C feels like frost to us, but on the absolute scale it is still 263 K. Air molecules still rush around at hundreds of meters per second. A heat pump only needs a refrigerant colder than the air — in the evaporator the refrigerant is at, say, −20 °C, so heat flows from the “freezing” air into the even colder refrigerant.

Bar length: the average kinetic energy of the molecules relative to absolute zero (proportional to T in kelvins). Air at −10 °C has only about 10% less of it than air at +20 °C.

Air or ground?

An air-source heat pump takes heat from the air: simple to install, but in the hardest frosts, when the most heat is needed, it works least efficiently and has to defrost its evaporator. A ground-source heat pump draws heat from the earth or groundwater through boreholes or a horizontal collector. A few to a dozen or so meters underground the temperature is almost constant all year (in Poland about 8–10 °C), so in winter the source is warmer than the air and the SCOP is usually higher — at the price of more digging.

The invention has a long history: William Thomson (Lord Kelvin) described the principle of a “heat multiplier” in 1852, and the first devices of this kind were built a few years later.

The p–V diagram: work is the area inside the loop

Hot reservoir T600 K

Cold sink T300 K

Isothermal expansion V₂/V₁2.0

Efficiency η: 50.0%
Carnot limit: 50.0%
Work W per cycle: 1.73 kJ
Heat taken in Q_h: 3.46 kJ

A Carnot cycle for 1 mole of a monatomic ideal gas: two isotherms (gas in contact with a reservoir, constant temperature) and two adiabats (gas insulated, changing only at the expense of its own energy). It is the most efficient engine possible between two temperatures.

Real engines and power plants

typical efficiency (rough)Carnot limit for their temperatures

A coal power plant heats steam to about 540–600 °C and condenses it at about 30 °C — Carnot would allow about 63–65%, while in practice it reaches about 35–45%. The rest of the heat has to go somewhere: hence cooling towers and warm rivers next to power plants. Details in Power plant, Steam engine and Four-stroke engine.

The second law of thermodynamics — in plain words

Heat does not flow by itself from a colder body to a warmer one (Clausius). A fridge and a heat pump do exactly that — but only because the compressor does work.

You can’t build an engine that turns all the heat it takes in into work (Kelvin, Planck). Some heat must always be dumped into a cold sink. The best possible result is:

η_max = 1 − T_cT_htemperatures in kelvins; e.g. 600 K and 300 K → 50%

An engine and a heat pump are the same cycle run in opposite directions: an engine takes heat from the hot reservoir and gives work, a pump takes work and moves heat from cold to hot. That’s why a pump’s Carnot COP is exactly the inverse of an engine’s Carnot efficiency: T_h/(T_h − T_c).

Energy and its transformations Gas laws

Gas in a box: why the molecules don’t go back

Number of molecules N40

Left / right half: 40 / 0
Microstates W of this split: 1
Entropy S / k = ln W: 0.0
P(all on the left): 9.1·10⁻¹³

Number of molecules N	Probability 2^−N	How often?
4	1/16 = 6.25%	every sixteenth glance
10	1/1,024 ≈ 0.098%	about once in a thousand
100	≈ 7.9·10⁻³¹	checking a billion times a second — about once in 4·10¹³ years, thousands of times longer than the Universe has existed
10²³ (a handful of gas)	≈ 10^−3·10²²	a number with about 3·10²² zeros after the decimal point — practically never

Every specific arrangement of the molecules is equally likely. But there are astronomically more “roughly half and half” arrangements than “all on the left” ones. Entropy measures exactly this number of possibilities W: S = k·ln W. A system moves on its own toward states of higher entropy simply because there are the most of them — and that’s why a film of gas mixing looks unnatural when played backward.

The arrow of time in everyday life

Coffee cools down

When 1 kJ of heat flows from coffee (350 K) into the room (293 K), the coffee loses 1000/350 ≈ 2.86 J/K of entropy and the room gains 1000/293 ≈ 3.41 J/K. Total: +0.56 J/K. The other way round (the coffee warming itself up) entropy would decrease — and we never see that.

ΔS = QT

Ice melts

Melting 1 kg of ice at 0 °C raises the ice’s entropy by 334,000/273 ≈ 1,223 J/K. A room at 20 °C loses only 334,000/293 ≈ 1,140 J/K. Balance: about +83 J/K. An ice cube won’t refreeze by itself in a warm drink.

Shuffling cards

A deck has 52! ≈ 8·10⁶⁷ possible orders. Only one is “in sequence”, so shuffling almost certainly ruins it.

All 52 cards in their places.

Ludwig Boltzmann linked entropy to the number of microstates. His tombstone in Vienna bears the formula S = k·log W (it was Max Planck who later wrote it in this form).

Game: be Maxwell’s demon

Left chamber: 300 K
Right chamber: 300 K
Temperature difference: 0 K
Demon’s decisions (bits): 0

Hold the button, the D key, or press the game area to open the door. Goal: the right chamber at least 100 K warmer than the left (with at least 8 molecules in each). In 1867 James Clerk Maxwell imagined such a “demon”, which sorts fast and slow molecules without doing any work — and breaks the second law.

The solution to the puzzle: the demon has to measure speeds and remember the results. Its memory is finite, so eventually it must erase it, and according to Landauer’s principle (1961) erasing 1 bit costs at least k·T·ln 2 of energy (about 2.9·10⁻²¹ J at 300 K) and dumps the corresponding entropy into the surroundings. The total never goes down — the second law wins.

Simulated thermal camera

Camera shows: —
Emissivity at point: —
Actual temperature: —
What it is: —

The camera measures infrared radiation (usually 8–14 µm) and converts it to temperature, assuming an emissivity close to 1. Shiny metal has an emissivity of about 0.05–0.2 — it glows weakly and reflects its surroundings, so it looks “cold” in the image even when it’s scalding. The scenes are simulated, with simplified temperatures.

Wien’s law: what color does a body glow?

Body temperature310 K

Spectrum peak λ_max: 9.35 µm
Temperature: 36.9 °C
Power from 1 m² (σT⁴): 524 W

λ_max = bTb ≈ 2.898·10⁻³ m·K. For the human body (about 310 K): λ_max ≈ 9.35 µm — deep infrared, which is why thermal cameras work in the 8–14 µm range.

Emissivity: who glows and who reflects

Human skin	approx. 0.98
Water, ice	approx. 0.95–0.97
Glass, glazed ceramics	approx. 0.9
Black electrical tape	approx. 0.95
Stainless steel (shiny)	approx. 0.1–0.2
Polished aluminum	approx. 0.05

Glass doesn’t let through infrared in the camera’s range — you can’t look into a room through a window with a thermal camera, and glasses show up on a thermogram as dark patches. Thin polyethylene film (a plastic bag), on the other hand, is almost transparent to this infrared. Thermal imaging is used to find leaks in houses and overheated electrical contacts, and also in medicine, rescue work and for spotting animals at night.

Heathow to pump heat uphill

Thermal contact: energy flows from hot to cold

Temperature, heat, internal energy

Scale converter

Latent heat: ice → water → steam

Who heats up fastest? Specific heat

A house in winter: where does the heat go?

Three ways heat escapes

Conduction

Convection

Radiation

Thermos

Mini-experiment: what keeps tea hot the longest?

The refrigeration cycle: heat “uphill”

COP vs. weather and flow temperature

SCOP — the average over a Polish winter

Comparison: how much heat from 1 kWh of purchased energy

CO₂ footprint per 1 kWh of heat

Where does freezing air get its heat?

Air or ground?

The p–V diagram: work is the area inside the loop

Real engines and power plants

The second law of thermodynamics — in plain words

Gas in a box: why the molecules don’t go back

The arrow of time in everyday life

Coffee cools down

Ice melts

Shuffling cards

Game: be Maxwell’s demon

Simulated thermal camera

Wien’s law: what color does a body glow?

Emissivity: who glows and who reflects

How does it work?

Heat and temperature

The first law of thermodynamics

Heat loss of a house

Heat pump and engine

Entropy

Thermal radiation

Challenges

Did you know…?

Keys