Every cook in India knows to keep a hand away from the whistle of a pressure cooker. A jet of steam there burns far worse than a splash of boiling water from a pan. Both are at about 100 °C. So the temperature cannot be the whole story.
Heat goes in, the temperature stays still
Put a thermometer into a glass of crushed ice and water on a warm day. The ice melts slowly, and the room keeps giving heat to the glass all the time. Yet the thermometer stays at 0 °C until the last piece of ice has gone. Only then does the water begin to warm. The same thing happens at 100 °C when water boils. The flame keeps heating, but the temperature does not rise until all the water has boiled away.
A change of solid to liquid, or liquid to vapour, is a change of state. It happens at a fixed temperature for a given pressure: the melting point for melting, the boiling point for boiling. The heat taken in during the change was called "latent", meaning hidden, because no thermometer shows it.
The latent heat \(L\) is the heat needed to change the state of one kilogram of a substance at a constant temperature. The latent heat of fusion \(L_f\) is for melting a solid; the latent heat of vaporisation \(L_v\) is for boiling a liquid. The unit is J/kg.
For water, this chapter uses \(L_f = 3.34 \times 10^5\,\text{J/kg}\) and \(L_v = 2.26 \times 10^6\,\text{J/kg}\). In older units they are 80 cal/g and 540 cal/g. Multiplying by 4.2 J/cal gives \(3.36 \times 10^5\) and \(2.27 \times 10^6\,\text{J/kg}\), within one per cent of the measured values. Problems set in calories use 80 and 540, with 1 cal/(g °C) for water and 0.5 cal/(g °C) for ice.
Fig 2.1 puts the numbers side by side. Melting a kilogram of ice takes 334 kJ, as much as warming the melted water by about 80 K. Boiling it away takes 2260 kJ, far more than anything else on the chart. The same heat comes back out when steam condenses to water, or water freezes to ice.
Why steam scalds
Now the cooker's whistle. A gram of boiling water that lands on the skin cools from 100 °C to 37 °C. It gives out \(0.001 \times 4200 \times 63 \approx 265\,\text{J}\). A gram of steam first condenses, giving out its latent heat, \(0.001 \times 2.26 \times 10^6 = 2260\,\text{J}\). Then the water it forms cools by the same 63 K and gives another 265 J. In all the steam gives about 2525 J, nine and a half times as much. It gives most of it the instant it touches the skin.
\(50\,\text{g}\) of steam at \(100\,\text{°C}\) condenses to water and the water cools to \(40\,\text{°C}\). How much heat does it give out? Take \(L_v = 2.26 \times 10^6\,\text{J/kg}\) and \(c = 4200\,\text{J/(kg K)}\).
Condensing at 100 °C: \(Q_1 = mL_v = 0.05 \times 2.26 \times 10^6 = 113\,000\,\text{J}\).
Cooling from 100 °C to 40 °C: \(Q_2 = mc\,\Delta T = 0.05 \times 4200 \times 60 = 12\,600\,\text{J}\).
Total: \(113\,000 + 12\,600 = 125\,600\,\text{J}\), about 126 kJ. Nine tenths of it is latent heat.
A \(1000\,\text{W}\) heater is placed in a large block of ice at \(0\,\text{°C}\). How long does it take to melt \(300\,\text{g}\) of ice, if all its heat goes into the ice? Take \(L_f = 3.34 \times 10^5\,\text{J/kg}\).
A block of ice at \(0\,\text{°C}\) absorbs sunlight at a rate of \(334\,\text{W}\). What mass of ice melts in one hour? Take \(L_f = 3.34 \times 10^5\,\text{J/kg}\).
Where the latent heat goes
Temperature is a measure of how fast the molecules of a body move on average: the higher the temperature, the greater their kinetic energy. The chapter on kinetic theory makes this exact. In a solid the molecules are held in place by strong attractions, and they vibrate about fixed positions. To melt the solid, those attractions must be partly overcome. Work must be done to pull the molecules apart.
So during melting the heat goes into the potential energy of the molecules, not into their kinetic energy. The molecules do not move faster, so the temperature does not rise. Boiling is the same, only more so. In steam the molecules are so far apart that they hardly attract one another at all. Pulling them that far apart is why \(L_v\) is nearly seven times \(L_f\).
Think of lifting bricks onto a shelf. While you lift them, the bricks gain energy, but none of them moves any faster once it is on the shelf. The energy is stored in their new positions. Latent heat is stored the same way, in the new, wider spacing of the molecules.
Heat is supplied steadily to ice at 0 °C, and the thermometer stays at 0 °C until all the ice has melted. Explain where the heat went, and why the temperature did not rise.
The temperature measures the average kinetic energy of the molecules. While the ice melts, the heat does work against the attractions that hold the molecules in the solid, pulling them apart. So the heat raises their potential energy and leaves their kinetic energy unchanged. The molecules move no faster, so the temperature stays at 0 °C until all the ice has turned to water.
A farmer in Punjab sprays water on his crops on a night when frost is expected. As the water freezes on the leaves, what does it do?
Evaporation cools
Water need not reach 100 °C to turn into vapour. Wet clothes dry on a line at 30 °C. This slow escape from the surface, at any temperature, is evaporation. The molecules of a liquid move at many speeds. Only the fastest ones near the surface have enough energy to break free. When they leave, the average energy of those that remain falls, and the liquid cools. Each kilogram that evaporates takes about \(L_v\) with it.
Sweating works this way. When one gram of sweat evaporates, it takes about \(0.001 \times 2.26 \times 10^6 = 2260\,\text{J}\) from the body. That heat would cool \(2260/4200 = 0.54\,\text{kg}\) of the body by one degree.
A clay pot (matka) holds \(5\,\text{kg}\) of water. Water seeps through its walls and evaporates from the outside. Roughly how much must evaporate to cool the water by \(1\,\text{°C}\)? Take \(L_v \approx 2.26 \times 10^6\,\text{J/kg}\).
Boiling and the pressure above
Boiling is different from evaporation. At the boiling point bubbles of vapour form inside the liquid, rise and burst. A bubble can grow only if the vapour inside it pushes out as hard as the air and water press in. So the boiling point depends on the pressure above the liquid. At sea level, at one atmosphere, water boils at 100 °C. At a lower pressure it boils at a lower temperature, and at a higher pressure at a higher one.
In Leh, about 3500 m up in Ladakh, the air pressure is about two thirds of that at sea level. Water there boils at about 88 °C, as Fig 2.3 shows. Dal and rice cook slowly, because the boiling water is 12 degrees cooler. A pressure cooker does the opposite. Its lid traps the steam until the pressure inside is about twice the air pressure. The water then boils only at about 120 °C, and food cooks much faster.
A traveller in Leh finds that dal takes much longer to cook in an open pan than at home in Chennai. Why?
During a cricket match a player loses \(0.5\,\text{kg}\) of sweat by evaporation. How much heat does this carry away from his body, in megajoules? Take \(L_v = 2.26 \times 10^6\,\text{J/kg}\).
Joseph Black, a Scottish chemist, noticed that snow does not vanish the moment the air warms above freezing. It lingers for days. He measured how long ice in a warm room took to melt, and how long the same mass of water took to warm. He concluded that melting ice swallows a large amount of heat without getting warmer, and he called it latent heat. His student James Watt used Black's ideas about steam to build a far better steam engine.
A matka is made of porous clay, and water seeps slowly through its walls. On the outside the water evaporates, taking its latent heat from the pot and the water inside. In a dry, breezy place the water stays several degrees cooler than the air, with no electricity at all. A kulhad of unglazed clay cools tea the same way, which is why chai in a kulhad never scalds for long.
- Thinking the temperature must rise whenever heat is supplied.During a change of state the heat raises the potential energy of the molecules, and the temperature stays fixed.
- Forgetting the latent heat when steam or ice is involved.Always ask whether the substance changes state. If it does, add \(mL\) for that step.
- Thinking a liquid can turn to vapour only at its boiling point.Evaporation happens from the surface at every temperature; boiling happens throughout the liquid at one temperature.
- Believing water always boils at 100 °C.100 °C is the boiling point at 1 atm. It is lower on mountains and higher in a pressure cooker.