Thermal equilibrium and conduction

A hot drink left on the table cools down, and a cold can from the fridge warms up, until both reach the temperature of the room. Energy transferred by heating moves from the hotter region to the colder one. One way it travels is by conduction, passed from particle to particle through a solid. The explanation uses the particle picture from the Kinetic particle model: the particles of a solid are closely packed and vibrate about fixed positions, and a higher temperature means a greater average kinetic energy of the particles.

Heating goes from hotter to colder until the temperatures are equal

When two regions have different temperatures, energy is transferred by heating from the region at the higher temperature to the region at the lower temperature. If neither region changes state and there is no other heating or cooling, the hotter region cools and the colder region warms. They approach the same temperature. During a change of state, energy can be transferred without a temperature change.

Two regions at the same temperature are in thermal equilibrium: there is no net energy transfer by heating between them.

1. The regions start at different temperatures.
2. Energy is transferred by heating from the hotter region to the colder region.
3. The temperature difference shrinks.
4. The temperatures become equal: thermal equilibrium. The net transfer stops.

Thermal equilibrium means equal temperature, not equal internal energy. A bathtub of water at 30 °C and a cup of water at 30 °C are in thermal equilibrium, but the bathtub has far more internal energy because it contains far more water. The particles in both keep moving at equilibrium; only the net transfer between them has stopped.

Worked example 1

A drink left in a room

Problem

A covered cup of tea at 80 °C is left on a table in a room at 25 °C. Assume evaporation is negligible and the room stays at 25 °C. Describe what happens to the temperature of the tea and state its final temperature.

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1. Compare temperatures

   Method

   Find the hotter and the colder region.

   Reason

   Energy is transferred by heating from a hotter region to a colder one.

   Working

   Answer

   The tea (80 °C) is hotter than the room (25 °C).

2. Find the direction

   Method

   Say which way energy is transferred.

   Reason

   The tea is the hotter region.

   Working

   Answer

   Energy is transferred from the tea to the cooler air and table around it, so the tea cools.

3. Find the end point

   Method

   Say when the net transfer stops.

   Reason

   There is no net transfer between regions at the same temperature.

   Working

   Answer

   The room is very large, so its temperature barely changes. The tea ends at about 25 °C, in thermal equilibrium with the room.

4. Check

   Working

   Answer

   The tea does not cool below 25 °C. Below room temperature it would be the colder region, and energy would flow into it. An uncovered drink can also lose energy by evaporation; that process is excluded from this model.

A can of cold drink at 5 °C taken from the fridge into the same room follows the same rule the other way: energy is transferred into the can until it reaches about 25 °C. Nothing called “cold” flows out of the can; the room is the hotter region.

Check your understanding 1

Water at 60 °C is mixed with water at 20 °C in an insulated container. Assume only the two water portions exchange energy. Which range contains their final equilibrium temperature, and which way does energy transfer?

Final temperature

Between 20 °C and 60 °CCorrectIncorrectHigher than both 20 °C and 60 °CCorrectIncorrectLower than both 20 °C and 60 °CCorrectIncorrect

Net energy transfer

From the hotter water to the colder waterCorrectIncorrectNone: insulation prevents transfer between the watersCorrectIncorrectFrom the colder water to the hotter waterCorrectIncorrect

Check answer

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Energy flows from the hot water to the cold water, so the hot water cools and the cold water warms. Only the two water portions exchange energy, so the energy lost by one is gained by the other. The mixture reaches one common temperature between 20 °C and 60 °C. It cannot rise above the hotter starting temperature or fall below the colder one.

Check your understanding 2

A thermometer at 25 °C is put into hot soup. Why must you wait before reading it?

Your waiting reason and condition for a valid reading

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At first the thermometer is colder than the soup, so energy is transferred from the soup into the thermometer and its reading rises. Only when the bulb reaches the soup’s temperature, in thermal equilibrium with it, does the thermometer show the soup’s temperature.

Conduction: passing energy through a solid

Conduction is the transfer of energy through a material without the material itself moving from place to place. In a solid it happens in two ways.

1. Vibrating atoms and molecules (all solids). When one end of a solid is heated:

1. The particles at the hot end gain kinetic energy and vibrate more vigorously about their fixed positions.
2. They collide with and push on their neighbours, which then vibrate more vigorously too.
3. Particle by particle, energy is passed along towards the colder end.

Each particle vibrates about its fixed position rather than flowing along the solid. Energy is passed between neighbouring particles.

2. Moving electrons (metals). A metal also has many free electrons that can move through it.

1. Free electrons at the hot end gain kinetic energy.
2. They move quickly through the metal, far from where they gained the energy.
3. They collide with atoms and other electrons in the colder parts and transfer energy there.

The electrons move randomly in different directions. Energy is transferred overall from the hotter region to the colder region; the rod does not need to carry an electric current. The solid atoms stay near their fixed positions.

Metals conduct by both methods and are usually good conductors. Wood, plastic and glass have no freely moving electrons; they conduct poorly and are useful insulators. Conductivity depends on the material, so vibration alone does not always mean poor conduction: diamond, for example, conducts very well.

Check your understanding 3

A copper rod and a glass rod of the same size each have a bead of wax stuck to one end. Their other ends are held in the same flame. Predict which bead melts first, and explain the difference.

Your first melting bead and microscopic conduction comparison

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The bead on the copper rod melts first. Glass conducts only by atomic vibrations, passing energy slowly from neighbour to neighbour. Copper also has free electrons, which gain kinetic energy at the hot end, move rapidly through the metal and transfer energy to atoms near the wax in collisions.

Worked example 2

A metal spoon in hot soup

Problem

A steel spoon is left standing in a bowl of hot soup. Explain why the handle becomes hot. Can energy still flow through the spoon when its temperature stops changing?

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1. Find the direction

   Method

   Compare the soup and the spoon.

   Reason

   Energy is transferred from hotter to colder.

   Working

   Answer

   Energy is transferred from the soup to the part of the spoon in the soup.

2. Lattice vibrations

   Method

   Describe the atoms in the hot part.

   Reason

   Vibrating atoms pass energy to their neighbours.

   Working

   Answer

   The atoms in the hot part vibrate more vigorously and pass energy to their neighbours along the handle.

3. Free electrons

   Method

   Describe the free electrons in the metal.

   Reason

   Metals have free electrons, which make them good conductors.

   Working

   Answer

   Free electrons in the hot part gain kinetic energy, move through the steel and transfer energy to atoms in the cooler handle in collisions.

4. Steady temperature

   Working

   Answer

   The handle also transfers energy to the cooler air. Its temperature can become steady when the energy entering each second equals the energy leaving. Energy still flows from the soup, along the spoon and into the room: a steady temperature does not necessarily mean thermal equilibrium. Different parts of the spoon can stay at different temperatures.

Check your understanding 4

A metal rod is not connected to an electrical supply. One end is heated and the other becomes warmer. A student says this is impossible because free electrons cannot transfer energy unless a current flows. Explain the error and state what happens to the solid atoms.

Your electron-motion explanation and solid-atom motion

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An electric current is not needed for thermal conduction. Free electrons move randomly; those gaining energy near the hot end transfer energy to cooler parts in collisions, giving a net energy transfer from hot to cold without a net charge flow along a circuit. The solid atoms remain near their fixed positions and also pass energy to neighbours through vibrations. The rod can therefore conduct energy while electrically disconnected.

Good conductors, poor conductors and trapped air

Air is an especially poor conductor. Its particles are far apart, so they collide with one another much less often than the closely packed particles of a solid, and energy is passed along very slowly.

Many insulating materials work by trapping air in small spaces: wool and fleece clothing, a foam cup, bubble wrap, feathers and the gap in double glazing. The air conducts energy very poorly, and the small spaces stop it from moving around and carrying energy away.

An insulator does not stop the transfer or make warmth. It slows the transfer. A blanket keeps you warm because it slows the transfer of energy from your warm body to the colder room.

Check your understanding 5

Why is a saucepan made of metal but its handle made of plastic or wood?

Your useful and unwanted conduction paths

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The metal base is a good conductor, so energy from the stove is conducted quickly through it to the food. The handle is a poor conductor, so energy is conducted along it only slowly, so the end you hold warms more slowly. It can still become hot after prolonged heating.

Why metal feels colder than wood at the same temperature

A metal bench and a wooden bench have stood side by side in the shade all afternoon. Both are at the air temperature, say 28 °C: they are in thermal equilibrium with the air. Yet the metal bench feels colder when you touch it.

1. Your hand (about 33 °C) is warmer than both benches, so energy is transferred from your hand into each bench.
2. The metal is a good conductor. It conducts energy away from the spot you touch quickly, into the rest of the bench, so your skin there cools quickly.
3. The wood is a poor conductor. The wood you touch warms up and the energy spreads away from it only slowly, so your skin loses energy more slowly.
4. Your skin senses its own temperature falling, not the temperature of the bench. It falls faster on the metal, so the metal feels colder.

On a very hot day, when both benches are at 50 °C, the metal feels hotter: now the benches are the hotter region, and the metal conducts energy into your hand faster.

Check your understanding 6

Two identical ice cubes are placed at the same moment, one on a metal tray and one on a plastic tray. Both trays have been in the same room all day. A student predicts that the cube on the metal tray will melt more slowly, because metal feels colder. Is the student right? Explain.

Your corrected melting prediction and temperature-versus-rate reason

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No: the cube on the metal tray melts faster. Both trays are at room temperature, warmer than the ice, so energy is transferred from each tray into its ice cube. Metal is a good conductor, so energy from the rest of the metal tray reaches the ice quickly. The plastic conducts poorly, so the spot under its ice cube cools and energy reaches that cube only slowly. The metal feels colder for the same reason: it conducts energy quickly, out of your warmer hand.

Check your understanding 7 Steady readings can accompany energy passing through

A metal rod connects a hot reservoir to a cold reservoir. Its temperatures stay constant along a temperature gradient. Each second, 50 J enters and 50 J leaves the rod. Find its net energy gain per second. Is the rod in thermal equilibrium?

Net energy gain each second

JUnit: J

Thermal equilibrium?

No: its ends remain at different temperaturesCorrectIncorrectYes: no electric current flows through itCorrectIncorrectYes: each part stays at a steady temperatureCorrectIncorrect

Check answer

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The rod is not in thermal equilibrium throughout: its ends have different temperatures, so energy continues to pass from hot to cold. The net change in the rod’s internal energy is 50 − 50 = 0 J each second, because input equals output. Zero accumulation does not mean zero transfer. Atoms vibrate about their positions and pass energy to neighbours; free electrons also transfer energy through collisions in the metal. The maintained temperature difference keeps energy passing through while the solid stays in place.

Common mistakes

• Saying cold flows from the ice into the drink. Energy is transferred from the warmer drink to the colder ice.
• Saying two objects in thermal equilibrium have the same amount of energy. They have the same temperature.
• Saying hot atoms travel along a rod. Atoms in a solid only vibrate about fixed positions; vibrating atoms pass energy to their neighbours, and free electrons carry energy through metals.
• Saying only metals conduct. All solids conduct by atomic vibrations; free electrons add a faster route in metals.
• Saying a metal bench is colder than a wooden one beside it. Both are at the air temperature; the metal conducts energy away from your hand faster.
• Saying a jacket gives out warmth. It produces no energy; its trapped air slows the transfer of energy from your body.