Thermal Physics Tutorial 1 - Heat Flow

 Contents

You will be familiar with conduction, convection, and radiation from GCSE.  You did it in Year 10 (and you probably weren't listening?).  If you want to revise the topic, click HERE.

Conduction involves the flow of heat energy in a solid material.  As well as the vibration of molecules, electrons are involved.  This explains why metals are good conductors of heat.  Materials that do not conduct electricity do not conduct heat well.  Liquids and gases can conduct heat, but are very poor at doing so.

Models of conduction are not easy and are well beyond the scope of these notes.

This occurs in fluids (liquids and gases).  It cannot happen in solid materials.  The particles in a fluid near a heat source vibrate more strongly.  Therefore they move apart, to occupy more space.  Therefore the density decreases, and the fluid rises.  As the particles rise higher, the heat is transferred to cooler particles.  Therefore the space occupied by the molecules decreases and the density rises.  The denser fluid falls.

 The particles themselves do not change density, just the fluid which consists of the particles.

All hot bodies radiate heat energy in the form of electromagnetic radiation.  Thermal (infra-red) radiation behaves just like light:

• It consists of a stream of photons;

• It travels in straight lines;

• Its speed is 300 million m s-1 in a vacuum;

• It can be reflected and refracted.

Infra-red radiation has a wavelength of above 650 nm.

Shiny surfaces reflect infra-red radiation.  Matt-black surfaces emit and absorb radiation particularly well.

Heat pipes are often found in computers to transfer the heat generated by the processors to the cooling fan.  Here is a picture of some heat pipes in a laptop computer.

They use all three processes of heat transfer.  If you want to find out more, click HERE.

Heat is the transfer of energy, a flow of energy from a hot body to a cooler body.  Heat is conducted by the movement of electrons as well as the vibration of moleculesGood conductors of electricity are good conductors of heat.  Heat must not be confused with internal energy, caused by the vibrations of atoms or molecules within a body.

If the heat source is removed, the object will cool by transferring heat energy to the surroundings.

Thermometry (Irish and CIE Syllabuses)

Temperature is a representation of internal energy, not heat flow.  The more internal energy there is, the higher the temperature.

The measurement of temperature is called thermometry Temperature is remarkably difficult to measure, and it was only in the seventeenth and eighteenth centuries that a reliable method was established.  Measurement depended on the even expansion of a liquid between two reference points.  The most logical reference points are the triple point of water (when water exists as liquid water, ice, and water vapour) and the boiling point of water.  These were proposed in 1742 by a Swedish astronomer and physicist, Anders Celsius (1701 - 1744).  There were 100 steps between the two reference points - hence the term centigrade (Latin - one hundred steps).  Originally the zero reference point was at the boiling point of water, and the triple point of water was 100.  The scale was reversed by Carl Linnaeus, who was the zoologist that gave us the system of naming species of animals and plants, to make the scale more practical.

Gabriel Daniel Fahrenheit (1686 - 1736) lived in The Netherlands, and had an enviable reputation as an expert glass-blower.  He used different reference points.  His low temperature reference point was a mixture of ice, water, and ammonium chloride, which was the coldest thing he knew.  His high temperature reference point was the boiling point of mercury.  He had 300 steps between the two.  Other reference points were the triple points of water (32 F) and the boiling point of water (212 F).  There were 180 steps between the two.

For its practicality and simplicity, the Celsius steps are used by all scientists.  The Fahrenheit scale is still used in the United States and some backward-looking news media in the UK.  Most people in the UK and Ireland are now used to the Celsius scale.

Physicists now use the Kelvin scale, devised in 1848 by William Thomson, Lord Kelvin (1824 - 1908).  It is an absolute scale, with the low temperature reference point being the temperature at which all molecular movement ceases.  The Kelvin scale does not depend on any physical property.  The absolute zero is written as 0 K which is -273.17 oC.  Note that Kelvin temperatures are never written "oK".

Temperatures are never below absolute zero.  It's impossible to have a temperature of -1 K (-274 oC).

Note that for most purposes, Celsius is the practical temperature scale, especially as we often take the temperature difference.  Celsius temperatures are often given the code q ("theta", a Greek lower case double letter 'th').   In certain cases, we have to use absolute temperatures.  In this case we have to use the Kelvin Scale, given the physics code T:

T = q + 273.15

The picture shows a number of different kinds of thermometer.  In this picture, there is a spirit in a glass tube thermometer that is widely used in school and college laboratories.  There are also two weather stations that have an external thermometer.

And this picture shows the sensors for each base station.  They are in the same place on a sheltered bird table in the garden.  They connect to the weather stations using radio waves.

Notice that the readings from the outside sensors in the top picture are not quite the same.  They are not too far out, but the picture shows that any two thermometers may not necessarily give the same reading.  This can increase the uncertainty in the results of a temperature dependent quantity.  These sensors can be calibrated to a reference to give as close to a true reading as possible.  Thermometers that are accurately calibrated tend to be quite expensive.

Thermometers such as these depend on the variation of the resistance of a thermistor with temperature (see Electricity Tutorial 6).

Another electrical method of measuring temperature is to use a thermocouple.  The thermocouple generates a voltage when exposed to a temperature difference.  A thermocouple is simply two wires of dissimilar metals that are joined together, as shown in the diagram:

The wires are joined together at the hot junction to ensure electrical contact.  The heat is applied at the hot junction.  The cold junction is left at room temperature.  Along their length, the wires are insulated with a material that will resist the temperatures generated by the heat source.  A voltage is generated, depending on the temperature.  The voltage is shown on a millivoltmeter, of which the scale is calibrated not in volts, but in Celsius.

If both junctions are heated by the heat source, the voltage will be zero.

The sketch graph shows the behaviour of two different thermocouples when exposed to temperatures between 0 oC to 1000 oC.

The pairs of metals that are used include:

• Nickel-chromium and nickel alumel;

• Iron and constantan.

We can compare the advantages and disadvantages of thermocouples when compared with thermistors:

• Thermocouples have a wide range of temperatures at which they can operate (-200 oC to + 350 oC, or 100 oC to 1200 oC, depending on the type), while the temperature range of a thermistor is about -55 oC to 150 oC.

• Thermocouples tend to have a linear characteristic, making calibration easy, while thermistors are non-linear (the resistance can decrease exponentially with temperature).

• Thermistors are sensitive a small temperature changes, while thermocouples require a large temperature change to show a difference.

• Thermocouples are active devices; they generate a voltage.  Thermistors are passive devices which require an external power supply to make them work.

• Thermocouples are accurate, and once calibrated, will retain their calibration.  Thermistors will be accurate to +/- 1oC, which leads to uncertainty.  They can be calibrated accurately, but this is a time consuming process, therefore expensive.

• Thermocouples don't need amplification circuitry while thermistors do.

Another important thing to do is to ensure good thermal contact, to ensure that you are measuring a temperature that is as close as possible to the true temperature.

Temperature sensors are used with data-loggers, which store the variation of temperature, and display the data on a graph.  Other examples of practical thermometers include:

• clinical thermometers;

• the thermometer on a coffee machine;

• the temperature gauge in a car.

Suppose we heat a rod at one end, and see how the temperature rises at the other.  At the start, the rod will be hot at one end and cold at the other end.  If we measure the temperature at the other end with a data-logger, we will see a graph like this:

When the heat flow into the rod is balanced by the heat flow out of the rod, we say that the system is in steady state.  The temperature in the rod varies in a linear way along the length of the rod.  This is called Thermal Equilibrium.  The heat flowing into the system is the same as the heat flowing out.

Heat flows can be modelled in the same way as electric circuits:

• The temperature difference is equivalent to the voltage;

• The heat flow is equivalent to the current;

• The thermal resistance is equivalent to the electrical resistance.

We can increase the internal energy by:

• Transferring heat into the body

• Doing a job of work on the body.

Some houses have electric central heating using night storage heaters.  These consist of a large metal box full of bricks, in which there are electrical elements.  The elements are switched on at night.  Electricity is cheaper at night, as power stations running at half load run less efficiently.

The elements transfer electrical energy to heat.  The heat flows into the bricks, increasing their internal energy.  The temperature rises.  When the electricity is turned off in the morning, the internal energy in the bricks is used to heat the room.

The rate of heat flow into the room can be controlled with flaps.

 Professor Turner warned his first year university class, “…it is internal energy.  If I catch any of you calling it ‘heat’, I will personally come out and thump you.”  Can you explain the difference so that none of his students will get into trouble with the professor?

Internal energy is the total energy in the movement of the molecules in a body.  The body itself might not seem to have potential or kinetic energy.  If  we look on the microscopic scale at an ideal monatomic gas, we can see that there are atoms travelling around in random directions at hundreds of metres per second.  The energy is translational kinetic energy.

If we have gases that consist of molecules, we will also observe:

• translational kinetic energy of vibrations of bonds;

• rotational kinetic energy of the molecules as they spin.

If there are liquid and solid materials in the system, we have to take into account the potential energy contained within the bonds.

The internal energy is all of these energies, potential and kinetic, added together.  The distribution of the energies in the molecules is random.

Brownian motion can be used as a model to show the vibration of molecules.

If you look at smoke particles in a cell under the microscope, you can see the particles jiggling about randomly as they are bombarded by air molecules.

Specific Heat Capacity is the quantity of heat required to raise the temperature of a unit mass through a unit temperature rise.  It is given the physics code c.   The formula associated with this is:

Heat flow (J) = mass (kg) × specific heat capacity (J kg-1 K-1) × temperature change (K)

• DQ is the flow of heat energy.

• Dq is the temperature change.  It doesn’t matter whether temperatures are in Kelvin or degrees Celsius.  Since the steps are the same, the difference will be the same.

• Units for specific heat are joules per kilogram per kelvin (J kg-1 K-1).  It is important that the masses are in kilograms.  Sometimes you might come across c in non-SI units of J g-1 K-1, but it is important that it is converted.  A material with a heat capacity of 1 J g-1 K-1 will have a capacity of 1000 J kg-1 K-1 in SI units.

Water has a specific heat capacity of 4200 J kg-1 K-1.  What is the amount of energy that is needed to bring 1.5 kg water to the boil from 20 oC

Here are the results of an experiment:

 Joulemeter reading at the start 31225 J Joulemeter reading at the end 43120 J Temperature at the start 23 oC Temperature at the end. 58 oC Mass of the material 1.25 kg Time taken 250 s

(a) Which piece of data is irrelevant?

(b)  What is the specific heat capacity of the material?

In the exam:

Watch out for conversions.  Make sure that your mass in the equation is in kilograms.  It does not matter whether the temperature is in Celsius or Kelvin.  The steps are the same, and it’s the difference that matters.

 Missing these conversions is a very common bear trap

We can measure the specific heat capacity of a metal in a simple experiment as shown in the diagram below:

We measure the voltage and current, which gives us power.  We should keep the voltage and current steady.  We then measure the temperature and time.

We can easily work out the energy supplied by:

E = VIt

We then need to plot the graph of temperature change against energy:

From this graph we can work out the specific heat from the gradient.

Specific heat with two materials

You may well find yourself having to work with more than one material.  An example of this may be:

• dropping a hot lump of metal into a beaker of water;

• taking into account the specific heat of a container, as oil is heated.

Although this may appear more complex, it's not that much harder than with a single material.

Suppose we have mass m1 of a liquid of specific heat c1 in a calorimeter (metal beaker) of mass m2 and specific heat c2. We want to change the temperature by Dq. To do this we need to supply DQ joules of energy.

For the liquid:

For the calorimeter:

The total energy supplied is:

To reduce uncertainty, the calorimeter needs to be well insulated around its sides.  It also needs to have a lid to reduce the heat loss by convection.

Here is a table of specific heat capacities of some materials:

The picture shows some oil in a copper calorimeter (a metal beaker).   As the oil gets hot, so will the calorimeter, so some energy will go to the oil, while some goes to the calorimeter.  We will use this for the worked example:

 Worked Example A copper calorimeter has a mass of 0.20 kg.  Oil, of mass 0.12 kg, is placed into the calorimeter.  The temperature of both the oil and the calorimeter is 20 oC.  15 kJ of energy is supplied to the oil and calorimeter, and the final temperature is 50 oC.  What is the specific heat capacity of the oil?  (c for copper = 381 J kg-1 K-1) Answer Formula:   DQ = mcDq Calculate the energy taken by the calorimeter: DQ = 0.20 kg × 381 J kg-1 K-1 × 30 K = 2286 J   The remaining energy is used to heat the oil.  So take 2286 J from 15000 J   Q = 15000 J - 2286 J = 12714 J   Now use this to calculate the specific heat capacity of the oil c' ("c-prime"):   c' =    Q       =    12714 J           mDq        0.12 kg × 30 K   c' = 3531 = 3500 J kg-1 K-1 (data are to 2 significant figures, so 2 s.f. is appropriate)

Now let us see what happens if we drop a hot lump of metal into some cool liquid.  The liquid will have a low starting temperature, while the metal has a high starting temperature.  The key to solving a problem like this is the energy flow DQ from the metal is the same as the energy flow into the liquid.  Let us look at that in the next example:

 Worked Example A lump of iron of mass 1.0 kg has a temperature of 500 K.  It is dropped into 10 kg water which has a temperature of 300 K.  Neglecting the heat capacity of the container, and assuming that the mass of water lost to steam is negligible, what is the final temperature of the iron and water?  (c for iron = 438 J kg-1 K-1.  c for water = 4200 J kg-1 K-1.) The temperature change is NOT 200 K.  The energy from the hot iron raises the temperature of the water.  The energy going out of the iron means its temperature will fall. Answer The same energy leaves the iron as enters the water (Conservation of energy).   DQ = 1.0 kg × 438 J kg-1 K-1  × -Dq1 = 10 kg × 4200 J kg-1 K-1 × Dq2   Now the end temperature is T.  It will be the same for the iron and the water.   For the iron: -Dq1 = T - 500 K (it will be negative because it's a temperature drop);   For the water: Dq2 = T - 300 K (it will be positive because it's a temperature rise) Now substitute: 1.0 kg × 438 J kg-1 K-1× -(T - 500 K) = 10 kg × 4200 J kg-1 K-1 × (T - 300K)   438 J K-1 ×  (-T + 500 K) = 42000 J K-1 × (T - 300K)   -438 T + 219 000 J = 42000 T - 1.26 × 107 J   -42000 T  - 438 T = -1.26 × 107 J - 2.19 × 105 J   42438 T = 1.2819× 107 J   T = 302 K (to 3 significant figures).

In this case, it is important to use absolute temperatures.  You will notice that the change in the temperature of the water is not very much.  Water has a particularly high specific heat capacity, which means you need to put in a lot of heat to get even a small rise in temperature.  That is why water is good at cooling things.

From your GCSE work, you will know that solid materials have a long range structure that follows a regular pattern.  There are bonds between the all atoms that determine the crystal structure.  In liquids the molecules go about in small groups  that can easily get past each other.  In gases the molecules are on their own with a large distance between their neighbours.

When we heat a solid material, the temperature rises with the heat flow into the body of the material.  However when the material reaches its melting point, the temperature remains the same until all the solid has melted.  The energy is still going in to break bonds.  Once all the material is liquid, the temperature rises until the boiling point is reached.  Then the temperature remains the same while all the bonds in the liquid are broken, and the liquid has boiled.

A change of state graph shows the idea of latent heat.

The rise in temperature is a reflection of the increase in internal energy.  In simple terms, internal energy is about vibration of molecules; the greater the vibration of molecules, the higher the internal energy.  When materials change state, work has to be done to break bonds between molecules.  Also most materials expand, so work has to be done against external forces.  So the latent heat represents the sum of these two:

• breaking of bonds;

• expansion of the material.

While the material is changing state, there no increase in internal energy, so the temperature stays the same.

The reverse is true.  As a gas condenses, heat is given out.  This heat can be made to do a job of work.  However, as the gas condenses, there is no change in temperature.  Nor is there a change in temperature as the liquid freezes to a solid.

While most work has been done on water (as it’s very common and convenient to use), the same models can be applied to other substances.  Investigating iron would not be easy, as it melts at over 1000 oC and boils at about 3000 oC.

The specific latent heat is the energy to change the state of a unit mass of liquid without a temperature change.  There is a value for specific latent heat during:

• fusion, or melting

• vaporisation, or boiling.

Whichever of these latent heats we are using, the calculation is the same.  The code for latent heat is L.

Energy flow = mass × specific latent heat.

The units are Joules per kilogram (J kg-1).

For water, the specific latent heat of fusion, lm = 334 000 J/kg.  The specific latent heat of vaporisation is rather higher, lv = 2.3 x 106 J/kg

 (Harder) Calculate how much energy is needed to melt, bring to the boil, and boil away 0.5 kg water.  How long would this take a 2 kW kettle?

Hot materials can do jobs of work, which is studied in detail in a discipline called Thermodynamics (see the option Applied Physics).  The steam engine below is an example of how hot fluids are used to do very large amounts of work.

You can also see that a considerable amount of heat is being lost as waste steam.