An introduction to thermodynamics.

Lesson Summary

• Discussion: The First Law and energy conservation (15 minutes)
• Discussion: Understanding the equation (15 minutes)
• Demonstrations: Expanding and compressing gases (20 minutes)
• Discussion: Adiabatic and isothermal changes (10 minutes)
• Student questions: On the second law (20 minutes)

Discussion: The first law and energy conservation

It is very easy to trivialise the ideas here and forget just how important thermodynamics has been in the development of physics. It is often worthwhile exploring this issue from two angles.

The first is historical and the following brief outline should serve as the basis for further study: In the late 18^th century Benjamin Thompson (Count Rumford) asserted that heat was not a fluid (caloric) stored within materials but was associated with motion in some way. This was deduced from seeing that apparently limitless amounts of heat were generated when boring a cannon, especially as the drill became blunter. However, as this predates a good knowledge of atoms and molecules it is unclear what is moving. In the early 19^th century James Joule performed quantitative measurements to compare the amount of mechanical work and heat that would raise the temperature of a known quantity of water by the same amount. This is the basis of the first law of thermodynamics. Recognition is also given to Julius Mayer, a physician, who noted on trips to the tropics that sailors’ venous blood was redder than when in colder climates, and so contained more oxygen. He deduced that, in the hotter climate, less energy was needed to keep warm and so less oxygen was used from the blood. He was able to link this to the amount of food needed and so made major headway in our understanding of work and energy.

The second approach is more philosophical. Students have had the idea of conservation of energy drilled into them from an early age but few question why we believe it. Ultimately it rests on experiment and there were at least two occasions in the 20^th century when the advent of quantum theory and the discovery of new particles made even top scientists question the validity of the law at a level of atoms and below. The first instance involves Compton Scattering, where light scatters off an electron and changes energy and momentum. In the early versions of the quantum theory, conservation of energy and momentum did not seem to both hold at the same time. Even Niels Bohr was willing to sacrifice conservation of energy. It turned out, of course, that the problem lay in the old quantum theory. A second example comes later in the century. In beta decay the beta particle can carry off a variable amount of energy and some appears to be lost. Again scientists were willing to conclude that energy is not conserved at a microscopic level. However, the discovery of the neutrino restored the energy balance.

So what does the First Law say? In words, the internal energy of a body (such as a gas) can be increased by heating it or by doing mechanical work on it. In symbols:

Δ U =  Δ Q +  Δ W

Note that internal energy U is not the same as the energy stored thermally. They are quite distinct concepts. Many text books tend to imply that energy stored thermally, what used to be referred to as heat and 'internal energy' are equivalent.

The sign convention here is that if DU is positive the amount of internal energy increases. This means that Δ Q stands for the heating of the system and Δ W for the work done on the system. This is known as the physicists' convention. You may come across text books that use the engineers' convention. Energy put into a system is positive, and the work output is also taken as positive. Providing you are consistent which convention you apply, all will be well.

Episode 605-1: Thermodynamics (Word, 47 KB)

Discussion: Understanding the equation

Practise using the equation and sign convention. Ask what happens to the internal energy as a gas is compressed or if it expands against an external pressure (such as air).

To compress a gas, you have to do work on it. This transfers energy to its particles, so they move faster – the gas is hotter.

If a gas expands against the atmosphere, it must do work to push back the atmosphere. Its particles lose energy and move more slowly, so its temperature falls.

How would such changes be observed? That is, what does a change in U represent? Note that it affects not just temperature but also possibly the state of a gas.

Demonstration: Expanding and compressing gases

There are a number of possible demonstrations of the First Law. Simple and dramatic ones include commercial devices that let you compress a cylinder of air rapidly and ignite a small wad of cotton. A more conventional alternative is to compress the air in a bicycle pump and to observe the rise in temperature.

Episode 603-1: Warming up a gas by speeding up its particles (Word, 46 KB)

The reverse effect is to demonstrate the formation of dry ice from a CO₂ cylinder, letting the gas expand against air pressure. You will need to consult the relevant safety documents for this.

Episode 605-2: Formation of dry ice from a carbon dioxide cylinder (Word, 79 KB)

You may also have the equipment for a quantitative analysis. This usually involves a friction drum or wheel and compares mechanical work done against a friction force to the rise in temperature of the system. An alternative system allows mechanical work to be compared with energy supplied electrically.

Episode 605-3: Doing mechanical work (Word, 48 KB)

Episode 605-4: Mechanical and electrical heating (Word, 50 KB)

Discussion

If your specification requires it you could then go on to look at examples of adiabatic changes (in which no energy flows in or out – either an insulated system or one, like the bicycle pump, where the change is fast), isothermal changes (where the temperature is kept constant, so Δ U is zero, usually involving a heat bath to extract or supply energy) and constant volume or isochoric (so no work can be done, but energy can flow in or out).

This simulation of an adiabatic change may be useful, but beware the different notation used for internal energy:

Boston University Physics Department

Student questions

The conservation of energy underlies our modern understanding of physics but also has important implications for our use of energy resources. In particular, the idea that mechanical working disspates to the surroundings is a very practical issue. With little quantitative work in this section, students could be set questions on sensible use of energy resources or the mechanics of power stations. Try to get them to emphasise the difference in meaning between conserving energy as in not wasting it and the scientific meaning of conservation.

This material is only relevant to some specifications so check carefully before covering it.

Lesson Summary

• Discussion and demonstration: Steam engines (10 minutes)
• Student activity: Finding out about thermal efficiencies (20 minutes)
• Discussion: The Second Law and efficiency (15 minutes)
• Worked examples and student questions: Calculating efficiency (20 minutes)

Discussion and demonstrations: Steam engines

If you have a model steam engine it would be excellent to show this and consider the changes to the way energy is stored when it is used. Emphasise that energy is dissipated (wasted) when, for example, steam emitted from the chimney heats the surroundings. If you do not have a steam engine, try to find a video or other resources relating to a power station. In all cases there should be energy stored in the fuel (and oxygen) that (usually) generates steam.

The steam drives a turbine and then is condensed and returned to the boiler. A significant proportion of the energy stored thermally in the steam is not delivered to the turbines and it is then wasted in heating up the coolant in the condenser. That coolant may then itself be used as a source of energy (in Combined Heat and Power stations) improving the overall efficiency of the power station.

Student activity: Finding out about thermal efficiencies

Use the web or printed information from energy companies to find out about the efficiency of different types of power stations. Use these to discuss the meaning of the definition of efficiency = useful energy outtotal energy in × 100 %.

Clearly without the word useful the efficiency would be 100% in all systems. (Why? Because energy is conserved)

Can students think of a system that is 100% efficient? If not, lead them to thinking about systems where heating is the intended change – such as a radiator or electrical fire. How does a coal fire compare (energy loss up the chimney)?

Can they draw up a quantitative Sankey diagram for a power station?

Episode 606-1: Power station efficiencies (Word, 30 KB)

Episode 606-2: Sankey diagrams (Word, 158 KB)

Discussion: The Second Law of Thermodynamics and efficiency

Ultimately, justification of the Second Law of Thermodynamics rests on an understanding of entropy. Without a full-blown mathematical proof, which is inappropriate at this level, it is necessary to rely on assertion and justification by reason. A simple statement of the Second Law is that you cannot have a process whose only effect is to use energy stored thermally to do work. If you could, you could build a car which extracted energy from the air and drive along without needing petrol. This limitation is fundamental, not merely a practical constraint.

In a power station the working fluid (water or steam) is allowed to expand through the turbines and so drive them. Afterwards the expanded steam needs to be returned at low pressure by cooling in order to complete a cycle – to put it back as it was before it entered the boiler. Hence the need to cool the steam in the condenser. This energy is not useful. Thus, although we can use some energy stored thermally to do work we cannot extract all of it.

Consideration of cycles like those in a power station (a heat engine ) shows that the maximum efficiency of such a device is given by T_hot  −  T_coldT_hot. In this equation T is in K, the absolute temperature .

Finally, you may need to mention the heat pump. This is simply a heat engine operated in reverse. Work shifts energy stored thermally in a cold reservoir to energy stored thermally in a hot one. The details are not needed, but a refrigerator is an example. Heat pumps are sometimes used to heat houses in cold climates. They can be very effective.

Worked examples: Calculating efficiency

1 Calculate the maximum theoretical thermal efficiency of a coal-fired power station that heats steam to 510 ° C and cools it in a condenser at 30  ° C.

Answer:

Maximum efficiency = T_hot  −  T_coldT_hot

Maximum efficiency = (510+273) K  −  (30+273) K(510+273) K

Maximum efficiency = 0.61, or 61%.

2 The temperature of the gases in a car engine during combustion is 1800  ° C. The exhaust is expelled at 80  ° C.

Calculate the maximum theoretical thermal efficiency of the engine.

Answer:

Maximum theoretical efficiency = T_hot  −  T_coldT_hot

Maximum theoretical efficiency = (1800+273) K  −  (80+273) K(1800+273) K

Maximum efficiency = 0.83, or 83%.

Of course, in both case, the actual efficiency will be smaller. Students should consider why.

Episode 606-3: Student questions; calculating efficiency (Word, 27 KB)

Energy must be transferred to (or from) a material to increase (or decrease) its temperature. Here is how to calculate how much.

Lesson Summary

• Discussion: Energy and change of phase (15 minutes)
• Student experiment: Measuring specific heat capacities (40 minutes)
• Worked example: Calculation involving c (10 minutes)
• Student questions: Calculations (30 minutes)

Discussion: Energy and change of phase

Up until this point the link between internal energy and temperature has been qualitative, except for gases. In order to extend the discussion to solids and liquids we need to get more quantitative in two ways. One is to discuss how much the temperature of a body changes when its internal energy in increased by a certain amount. The other is to ask what happens when a substance changes phase from a solid to a liquid or liquid to a gas.

Start by introducing the equation for specific heat capacity c (SHC) and defining the terms. The word specific is an old fashioned way of saying per unit mass. Work through a simple calculation.

Understanding this equation will help solidify ideas about temperature and energy and how they differ. A possible analogy was supplied by Richard Feynman. He suggested thinking of energy as being like water, and temperature as wetness. A towel can have different amounts of fluffiness, so take more or less water to make it wet. When we dry ourselves, we dry until the towel is as wet as we are (same temperature).

The anomalously large SHC of water should also be discussed as it is particularly important for the development and maintenance of life on Earth.

NB nomenclature: there isn’t any agreed way to name c. Some use specific thermal capacity , others favour specific heating capacity to emphasise the fact that heat is not an entity but a short hand name for a process (heating as oppose to working). Perhaps the most common is specific heat capacity.

Another source of confusion is treating state and phase as synonyms (as in changes of state / phase). Solids, liquids and gases are three of the different phases of matter (superfluids and plasmas are two others. NB Here, by a plasma, we mean an ionised gas, not a biological fluid). Thus melting, boiling etc are changes of phase. Each phase can exist in a variety of states depending upon e.g. the temperature and pressure. Thus the Ideal Gas Equation of State PV = nRT summarises the physically possible combinations of P, V and T for n moles of the ideal gas.

Student experiment: Measuring specific heat capacities

Students should carry out an experiment to measure the specific heat capacity of a solid and/or a liquid very soon after meeting the expression. There are a number of points to note here:

• If specific heat capacity is constant, the temperature will rise at a uniform rate so long as the power input is constant and no energy is dissipated (stored thermally in the surroundings)
• If the substance is not well insulated a lot of energy will be dissipated. This can be accounted for but in most cases students will not do so quantitatively
• They should calculate their value and make a comparison with data book values. They should be able to think of a number of reasons why their value does not match that in the data book.

Several different methods of determining the SHC of liquids and solids are given in the links below. Choose those best suited to your pupils and available equipment.

Episode 607-1: Measuring the specific heat capacity of a metal (Word, 33 KB)

Episode 607-2: The specific heat capacity of water and aluminium (Word, 37 KB)

It is useful to compare electrical methods of measuring the specific heat capacity of a solid and liquid including the continuous flow calorimeter for a liquid.

Episode 607-3: Measurement of specific heat capacities (Word, 56 KB)

Worked example: Calculation involving C

This example deals with the mixing of liquids at different temperatures.

1.5 l of water from a kettle at 90 ° C is mixed with a bucket of cold water (10 l at 10 ° C) to warm it up for washing a car. Find the temperature of the mixed water, assuming no significant dissipation during the mixing. c of water is 4.2  kJ^kg-1 °^C-1.

Answer: If there is no energy dissipated and no work done then the total energy of the system at the end is the same as at the start. Working with temperature differences from 10 ° C we have:

initial energy = mc Δ T

as 1.5 l has mass of 1.5 kg:

Energy stored initially due to rise in temperature of the water in the kettle = 1.5 kg  ×  4.2  kJ kg^-1 ° C^-1  ×  (90-10)  ° C

Energy stored initiallly = 504 kJ

This is the amount of energy available to raise the temperature of all 11.5 l of water. Hence:

504 kJ = 11.5 l  ×  4.2  kJ kg^-1 ° C^-1  ×  (T_final-10) ° C}

(where T_final is the final temperature)

T_final = 20.4  ° C.

You can invent similar problems using a mixture of substances – a hot brick in water for example. However, be careful to consider realistic situations where not too much is dissipated (due to production of steam).

Student questions: Calculations

A range of questions involving specific heat capacity.

Episode 607-4: Specific heat capacity: some questions (Word, 30 KB)

Episode 607-5: Thermal changes (Word, 46 KB)

Energy is involved in changes of phase, even though there is no change of temperature.

Lesson Summary

• Discussion: Defining specific latent heat (10 minutes)
• Demonstration: Boiling water (15 minutes)
• Student experiment: Measuring l (30 minutes)
• Student experiment: Cooling curves (30 minutes)
• Worked example: Latent and specific heat (5 minutes)
• Student questions: Involving c and l (40 minutes)

Discussion: Defining specific latent heat

The final point in this topic is to return to the original definition of (internal energy) as being a combination of both the energy stored kinetically (kinetic energy) and energy stored chemically i.e. in the motion of the particles and the bonds between them. In talking about ideal gases all the energy was assumed to be stored kinetically because there were assumed to be no bonds between the atoms. However, in a solid or liquid there are bonds and clearly some energy is needed to break those bonds. That means that, in melting a solid or boiling a liquid, a substantial amount more energy needs to be transferred which does not raise the temperature. This is the hidden heat or latent heat.

The energy you need to transfer to a mass m of a substance to melt it is given by

Δ E = m × L

Or the 'specific latent heat' is the energy you need to transfer to change the unit mass from one phase to another.

Demonstration: Boiling water

Ask your class to watch some water boiling and think about what is going on. Energy is being transferred, but the temperature is not rising. Intermolecular bonds are breaking, and, as a physicist would say, work is being done to separate the particles against intermolecular attractive forces.

The key point from these is that, for certain materials, there is a phase transition where the energy transferred no longer raises the temperature (adds to each molecule's kinetic energy) but instead breaks bonds and separates the particles. This should be made quantitative. Likewise, the reverse processes involve energy being transferred from the substance. So evaporating liquids are good coolants and freezing water to make ice is considerably more of an effort than cooling water to 0  ° C.

Episode 608-1: Examination of boiling (Word, 34 KB)

Student experiment: Measuring L

It is useful to have measured a specific latent heat – for example, that of melting ice.

Episode 608-2: The specific latent heat of fusion of ice (Word, 67 KB)

Student experiment: Cooling curves

If you have a class set of data-loggers for recording temperature, determination of the cooling curve of stearic acid, naphthalene or lauric acid is worthwhile. Even as a demonstration this is good and can be left running in the background while the students work on calculations.

Episode 608-3: Heating and cooling curves (Word, 52 KB)

Worked examples: Latent and specific heat

Scalds from water and steam

We assume that our hand is at 37  ° C, and that we put 10 g of water at 100  ° C accidentally on our hand. The water will cool to 37  ° C. Assuming that all the energy lost by the water will be gained by our hand:

Energy shifted from water = mc Δ T

Energy shifted = 1.5 kg  ×  4.2  kJ kg^-1 ° C^-1  ×  63  ° C

Energy shifted = 2 646 J.

But if the 10 g had been steam then the steam would first have to condense.

Energy shifted in condensing = mL

Energy shifted in condensing = 1.5 kg  ×  2 260 kJ kg^-1

Energy = 22 600 J

So the energy lost in 10 g of steam turning to water at 37  ° C is 25,246 J.

This is nearly ten times as much as the water alone!

The worked example is based on one from Resourceful Physics.

Student questions: Involving c and L

Practice in situations involving specific heat capacity and specific latent heat.

Episode 608-4: Questions on specific heat capacity and specific latent heat (Word, 27 KB)

Episode 608-5: Further specific and latent heat questions (Word, 30 KB)