Thermal Physics Tutorial 4 - The Laws of Thermodynamics

 Contents

History of Thermodynamics

Thermodynamics is a branch of physics that links matter and energy.  The science of thermodynamics has common rules that govern the interactions between matter and energy from the very large (macroscopic), for example a nebula formed by a supernova, to the everyday, for example the way a car uses its petrol or diesel, right down to the very small (microscopic), for example the energy in an atom.  Thermodynamics was studied in the Industrial Revolution to find ways of making ways of making steam engines more efficient.

The first studies were carried out by a French engineer, Nicolas Léonard Sadi Carnot (1796 - 1832) and published in 1824.  He introduced a concept of caloric, in which heat was an invisible fluid of zero mass.  Caloric ran downhill from the boiler through a steam engine to the condensers.  Energy could be extracted from the downhill flow, like a water mill extracting energy from the downhill flow of water in a river.  Although caloric does not exist,  Carnot's Principle is still valid.  Its results are that there is a maximum efficiency for any heat engine (steam, turbine, or internal combustion engine.   The maximum efficiency is determined by the difference in temperature of the source and the temperature of the coolant.   Engineers often call this the temperature gradient.

Quantitative studies of heat flow was carried out by James Prescott Joule (1818 - 1889) in his father's brewery in Manchester.  He disproved the existence of caloric, and produced much of the groundwork for our understanding of the link between work, energy and temperature.  Joule's work also laid the foundation for the Molecular Kinetic Theory.

Joule's work was developed further by William Thomson (1824 - 1907), better known as Lord Kelvin.  Thomson's work was then taken even further by a German physicist, Rudolf Julius Emmanuel Gottleib (1822 - 1888), who is usually known by a Latin name, Clausius (it was a fashionable thing to do in those days).  Both concluded that the quantity of energy remained the same in a steam engine, but its distribution changed irreversibly. Both Gottleib and Thomson's work led to two versions of the Second Law of Thermodynamics.

Ludwig Eduard Boltzmann (1844 - 1906) explained how the behaviour of particles on the microscopic scale could affect our understanding of flows of energy on the macroscopic scale.  It was Boltzmann's work that led to the Molecular Kinetic Theory that we have seen in Thermal Physics Tutorial 3.

Language of Thermodynamics

We need to be aware of some of the key ideas for us to make sense of Thermodynamics:

• Energy is the ability to do work.

• Internal energy is the result of motion of molecules.  This can be due to movement of molecules (translational kinetic energy), and/or the vibration of intermolecular bonds.

• Heat is a flow of energy from a region of high internal energy to low internal energy.

• Temperature is a measure of internal energy, not heat.

• A hot reservoir is a region of high temperature.

• A cold reservoir is a region of low temperature.

• Absolute zero is the temperature at which all molecular movement ceases (-273.15 oC, or 0 K).  It is impossible to have -1 K.

• Entropy is a concept in which energy becomes less useful.  As entropy increases, energy gets more spread out and less useable.

• Temperature gradient the temperature per unit length.  Units are K m-1.

• Thermal Equilibrium is where the energy flow (heat) is zero between two reservoirs.

• Efficiency is the ratio between the work got out and the energy put in.

• A diathermal system allows heat to flow in and out.

• An adiabatic system keeps the heat in, so that there is no heat flow.

The idea of a temperature gradient is shown below:

We can write an equation:

Principles of Thermodynamics

• Heat flows from a hotter object to a colder object.  It never happens the other way round.

• Work can be extracted from the heat flow.

• Total entropy must always increase.  This means that energy gets more spread out.

• If entropy is decreased in one region, it must be at the expense of an increase in entropy elsewhere.

• There are three main laws of Thermodynamics.

• There can be no temperature lower than absolute zero.  Absolute zero cannot be reached.

The Zeroth Law of Thermodynamics

The Zeroth Law of Thermodynamics was formulated after the First and Second Laws.  It is considered to be more fundamental than the First and Second Laws.  Hence it was called the Zeroth Law.  It states:

If two bodies are each in thermal equilibrium with a third body, then they must be in thermal equilibrium with each other.

Consider the picture below:

A is in thermal equilibrium with C, and B is in thermal equilibrium with A.  Therefore B and C must be in thermal equilibrium.  All will therefore be at the same temperature.

The First Law of Thermodynamics

Thermodynamics is the study of heat flows and how they can be put to work.  Engines work by converting heat energy into movement energy, which can then do useful jobs of work for us.  We need to look at a couple of key words.  A system is the object of interest whose behaviour we are monitoring in relation to its surroundings.  A flask containing gas is a system; the water bath in which the flask is placed is its surroundings.  The diagram below helps to show the idea:

The Laws of Thermodynamics were the results of work by nineteenth century physicists.  Ironically the Second Law came before the First Law.  Then a more fundamental law, the Zeroth Law was worked out.

In words the First Law of Thermodynamics is:

The change in internal energy of a system is equal to the sum of energy entering the system through heating and energy entering the system through work done on it.

 Question 1 What is internal energy?

We can write the first law in code:

[DQ = heat entering the system; DU = increase in internal energy; DW - work done by the system]

The diagram here explains the idea:

 Worked Example A lump of lead of mass 0.50 kg is dropped from a height of 20 m onto a hard surface.  It does not bounce but remains at rest. What are DQ, DW, and DU? Answer DQ = 0 J as zero heat is supplied to the system DU = mgDh = 0.5 kg × 9.81 m s-2 × 20 m = 98 J DW = - 98 J as work is done on the system rather than by the system

 Question 2 Some air in a bicycle pump is compressed so that its volume decreases and its internal energy increases.  If 25 J of work are done by the person compressing the air, and if 20 J of thermal energy leave the gas through the walls of the pump, what is the increase in the internal energy of the air?

If we compress a gas in a bicycle pump, we find it gets hot.  Then we let the pump cool down, without releasing any gas.

 Question 3 What happens if we release the pump?

Consider a cylinder of area A.  A fluid is admitted at a constant pressure, p.  It makes the piston move a distance, s.

We know that:

• Pressure (N m-2) = force (N) ÷ area (m2)

p = F/A

• Work done (J) = force (N) × distance moved (m)

W = FDs

Therefore

• Force (N) = pressure (N m-2) × area (m2)

F = pA

• Work done (J) = pressure (N m-2) × area (m2) × distance moved (m)

• Area (m2) × distance moved (m) = change in volume (m3)

So we can write:

Work done (J) = pressure (N m-2) × change in volume (m3)

In code:

DW = pDV

This can be shown in a graph:

 Question 4 A cylinder has an area of 0.125 m2 .  Steam is admitted at a pressure of 1.5 x 106 Pa.  The piston moves a distance of 0.20 m.  What work is done?

The First Law of Thermodynamics has important implication in heat engines.   These are discussed in Engineering Physics Tutorial 3, Engineering Physics Tutorial 4, and Engineering Physics Tutorial 5.

The Second Law of Thermodynamics

The Second Law of Thermodynamics states that it is impossible for any heat engine to be 100 % efficient:

No process is possible which results in the extraction of an amount of heat from a reservoir and its conversion to an equal amount of mechanical work.

This is the statement of the Second Law as described by Kelvin.

The Clausius (Gottlieb) version states:

No process is possible in which there is an overall decrease in the entropy of the universe.

The theory behind this is that entropy increases.  In other words all processes tend towards chaos (which might explain my physics lessons when I worked in schools).  If you drop a pack of cards, they will scatter and the chances of their landing in a meaningful order are very small indeed.

Most energy is lost to the surroundings as low grade heat.  We can show this in the diagram below:

In this diagram, called a Sankey Diagram, we can see that of 72 kW of power from the fuel, only 9 kW are used in actually driving a car along a road.  The rest is lost as low grade heat.  As we said before, getting energy out of heat is remarkably difficult.

All heat engines work by extracting mechanical energy from a temperature gradient.  A heat engine has to operate between the hot reservoir and the cold reservoir to satisfy the Second Law of Thermodynamics.  Heat flows from hot to cold, never the other way round:

Heat won't pass from a cooler to a hotter.

You can try it if you like,

But you far better notta,

Because the cold in the cooler

Will get hotter as a ruler,

And that's a physical law!

[Michael Flanders and Donald Swan]

The engineering implications of the Second Law of Thermodynamics are discussed in Engineering Physics Tutorial 6.

Diffusion is an example of how entropy increases.  Suppose we have two gas jars.  The one on the left has bromine (Br2) molecules, while the one on the right has air molecules.  We use bromine (an unpleasant smelling and toxic gas) as it is readily visible.

We then remove the cover slip and watch the way the molecules diffuse.  We assume that the molecules move randomly.

Eventually the molecules become evenly distributed:

It would be impossible to catch all the molecules and put them back all into the left-hand jar.  In this example, there are 29 molecules.  If there was just one molecule there would be a chance or probability of 1 in 21 (i.e., 0.5 or 2-1) of the molecule being in the left hand jar.  Similarly, if there are 4 molecules, the chances of all molecules being in just the left hand jar would be 1 in 16, or 2-4.

 If there are 29 molecules, what is the chance of all the molecules being in the left hand jar at the same time?

If there are W molecules, the chances of all of them being in the left hand jar would be 2-W.

 Suppose we watched the 29 molecules once a second, show that it would take about 17 years for all 29 to be in the left hand jar at the same time.

Your answer to Question 6 is an average.  Due to the random nature of molecular movement, the molecules could be all in the left hand jar in 10 seconds, or 300 years.  On average, it would take about 17 years (and when it happened, you weren't there).

If 1 gram of bromine was placed in the gas jar, a back of the envelope calculation would show us that there would be about 1022 molecules.  Thus the probability of there being all the molecules back in the left hand jar is 2 to the power -1022, which is infinitesimally small.

The entropy of a system is the property that tells us the number of possible arrangements of the molecules.  It is given the physics code S, and has the units J K-1.

S = k ln W

The constant k is the Boltzmann Constant:

k = 1.38 × 10-23 J k-1

 If there are 1022 molecules in the two gas jars, calculate the entropy.

The graph shows the even distribution of the molecules.  The curves show the normal distributions for N = 10, 100, and 1000 molecules.  For 10 molecules, there is a reasonable probability that most molecules are on the right.  The more molecules there are, the chances that most molecules are to the right are reduced.

The more molecules, the closer the distribution curve is to the central line.  Therefore with 1022 molecules, the molecules are evenly distributed.

Entropy and Energy

Like molecules, energy spreads out.  At high temperatures, the energy is concentrated.  For a useful job of work to be done, we have to have a high concentration of energy.  High concentrations of energy lead to high temperatures.  In the last tutorial, we saw that:

At high temperatures, molecules move about very quickly and can do a job of work.  A fuel reacts in a boiler or engine to make a very high temperature, so a lot of work can be extracted.  Then the energy becomes more spread out, eventually to heat up the environment.  All heat engines work by extracting mechanical energy from a temperature gradient.  A heat engine has to operate between the hot reservoir and the cold reservoir to satisfy the Second Law of Thermodynamics.  Heat flows from hot to cold, never the other way round.  We can show the heat flowing from a hot reservoir through a heat engine to a cold reservoir.

All heat engines give up their energy to a cold reservoir.  We can define the terms used on the diagram:

• Qin = the heat flow from the hot reservoir to the engine

• Qout is the heat flow from the engine to the cold reservoir.

• The work done by the heat engine is the difference between Qin and Qout.

Therefore:

W = Qin - Qout

Consider an engine in a car.

The fuel delivers energy in its most concentrated form, leading to a very high temperature when it is ignited by the spark plugs (this is a petrol car).  Some energy is extracted as the gases expand, and make the crankshaft turn.  Most of the energy is used to heat up the coolant, which is at a low temperature to maximise the temperature gradient.  This makes the energy more spread out.  Not much energy can be extracted from the coolant (some can be used to heat the car on a cold day).  The coolant needs have its energy extracted from it by the flow of air through the radiator.  The energy gets even more spread out, and is useless.  The exhaust gases are hot as they leave the engine.  They can be harnessed to drive a turbocharger to increase the pressure in the inlet manifold of the engine.  This makes the engine more powerful.  (However the downside to this is that turbocharged engines can be less reliable.)

The engine on this car does not have a turbocharger.  It has a different method of managing the fuel that is to be burned.  All the energy in the exhaust gases is lost to the environment.

Entropy and Cosmology

The entropy at the time of the Big Bang was extremely low.  All the energy in the Universe was concentrated into an infinitesimally small space.  Therefore the temperature was extremely high.  Since the lower the entropy, lower the probability of the event happening.  We saw that with the 1022 molecules in the left-hand gas jar.  Some theoretical physicists have concluded that since the entropy of the Big Bang was so low, the chances of the event happening was infinitesimally small.  But it did happen.  The evidence for the Big Bang is compelling (see Astrophysics Tutorial 7).

The energy from the Big Bang has spread out to give the Cosmic Background Radiation, which gives the Universe an average temperature 2.7 K.  Some places are hotter, like stars.

When all the stars in the Universe eventually die out and cool, astrophysicists believe they will evaporate into protons, electrons, photons, and neutrons.

 Why not neutrons?

Astrophysicists believe that this will happen in 1022 years, and that the temperature of the Universe will be down to 10-30 K.  Not an appealing prospect if you don't like the cold.  However by the time it happens, we will be long since gone and forgotten.

Heat is work and work's a curse,

And all the heat in the Universe

Is gonna cool down.

That will mean no more work,

And there'll be perfect peace.

That's entropy, man!

[Michael Flanders and Donald Swan]

The Third Law of Thermodynamics

Just above, we have seen that the temperature of the Universe will get down to 10-30 K.  It is very nearly absolute zero, but not quite.  The law states:

Absolute zero cannot be reached by a finite steps.

If we cool a body, there has to be an even cooler body to transfer the heat energy to.  Even if the process takes many millions of steps over each of the 1022 years, there is still a finite number of steps to be taken.