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Chapter 3

Statistical Mechanics

3.1Atoms

The Greek philosopher Democritus (460–370 BCE) proposed that material objects cannot be divided into smaller and smaller pieces ad infinitum, but that a stage would be reached when it would be impossible to divide matter any further. Thus all matter was made of very tiny grains which were so small that they could not be seen. Democritus claimed that these tiny grains could not be divided any further, and so he called such a grain an atom, which in Greek means “indivisible”.

What evidence is there for the existence of atoms?

Democritus argued that odors and aromas are caused by atoms breaking off from the material and flying through the air and eventually entering our nostrils. This explanation is valid even today, though we would use the word “molecule” rather than atom to describe the smallest particles of the substance that we are smelling.

The first visible proof of Democritus’ hypothesis of these atoms was the observation of the random erratic motion of pollen grains and saw dust grains suspended in water when seen through a powerful microscope. This motion is called Brownian motion, after the botanist R. Brown who reported his observation in 1827.¹ The pollen grains were seen dancing about in a zigzag haphazard manner. The explanation is that the tiny invisible particles of water were hitting the pollen grains from different directions at different speeds, and the pollen grains were reacting to these collisions by moving in random directions. Today we know that the smallest particles of water are not really atoms, but molecules. Brownian motion provided the first visible proof that matter is made of small indivisible particles.

The atomic theory of matter is able to explain the observations of chemistry. Chemistry investigates reactions whereby elements combine with other elements to form compounds, and compounds react with other elements or compounds to produce other substances, both elements and compounds. Around 1880 the chemist John Dalton found that when elements combine to form compounds they do so in fixed proportions by weight. As an example, 1 gram of hydrogen combines with 8 grams of oxygen to form 9 grams of water, and 2 grams of hydrogen combine with 16 grams of oxygen to form 18 grams of water. This seems to suggest that substances are made of atoms which have fixed masses, because this observation would make sense if each substance was made of small microscopic particles all having the same mass and that the masses of these fundamental particles varied from one substance to another. So Dalton’s observation offered a good indirect proof for the existence of atoms.

Another major victory scored by the atomic theory is that it has been able to explain our most basic experiences of heat and cold. Temperature is something everybody experiences. The application of heat to effect useful changes — particularly for cooking food — had been recognized since the discovery of fire. The industrial revolution took off when heat was recognized as a form of energy that could be harnessed to do useful work such as in the steam engine and the automobile engine. The study of the relationship between heat and mechanical energy is called thermodynamics. Physicists made considerable progress in discovering important laws of thermodynamics before they understood the nature of heat. The atomic theory finally explained that heat is nothing but the sum total of the energies of the individual atoms or molecules of a material object — solid, liquid or gas. Temperature is simply a measure of the average kinetic energy of the microscopic atoms or molecules of an object.

We shall do a quick review of the laws of thermodynamics and show how these laws make sense in the light of the atomic theory of matter.

3.2The laws of thermodynamics

The word “thermodynamics” is a combination of “thermo” meaning heat and “dynamics” meaning the effects of forces.

So in the study of thermodynamics we are interested in the interplay between heat and work. Let us recall that we learned earlier (Ch. 2) that work is done when a force causes a body to undergo a displacement.

When a dynamo is made to rotate swiftly, it produces electricity. One way of causing a dynamo to rotate is to attach paddles to it and place it under a waterfall. The falling water has mechanical energy due to the speed with which it falls from the upper level. We say that the water has potential energy by virtue of its elevation above the ground and that this potential energy is converted to kinetic energy as the water gathers speed in its descent. Both potential and kinetic energies are forms of mechanical energy. So in this example — known as hydroelectricity — the mechanical energy of the water is converted to mechanical energy of the rotating dynamo which in turn generates the electricity. Thus mechanical energy is converted to electrical energy.

If a waterfall is not available, but there is fuel available for burning, such as coal, gasoline or nuclear fuel, then these fuels can be burned and heat energy obtained. This heat energy can be used to drive a dynamo which would then produce electricity. So in this example — known as thermoelectricity — heat energy from the burning fuel is converted to mechanical energy in the rotating dynamo which then generates the electricity.

The obvious disadvantage of thermoelectricity is that the fuels will eventually be exhausted at some time in the future. There is also another disadvantage, and that is, in converting heat into work there is always a waste of heat energy. Not all the available heat can be converted into useful energy. And this follows from the laws of thermodynamics, which, as we shall see presently, follow from the atomic or particle nature of matter. We now turn to these laws.

The study of thermodynamics is classified under four laws: the zeroth law, the first law, the second law and the third law.

Brief explanations of these laws are laid out below:

Zeroth Law:

The zeroth law states that if two bodies are separately in thermal equilibrium with a third body, then they must be in thermal equilibrium with one another. By thermal equilibrium we mean simply that no heat flows between the two bodies. This law enables us to define the concept of temperature. Two bodies that are in thermal equilibrium with one another have the same temperature. So if body A and body B have the same temperature, and body A and body C have the same temperature, then by the zeroth law, body B and body C must have the same temperature. This means we can define temperature as an absolute quantity, regardless of the nature of the body that has the temperature. So thermometers can be built to measure the temperatures of objects having compositions totally different from that of the thermometer. We express temperature as a number. This number is commonly written using one of two common scales, Celsius and Fahrenheit. These scales are related by the formula

According to this formula 100° C = 212° F, which is the temperature at which water boils under normal atmospheric pressure.

Physicists who worked on the behavior of gases as they expanded with increase of temperature noticed that there seemed to be a lowest possible temperature which is −273.15° C. This is called Absolute Zero. One could think of Absolute Zero as the temperature of a body after all its heat has been taken away. So there is a lower limit to temperature. Since it is impossible to get colder than Absolute Zero, it makes sense to define a scale of temperature in which 0 corresponds to Absolute Zero. Such a scale is called the Kelvin or Absolute Scale.²

Exercise 3.1.

Which is colder: (a) 0° C or 0° F? (b) −20° C or −20° F? (c) −40° C or −40° F? (d) −60 ° C or −60° F?

First Law:

The first law is a statement of the conservation of energy. It states that when heat energy is given to a body, part of it goes to raise the temperature, and hence the internal energy of the body increases, and the rest goes to do external work, which is done by the expansion of the body. Since this is a law of conservation of energy, it can also be applied to a case when work is done on a gas by compressing it. In this case the gas would get heated up, and may give up some of its heat to the surroundings. Here also there is a balance of energy. No energy is created and no energy is destroyed.

If the amount of heat energy supplied to a body is ΔQ, the rise of internal energy of the body ΔU, and the external work done by the body ΔW, then the first law can be expressed as

Second Law:

The second law is a statement of irreversibility. It can be stated in many different ways. The simplest statement is that heat always flows naturally from a hotter to a cooler body.

Both the first and the second laws deal with the conversion of heat into work and work into heat. The first law tells us that heat and work are different forms of energy and that one can be converted into the other. The second law places restrictions on the conversion of heat energy into work. Heat and work (mechanical energy) are not reversible. Whereas mechanical energy can be converted entirely into heat energy, the reverse cannot take place. For example, when a meteor falls through the atmosphere, it has a large kinetic energy because of its speed, and this kinetic energy is entirely converted to heat energy as the meteor burns up in the atmosphere. When brakes are applied to a moving car, the kinetic energy of the car is entirely converted to heat energy in the wheels and the road. But the heat energy that is so generated in either of these examples cannot be converted back to kinetic energy.

Third Law:

The third law states that it is impossible to cool a body right down to absolute zero (0 K), even though it is theoretically possible to come closer to this temperature with each attempt.

All the laws of thermodynamics can be fully explained on the atomic or molecular theory of matter. Heat is a manifestation of the energies of the molecules of a body — the sum of all the kinetic and potential energies of all the molecules. Here the forces that give rise to the potential energy are not due to gravity but due to attraction and repulsion between molecules. The phenomenon of Brownian motion showed that the motion of the molecules in a liquid is erratic and random. The molecules move in all possible directions with a range of velocities that change in magnitude and direction each time a molecule collides with another or with the molecules of the walls of the container. Thus it is futile to try and follow the movements of any one molecule. The best we can do is to investigate the overall aggregate or statistical behavior of these molecules. The study of the behavior of matter in terms of the collective motion of the molecules is therefore called statistical mechanics.

3.3Statistical mechanics

Every substance — whether an element or a compound — is made up of molecules. Each molecule contains one or more atoms. The molecule of an element contains one or more atoms of the same kind. A molecule of a compound contains two or more atoms, which are not all of the same kind. Sulfuric acid has the formula H₂SO₄, which means a molecule of sulfuric acid contains two hydrogen atoms, one sulfur atom and four oxygen atoms. Because atoms — and molecules — are so small, there is a very large number of these particles in any observable piece of matter. A measure of this large number is Avogadro’s number N_A = number of molecules present in 1 mole of any element or compound. A mole is the molecular weight expressed in grams. 1 mole of hydrogen gas (H₂) has a mass of 2 grams. N_A = 6.02 × 10²³. So 1 gram of hydrogen gas (H₂) contains about 3.01 × 10²³ molecules. This means 1 hydrogen atom has a mass of about 1.7 × 10⁻²⁴ grams. This is a very small quantity. 16 grams of oxygen gas (O₂) contain about 3.01 × 10²³ molecules. Because of the very large number of molecules present in this small mass of oxygen, we need statistical mechanics to provide a reliable description of the observable behavior of the gas.

Exercise 3.2.

(a) The formula for water is H₂O. How many molecules are there in 1 gram of water?

(b) The molecular weight of a substance is the sum of the atomic weights of the atoms in a molecule of the substance. Given the following atomic weights: H = 1, O = 16, S = 32, find the molecular weight of sulfuric acid H₂SO₄.

The temperature of a body is a measure of the average kinetic energy of the molecules of the body. If two bodies are in thermal communion with each other, there will be a transfer of kinetic energy from the molecules of one body to the molecules of the other body, until both bodies have the same average kinetic energy of their molecules. This means they will have the same temperature. This is the explanation for the zeroth law. This also explains the second law, as we shall see further on.

The temperature of a gas is proportional to the average kinetic energy of its molecules. Consider a gas contained in a cylinder enclosed by a piston. If some heat is supplied to the gas, its temperature will increase, and so the molecules will have greater kinetic energy. This greater kinetic energy will mean that the molecules will pound on the piston with greater force, causing the piston to move outwards. So the gas expands and the force of this expanding gas does work on the piston. This is an illustration of the first law.

3.3.1One-dimensional gas

A helium molecule which consists of a single atom can be thought of as a rigid sphere. Diatomic molecules such as hydrogen and oxygen have a dumbbell shape. Molecules with three or more atoms have more complex shapes. Let us now limit our discussion to the simplest kind of gas — one consisting of identical monatomic molecules such as Helium, Argon or Neon. Each of these molecules can be modeled as a tiny rigid sphere.

Suppose all these rigid spheres were lined up along their line of centers, i.e. like a one-dimensional array of identical billiard balls.

And let us say that this array of balls is suspended within a zero gravity box somewhere in outer space. If the sphere A at the far left were set in motion towards the right, it would collide with the next one, which in turn would collide with the sphere next to it, and so on till the last sphere E moves forwards, hits the wall of the container, bounces back, hits the previous ball D, which in turn hits the ball C behind it, and so on until the ball A moves to the left wall, bounces back, hits the ball B, and the process continues indefinitely. As long as all the collisions are elastic, i.e. with no loss of kinetic energy, the process will be repeated forever. Moreover, the process is also time reversible. If we were to record the motion of the balls for a period of time and play the film backwards it would be impossible to find any essential difference between the forward time and backward time sequences of motion. What we have just described is a model of a one-dimensional gas, which of course does not exist in nature. But what is important for our purposes is that a one-dimensional gas in a gravity-free environment is a time reversible system.

The educational toy called “Newton’s Cradle” is an approximate illustration of this process. A typical Newton’s Cradle has about five identical metal balls suspended by strings from a horizontal support. The strings all have the same length and they are spaced apart in a straight line so that when the apparatus is stationary all the balls hang vertically at equal distances from their neighbors. If now the ball at one end is pulled away from the rest and released, it would fly like a pendulum and hit the next ball. There is a total transfer of momentum from the first ball to the second with the result that the first ball becomes stationary. The momentum is then communicated from the second to the third to the fourth to the fifth. The fifth ball flies away from the remaining balls which are now all stationary. The fifth ball makes a pendulum-like swing and returns and hits the fourth ball, which communicates the momentum to the neighboring ball and so on all the way to the first ball which now pulls away from the others and moves to the left like a pendulum before swinging back and hitting the second ball and so the sequence of movements is repeated all over again. There is loss of energy at each collision, as kinetic energy is converted to sound and heat energy, and there is air resistance. These factors will cause the process to slow down and eventually stop. An ideal Newton’s Cradle with zero energy dissipation would be a model for a one-dimensional gas.

A degree of freedom is a particular way in which a molecule is free to move. And because a molecule in this scenario can execute only one kind of motion, which is to move along a straight line, such a molecule has a single degree of freedom.

3.3.2Two-dimensional gas

Next we consider a two-dimensional gas. Again we consider a container in a gravity-free environment. Here the balls are floating at different points but their centers are all in the same plane. This time, the spheres are not all aligned in straight lines. Now, if one ball were given a push in any direction, it would hit another, which would hit another, and so on, but these collisions would not necessarily be head on collisions.

These collisions would be random. Eventually, the balls would be moving haphazardly in all directions, while remaining in the same twodimensional plane. But the kinetic energy of the balls has now been distributed evenly along two dimensions. The random statistical nature of the motion ensures that the average kinetic energy due to the motion in any one direction equals the average kinetic energy due to motion in any other direction. Every two-dimensional motion can be resolved into motion in two mutually perpendicular directions, and we call each such perpendicular direction a degree of freedom. So a monatomic molecule that is capable of moving in two dimensions has two degrees of freedom. Each degree of freedom has the same average kinetic energy. The average energy per molecule has been divided equally between its two degrees of freedom. Clearly, the dynamics of this two-dimensional gas are not time reversible. In forward time the energy gets distributed evenly between the two degrees of freedom. One does not observe the reverse happening in nature. One does not see a collection of billiard balls initially moving randomly gradually changing their motion until all the balls are moving in one direction only. Thus a two-dimensional gas is an irreversible system.

3.3.3Three-dimensional gas

Let us next picture a physically real cubic meter of helium gas inside a cube of sides 1 meter in a laboratory on earth. What are the molecules of helium doing? They are not stationary. If they were, they would all be lying at the bottom of the container like a lot of microscopic apples in a largely empty crate. But these molecules are constantly moving. They move fast like tiny bullets and so gravity does not play a perceptible role in their motion. As they move inside the cube they collide with the walls of the cube and change direction. They also collide with one another. With each collision, the molecules abruptly change velocity and exchange kinetic energy with each other, and with the molecules on the walls of the container. Since the collisions are entirely haphazard, at every collision each molecule undergoes a random change of momentum and energy. With about 10²⁴ molecules to deal with it is impossible to follow the motions of all of them within the gas. The energy and velocity of each molecule change constantly from collision to collision.

Because these molecules are constantly exchanging energy with each other, it makes better sense to talk about averages than actual values when dealing with such large numbers. When the molecules are moving in the most random fashion, the kinetic energy gets distributed equally among the three degrees of freedom of the three-dimensional gas. This is an example of the Principle of Equipartition of Energy. This Principle is simply a statistical consequence of the random nature of the motion of a very large number of particles which are constantly exchanging kinetic energy and momentum as they collide with one another and with the walls of the container.

3.3.4Third law of thermodynamics

The third law can be explained by an analogy. Suppose a moving sphere A collides head-on with an identical stationary sphere B. The first sphere will stop, and the second sphere will move with the same velocity possessed by the first sphere before the collision. The first sphere will not stop if the second had any velocity at all. Suppose A represents a gas that we are trying to cool to absolute zero. At absolute zero the kinetic energy of the molecules is zero. So if we want to reduce the temperature of a gas to zero, we must place it in contact with a gas that is already at absolute zero, with stationary molecules. Unless we can find such a gas somewhere in the universe — which is impossible, considering that the universe is constantly cooling down from a very hot initial state — it is impossible to reduce the temperature of any gas to absolute zero.

3.3.5Second law of thermodynamics

Suppose we have a thermally insulated container of gas with two compartments, one having gas A initially at 150° C and the other having gas B initially at 50° C and these compartments are separated by a wall that permits heat to flow through it. Since heat is due to the kinetic energies of the molecules of the gas, the average kinetic energy of the molecules of A is greater than the average kinetic energy of the molecules of B. Due to the randomness of the collisions of the gas molecules with the molecules of the wall that separates them, the molecules of A will gradually impart some kinetic energy to the molecules of the wall, which in turn will pass on the kinetic energy to the molecules of B. The result is that the molecules of A will gradually slow down, and the molecules of B will gradually speed up. This transfer of kinetic energy will continue until both compartments have the same average kinetic energy. Thus both the gases will have the same temperature, and heat has flowed from the hotter body A to the cooler body B. It is impossible for the reverse to happen. And thus the Second Law of Thermodynamics follows from the fact that any piece of matter is made of a very large number of particles in random motion.

Because heat cannot flow from a cold body to a hot body by itself, the Second Law of Thermodynamics provides a unique arrow of time. Suppose an ice cube were placed in a glass of warm water. A video recording will show the ice melting as it receives heat from the water. If the video were played backwards it would show a tiny piece of ice gradually becoming bigger until it acquired the shape of a cube floating on the warm water. It is evident that this sort of time reversal cannot occur in nature. The flow of time is like the flow of heat. It cannot be reversed. As we saw earlier in this chapter, when we have a large number of microscopic particles, no matter how orderly they are arranged in the beginning, once the system is set in motion, the random collisions will create a disorder from which the original order can never be retrieved. This has some very important consequences.

One consequence is the diminishing of available energy. An array of molecules all moving together can apply a concerted force which can therefore do a lot of work on an object and impart a corresponding energy to the object. But if the molecules are moving haphazardly, the force they can exert together is considerably less, and so the amount of energy that can be provided is less. Thus, in an irreversible process the amount of available energy decreases. So the Second Law can also be stated as: natural processes always take place in such a way that the amount of available energy decreases.

Another consequence is the collapse of orderliness. An array of molecules all moving with the same velocity parallel to each other is a highly orderly system. But as the system is left to itself, the degree of orderliness will gradually diminish until there is total randomness. So the Second Law can also be stated thus: natural processes will always take place in such a way that there is a loss of order. Disorderliness is also called entropy. So another formulation of the Second Law: Natural processes occur in such a way that the entropy increases. Yet another consequence is the loss of information. We could create different arrays of molecules which are all orderly, but not identical with each other. Let us say we have two boxes with the same number of molecules all moving parallel to each other. In one box we divide the molecules into two parallel arrays with a gap between them. In the second box we have the same number of molecules, all parallel to each other, but without a gap. We could label the first box 0 and the second box 1. The distinction between the two boxes allows us to store information. The simplest information is binary — yes or no. We could agree that 0 means yes and 1 means no, or vice versa. Now, suppose we allow both the boxes to stand for a while. After some time all the molecules in both boxes will be moving at random, and the gap between the molecules in the first box will vanish. And so the distinction between the two boxes has disappeared. We can no longer tell which is 0 and which is 1. The information is lost. So the Second Law can be stated thus: Natural processes tend to destroy information.

The Second Law explains our consciousness of the flow of time. Momentary experiences are instantly converted to memories which are constantly being stored in our brains. We remember the past but not the future because the past has left an imprint in our memories, somewhat like the sedimentary layers under the soil studied by archaeologists. As we acquire more knowledge, more information is stored in our brains. At first sight this appears to go against the Second Law, but it is not hard to see that the Second Law is not violated. The increase of information stored in the brain is accompanied by the loss of biological information which had been stored in the food that we digested and converted to energy. So overall, as human knowledge increases, it does so at the expense of the information that exists outside of our bodies. The storing of information in computer memories also requires energy which is ultimately obtained by the loss of information in the fuels that produce the energy. The constant supply of energy maintains the increase of information and order within our bodies, but even this process is not unending. The Second Law is also responsible for biological aging and the gradual erosion of our memories with time. The body is programmed to generate order and to decrease entropy through the intake of food and oxygen. But eventually the body will give up the fight against the tendency to greater disorder and higher entropy. Biological death is a consequence of the Second Law.

The most important thing to learn about the Second Law is that it is a statistically based law. Its validity rests on the very large number of atoms or molecules that make up a normal mass of matter. Maxwell — one of the pioneers of Statistical Mechanics — boldly declared: “The true logic of this world is in the calculus of probabilities.”³ Maxwell showed that Newton’s Laws do not forbid heat from flowing from a cooler body to a warmer body, but he pointed out that the probability of this occurring is microscopically small.⁴ So a law so fundamental as the Second Law of Thermodynamics is rooted in probability. This seemed an outright affront to the determinism that was engendered by Newton’s Laws, and many philosophers and physicists resisted this statistical explanation of a basic physical law. But eventually all objections were overcome with the establishment of the atomic or molecular structure of matter.

Maxwell’s dictum about the true logic of this world ultimately won the day in the triumph of quantum theory, of which he himself knew nothing. Maxwell died at the age of 48 in 1879, twenty one years before the birth of quantum theory. In quantum theory, it is probabilities that dictate the outcome of any process. The only real prediction we can make is the probability of observing one or another outcome. Maxwell’s prophetic insight into microscopic phenomena made it easier for scientists to embrace the counterintuitive notions of quantum theory.

3.4Summary

All matter is composed of indivisible particles called atoms. Atoms tend to combine with other atoms to form molecules. (Some molecules such as Helium and Argon consist of a single atom). The most important visible proof of the existence of such microscopic molecules is Brownian motion, the erratic dancing motion of pollen grains suspended in water. The laws of the combination of elements according to proportions of weight also reveal the atomic structure of matter.

The atomic structure of matter is especially discernible in the experience of heat and temperature. Heat is the energy possessed by the molecules of an object. The temperature of an object is proportional to the average kinetic energy of the molecules. Because there are a very large number of molecules in any normal quantity of a material, it is impossible to study the motions of all the molecules with precision. So we study their statistical behavior. This branch of physics is called statistical mechanics.

The study of the relationship between heat and mechanical energy is called thermodynamics. There are four important laws of thermodynamics:

a) the zeroth law which deals with objects in thermal equilibrium with one another,

b) the first law which states that heat can be converted to work and vice versa and there is no loss or gain of energy in this process,

c) the second law which states that heat can flow spontaneously only from a hotter object to a cooler object, and

d) the third law which states that it is impossible to cool down any object to absolute zero temperature in any finite number of steps.

The Second Law of thermodynamics is responsible for our subjective perception of time and memory. Time only flows forwards and never backwards. We remember the past but not the future. A fruit, an animal or a human being can only become older, never younger, as time progresses. Natural processes occur in such a way that there is a decrease in the total available energy in the universe. They also occur in such a way that information tends to get erased. We say that only those processes occur in nature that increase the total entropy of the universe.

¹For the record, though, this phenomenon had been observed and reported earlier by Jan Ingen-Housz in 1784, but the report was so brief that it escaped the notice of subsequent researchers. Though unfortunate for the memory of Ingen-Housz, this historical lapse is perhaps to the advantage of English speakers, since it is easier to say “Brownian motion” than “Ingen-Houszian motion”!

²The lowest temperature on this scale is 0 K, the melting point of ice is 273.15 K and the boiling point of water is 373.15 K. Notice that we do not put the degree symbol ° in the Kelvin scale.

³Lewis Campbell and William Garnett, The Life of James Clerk Maxwell: With Selections from his Correspondence and Occasional Writings (London: Macmillan, 1884) p. 97.

⁴In every gas the molecules are in random motion, which means not only the directions of their velocities but also the magnitudes of the velocities are random. In a hotter gas the average speed of the molecules is greater than in a cooler gas. But within each gas the molecules have a range of speeds. So the slowest molecule in a hotter gas could be much slower than the fastest molecule in a cooler gas. Maxwell devised a famous thought experiment in which a microscopic agent — called a demon — would allow the faster molecules from a cooler gas to pass into a warmer gas, and the slower molecules from the warmer gas to pass into the cooler gas. The result is that the average speed of the warmer gas increases, which means it gets hotter, and the cooler gas gets colder. Thus heat has been made to flow from a colder body to a hotter body without violating Newton’s laws of motion. But this inference is valid only if we assume that the demon itself is not subject to the laws of physics. A material demon would get bombarded by the molecules so that it would itself execute a random motion making it totally ineffective. Statistical mechanics would prevail and heat would flow from hot to cold.