Basic Physics Of Quantum Theory, The. Basil S Davis
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