Physics 53Thermal Physics 1Statistics are like a bikini. What they reveal is suggestive; what they conceal is vital.— Arthur KoestlerOverviewIn the following sections we will treat macroscopic systems that are in thermal equilibrium. Most of our discussion will center on the behavior of gases. We will use appropriate macroscopic variables, some of which are already familiar, such as density, volume and pressure. We will usually assume the CM of the system is at rest, and that molecular velocities are randomly distributed as to direction, so there is no bulk flow or velocity field to consider. There may be energy (kinetic or potential or both) internal to the molecules themselves, if they are not monatomic.The total kinetic energy of the random molecular motion, plus any energy the molecules may have internal to themselves, constitutes the internal energy of the system. This is one of the new macroscopic variables we will use in our description.Thermal equilibrium, temperature and the Zeroth LawMacroscopic variables used to describe the state of a system in thermal equilibrium are called state variables. Density, volume and pressure are state variables, as is internal energy. We will introduce two more as we go along: temperature and entropy.What is meant by thermal equilibrium is this:If the system is in thermal equilibrium, state variables all have definite values.Since the state variables all represent averages of some kind, what it means to have a “definite value” is that the fluctuations which might make the average uncertain are small enough to be ignored. In a statistical situation, such fluctuations are generally inversely proportional to the number of particles. In systems of macroscopic size that number (roughly Avogadro’s number) is extremely large, so the fluctuations are tiny. Most of the possible situations a system may be in are not states of thermal equilibrium. Nevertheless, it is an experimental fact that a system left to itself evolves toward thermal equilibrium. (How that happens is still not fully understood in detail.) The PHY 53! 1! Thermal Physics 1time it takes for a non-equilibrium state to evolve to equilibrium is called the “relaxation time”. It varies from case to case, but is often quite short by human standards.Two macroscopic systems can exchange energy with each other. This exchange can be accomplished by collisions among particles at a common interface, or by other methods such as emission and absorption of electromagnetic radiation. If energy exchanges can take place, we say the systems are in thermal contact. Such exchanges generally disrupt, at least temporarily, the states of equilibrium of the two systems. It may happen, however, that two systems brought into thermal contact undergo no disruption of equilibrium in either system. Then we say the systems are in equilibrium with each other.The properties of this “mutual thermal equilibrium” constitute a fundamental law:Zeroth Law of thermodynamicsIf systems A and B are both in thermal equilibrium with system C, then they are also in equilibrium with each other.This principle is the basis for the state variable called temperature. Two systems in equilibrium with each other are said to have the same temperature. If two systems with different temperatures are brought into thermal contact, there is exchange of energy until they come to mutual equilibrium at the same (new) temperature.Temperature is a scalar quantity. It is defined operationally by its measurement, using a device called a thermometer. When this (small) device is placed in thermal contact with the (large) system in question, some energy is exchanged between the two. (Because the thermometer is small, this energy exchange does not change appreciably the state of the large system.) Some physical aspect of the thermometer (for example, the length of a column of liquid) changes in an observable way on account of the energy exchange; when this aspect stops changing we know the system and the thermometer are in equilibrium. By calibrating the changes in the thermometer one makes a scale of temperatures. The first thermometer and scale was made by Fahrenheit around 1700.How is temperature related to average behavior of the microscopic states? We will find that temperature is one measure of the average energy of (random) motion of the molecules in the system.The ideal gas lawFor a gas the most commonly used state variables are pressure (P), volume (V) and temperature (T). For reasons to be made clear later, the temperature of a gas is usually specified using a scale with the lowest possible temperature chosen as zero. The most popular of these scales is that due to Lord Kelvin (William Thomson), which has 100 PHY 53! 2! Thermal Physics 1units (“Kelvins”, denoted by K) between the ice point and the steam point of water. In this scale the “absolute zero” of temperature is approximately 273 K below the ice point.Pressure, volume and temperature are not all independent of each other. An equation relating them is called an equation of state. It is generally written in the form T = f P,V( ). Much research has been done to find the function f (P,V) appropriate to particular gases. In the late 17th century it was found that for a wide variety of gases, at conditions of pressure and temperature easily obtainable in the technology of that time, there is a simple and universal function that describes them all. It is f (P,V) = (const) ⋅ PV. This formula is called the ideal gas law.In SI units, it is usually written in either of two equivalent forms:Ideal gas law PV = NkTIdeal gas law PV = nRTIn the first form N is the number of molecules in the gas and k is a universal constant, called Boltzmann's constant, which in SI units is k = 1.38 × 10−23. The second form specifies the quantity of gas in terms of the number of moles n, and R is the “universal gas constant”. R = 8.31 in SI units. The two forms are related through Avagadro’s number NA, the number of molecules per mole. We have N = nNA and so R = NAk. Experimentally NA≈ 6.02 × 1023. The ideal gas law is an approximation that gives good results for gases that are not very dense, at temperatures and pressures near those in which we live.Internal energyThe internal energy of a system, denoted by Eint, consists of the total energy of random motion of the particles, plus any energy associated with internal structure of the particles, such as rotation