Lectures on Thermodynamics and Statistical Mechanics

VP. Nair

CUNY City College of New York

1. Basic Concepts

Thermodynamics is the description of thermal properties of matter in bulk. The study of phenomena involving the transfer of heat energy and allied processes form the subject matter. The fundamental description of the properties of matter in bulk, such as temperature, heat energy, etc., are given by statistical mechanics. For equilibrium states of a system the results of statistical mechanics give us the laws of thermodynamics. These laws were empirically enunciated before the development of statistical mechanics. Taking these laws as axioms, a logical buildup of the subject of thermodynamics is possible.

1.1 Definition

Thermodynamic coordinates

The macroscopically and directly observable quantities for any state of a physical system are the thermodynamic coordinates of that state. As an example, the pressure p and volume V of a gas can be taken as thermodynamic coordinates. In more general situations, other coordinates, such as magnetization and magnetic field, surface tension and area, may be necessary. The point is that the thermodynamic coordinates uniquely characterize the macroscopic state of the system.

Thermal contact

Two bodies are in thermal contact if there can be free flow of heat between the two bodies.

Adiabatic isolation

A body is said to be in adiabatic isolation if there can be no flow of heat energy into the body or out of the body into the environment. In other words, there is no exchange of heat energy with the environment.

Thermodynamic equilibrium

A body is said to be in thermodynamic equilibrium if the thermodynamic coordinates of the body do not change with time.

Two bodies are in thermal equilibrium with each other if on placing them in thermal contact, the thermodynamic coordinates do not change with time.

Quasistatic changes

The thermodynamic coordinates of a physical can change due to any number of reasons, due to compression, magnetization, supply of external heat, work done by the system, etc. A change is said to be quasistatic if the change in going from an initial state of equilibrium to a final state of equilibrium is carried out through a sequence of intermediate states which are all equilibrium states. The expectation is that such quasistatic changes can be achieved by changes which are slow on the time-scale of the molecular interactions.

Since thermodynamics is the description of equilibrium states, the changes considered in thermodynamics are all quasistatic changes.

Work done by a system

It is possible to extract work from a thermodynamic system or work can be done by external agencies on the system, through a series of quasistatic changes. The work done by a system is denoted by . The amount of work done between two equilibrium states of a system will depend on the process connecting them. For example, for the expansion of a gas, the work done by the system is

(1.1)

Exact differentials

Consider a differential form defined on some neighborhood of an n-dimensional manifold which may be written explicitly as

(1.2)

where are functions of the coordinates . is an exact differential form if we can integrate A along any curve C between two points, say, and and the result depends only on the two end-points and is independent of the path C. This means that there exists a function in the neighborhood under consideration such that . A necessary condition for exactness of is

(1.3)

Conversely, if the conditions (1.3) are satisfied, then one can find a function such that in a star-shaped neighborhood of the points and .

The differential forms we encounter in thermodynamics are not necessarily exact. For example, the work done by a system, say, is not a n exact differential. Thus the work done in connecting two states α and , which is given by , will depend on the path, i.e., the process involved in going from state α to state .

1.2 The zeroth law of thermodynamics

This can be stated as follows.

The zeroth law of thermodynamics: If two bodies and are in thermal equilibrium with a third body , then they are in thermal equilibrium with each other.

Consequences of the zeroth law

Thermal equilibrium of two bodies will mean a restrictive relation between the thermodynamic coordinates of the first body and those of the second body. In other words, thermal equilibrium means that

(1.4)

if and are in thermal equilibrium. Thus the zeroth law states that

(1.5)

This is possible if and only if the relations are of the form

(1.6)

This means that, for any body, there exists a function of the thermodynamic coordinates , such that equality of for two bodies implies that the bodies are in thermal equilibrium. The function is not uniquely defined. Any single-valued function of , say, will also satisfy the conditions for equilibrium, since

(1.7)

The function is called the empirical temperature. This is the temperature measured by gas thermometers.

The zeroth law defines the notion of temperature. Once it is defined, we can choose variables as the thermodynamic coordinates of the body, of which only n are independent. The relation is an equation of state.

1.3 Equation of state

The ideal gas equation of state

In specifying the equation of state, we will use the absolute temperature, denoted by . We will introduce this concept later, but for now, we will take it as given. The absolute temperature is always positive, varying from zero (or absolute zero) to infinity. The ideal gas is then characterized by the equation of state

(1.8)

where denotes the number of molecules of the gas and is a constant, known as Boltzmann’s constant. Another way of writing this is as follows. We define the Avogadro number as . This number comes about as follows. The mass of an atom is primarily due to the protons and neutrons in its nucleus. Each proton has a mass of gm, each neutron has a mass of gm. If we neglect the mass difference between the proton and the neutron, the mass of an atom of atomic weight (= number of protons + number of neutrons in the nucleus) is given by gm. Thus if we take grams of the material, the number of atoms is given by. This is essentially the Avogadro number. The mass difference between the proton and neutron is not completely negligible and also there are slight variations from one type of nucleus to another due to the varying binding energies of the protons and neutrons. So we standardize the Avogadro number by defining it as , which is very close to the number of atoms in gm of the isotope of carbon (which was used to standardize the atomic masses).

If we have molecules of a material, we say that it has moles of the material, where . Thus we can rewrite the ideal gas equation of state as molecules of a material, we say that it has 𝓃 moles of the material, where . Thus we can rewrite the ideal gas equation of state as

( 1.9)

Numerically, we have, in joules per kelvin unit of temperature,

(1.10)

The van der Waals equation of state

The ideal gas law is never obtained for real gases. There are intermolecular forces which change the equation of state, not to mention the quantum nature of the dynamics of the molecules which becomes more important at low temperatures. The equation of state can in principle be calculated or determined from the intermolecular forces in statistical mechanics. The corrections to the ideal gas law can be expressed as series of terms known as the virial expansion, the second virial coefficient being the first such correction. While the method is general, the specifics depend on the nature of the molecules and a simple formula is not

Figure 1.1: The general form of the intermolecular potential as a function of distance

easy to write down. An equation of state which captures some very general features of the intermolecular forces was written down by van der Waals, in the 1870s, long before the virial expansion was developed. It is important in that it gives a good working approximation for many gases. The van der Waals equation of state is

(1.11)

The reasoning behind this equation is as follows. In general, intermolecular forces have a short range repulsion, see Fig. 1.1. This prevents the molecules from forming bound states. The formation of bound states would be a chemical reaction, so we are really considering gases where there is no further chemical reaction beyond the initial formation of the molecules. For example, if we consider oxygen, two oxygen atoms bind together to form the oxygen molecule , but there is no binding for two oxygen molecules (two ’s) to form something more complicated. At the potential level, this is due to a short range repulsion keeping them from binding together. In van der Waals’ reasoning, such an effect could be incorporated by arguing that the full volume V is not available to the molecules, a certain volume b around each molecule is excluded from being accessible to other molecules. So we must replace by in the ideal gas law.

Intermolecular forces also have an attractive part at slightly larger separations. This attraction would lead to the molecules coming together, thus reducing the pressure. So the pressure calculated assuming the molecules are noninteracting, which is the kinetic pressure , must be related to the actual pressure p by

(1.12)

The interaction is pairwise primarily, so we expect a factor of (which is number of pairings one can do) for molecules. This goes like , which explains the second term in (1.12). Using the ideal gas law for with the volume , we get (1.11). In this equation, a and are parameters specific to the gas under consideration.