The Zeroth Law of thermodynamics

TABLE 1.1 Common state variables and their units 

Pressure P Atmosphere, atm (= 1,01325 bar) Torricelli, torr (= atm) Pascal (SI unit) Pascal, Pa (= ioo]ooo bar) Millimeters of mercury, mmHg (= 1 torr) 

Volume V Cubic meter, m (SI unit) Liter, L (= m ) Milliliter, mL (= L) Cubic centimeter, cm (= 1 mL) 

Amount n Moles (can be converted to grams using molecular weight) 

Thermodynamics is based on a few statements called laws that have broad application to physical and chemical systems. As simple as these laws are, it took many years of observation and experimentation before they were formulated and recognized as scientific laws. Three such statements that we will eventually discuss are the first, second, and third laws of thermodynamics. 

The law of thermal equilibrium, the zeroth law of thermodynamics, is another important principle. The importance of this law to the temperature concept was not fully realized until after the other parts of thermodynamics had reached a rather advanced state of development hence the unusual name, zeroth law. 

To illustrate the zeroth law we consider two samples of gas. One sample is confined in a volume F, the other in a volume V2. The pressures are pi and P2, respectively. At the beginning the two systems are isolated from each other and are in complete equilibrium. The volume of each container is fixed, and we imagine that each has a pressure gauge, as shown in Fig. 6.1(a). 

Consider three systems A, B, and C arranged as in Fig. 6.2(a). Systems A and B are in thermal contact, and systems B and C are in thermal contact. This composite system is allowed sufficient time to come to thermal equilibrium. Then A is in thermal equilibrium with B, and C is in thermal equilibrium with B. Now we remove A and C from their contact with B and place them in thermal contact with each other (Fig. 6.2b). We then observe that no changes in the properties of A and C occur with time. Therefore A and C are in thermal equilibrium with each other. This experience is summed up in the zeroth law of thermodynamics Two systems that are both in thermal equilibrium with a third system are in thermal equilibrium with each other. 

The temperature concept can be stated precisely by (1) Systems in thermal equilibrium with each other have the same temperature and (2) systems not in thermal equilibrium with each other have different temperatures. The zeroth law therefore gives us an operational definition of temperature that does not depend on the physiological sensation of hotness or coldness. This definition is in agreement with the physiological one. 

The argument does not depend in the least on whether gases, real or ideal, or liquids or solids were chosen. 

---

I mentioned temperature at the end of the last chapter. The concept of temperature has a great deal to do with thermodynamics, and at first sight very little to do with microscopic systems such as atoms or molecules. The Zeroth Law of Thermodynamics states that Tf system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is also in thermal equilibrium with system C . This statement indicates the existence of a property that is common to systems in thermal equilibrium, irrespective of their nature or composition. The property is referred to as the temperature of the system. 

The Zeroth Law of Thermodynamics An extension of the principle of thermal equilibrium is known as the zeroth law of thermodynamics, which states that two systems in thermal equilibrium with a third system are in equilibrium with each other. In other words, if 7), T2, and 77, are the temperatures of three systems, with 7j - T2 and T2 = T2, then 7j = 7Y This statement, which seems almost trivial, serves as the basis of all temperature measurement. Thermometers, which are used to measure temperature, measure their own temperature. We are justified in saying that the temperature T3 of a thermometer is the same as the temperature 7j of a system if the thermometer and system are in thermal equilibrium. 

The zeroth law of thermodynamics says imagine three bodies, A, B and C. If A and B are in thermal equilibrium, and B and C are also in thermal equilibrium, then A and C will be in thermal equilibrium. 

Incidentally, this argument also explains why the mouth feels cold after the ice has melted, since the energy necessary to melt the ice comes entirely from the mouth. In consequence, the mouth has less energy after the melting than before this statement is wholly in accord with the zeroth law of thermodynamics, since heat energy travels from the hot mouth to the cold ice. Furthermore, if the mouth is considered as an adiabatic chamber (see p. 89), then the only way for the energy to be found for melting is for the temperature of the mouth to fall. 

We can predict whether an ice cube will melt just by looking carefully at the phase diagram. As an example, suppose we take an ice cube from a freezer at — 5 °C and put it straightaway in our mouth at a temperature of 37 °C (see the inset to Figure 5.1). The temperature of the ice cube is initially cooler than that of the mouth. The ice cube, therefore, will warm up as a consequence of the zeroth law of thermodynamics (see p. 8) until it reaches the temperature of the mouth. Only then will it attain equilibrium. But, as the temperature of the ice cube rises, it crosses the phase boundary, as represented by the bold horizontal arrow, and undergoes a phase transition from solid to liquid. 

Thermodynamics is a science in which the storage, transformation, and transfer of energy E and entropy S are studied. Thermodynamics is governed by four basic laws called (1) the zeroth law of thermodynamics, (2) the first law of thermodynamics, (3) the second law of thermodynamics, and (4) the third law of thermodynamics. 

Given this simple concept of thermal sameness or equilibrium, we can express the results of universal human observations in the following Inductive Law 3, also known as the zeroth law of thermodynamics ... 

When a hot body and a cold body are brought into physical contact, they lend to achieve the same warmth after a long lime. These two bodies are then said to be at thermal equilibrium with each other. The zeroth law of thermodynamics (R.H. Fowler) states that two bodies individually at equilibrium with a third are at equilibrium with each other. This led lo the comparison of the states of thermal equilibrium of two bodies in lei ms ol a third body called a thermometer. The temperature scale is a measure of state or thermal equilibrium, and tw-o systems at thermal equilibrium must have the same temperature. 

TEMPERATURE SCALES AND STANDARDS. That property of systems which determines whether they are in thermodynamic equilibrium. Two systems are in equilibrium when their temperatures (measured on die same temperature scale) are equal, The existence of the property defined as temperature is a consequence of the zeroth law of thermodynamics. The zerodi law of thermodynamics leads to the conclusion that in the case of all systems there exist functions of their independent properties j , such dial at equilibrium... 

The fundamental meaning of temperature may be described in terms of the zeroth law of thermodynamics. This states that when two bodies are each in thermal equilibrium with a third body they are then in thermal equilibrium with each other, i.e. 

The zeroth law of thermodynamics is in essence the basis of all thermometric measurements. It states that, if a body A has the same temperature as the bodies B and C, then the temperature of B and C must be the same. One way of doing this is to calibrate a given thermometer against a standard thermometer. The given thermometer may then be used to determine the temperature of some system of interest. The conclusion is made that the temperature of the system of interest is the same as that of any other system with the same reading as the standard thermometer. Since a thermometer in effect measures only its own temperature, great care must be used in assuring thermal equilibrium between the thermometer and the system to be measured. 

Heat capacities can be defined for processes that occur under conditions other than constant volume or constant temperature. For example, we could define a heat capacity at constant length of a sample. However, regardless of the nature of the process, the heat capacity will always be positive. This is ensured by the zeroth law of thermodynamics, which requires that as positive heat is transferred from a heat reservoir to a colder body, the temperature of the body will rise toward that of the reservoir in approaching the state of thermal equilibrium, regardless of the constraints of the heat-transfer process. 

This leads us to the Zeroth Law of Thermodynamics (which is really a statement of experience) which states that if we have an adiabatically insulated system (q = 0) in which a body A is in thermal equilibrium with a body B and B is, in turn, in thermal equilibrium with a third body C then A will be in equilibrium with C even though it may not be in direct contact. 

The zeroth law of thermodynamics involves some simple definition of thermodynamic equilibrium. Thermodynamic equilibrium leads to the large-scale definition of temperature, as opposed to the small-scale definition related to the kinetic energy of the molecules. The first law of thermodynamics relates the various forms of kinetic and potential energy in a system to the work which a system can perform and to the transfer of heat. This law is sometimes taken as the definition of internal energy, and introduces an additional state variable, enthalpy. 

Two systems in thermal contact eventually arrive at a state of thermal equilibrium. Temperature, as a universal function of the state and the internal energy, uniquely defines the thermal equilibrium. If system 1 is in equilibrium with system 2, and if system 2 is in equilibrium with system 3, then system 1 is in equilibrium with system 3. This is called the zeroth law of thermodynamics and implies the construction of a universal temperature scale (stated first by Joseph Black in the eighteenth century, and named much later by Guggenheim). If a system is in thermal equilibrium, it is assumed that the energy is distributed uniquely over the volume. Once the energy of the system increases, the temperature of the system also increases (dU/dT> 0). 

The Zeroth Law of Thermodynamics is based on a statement which emphasizes the transitive property that two bodies in equilibrium with a third are in equilibrium with each other. Here we restrict ourselves to the case where the mechanical variables, namely pressure P and volume V, suffice to describe a system at equilibrium however, this approach can easily be generalized. 

The Zeroth Law of Thermodynamics asserts that if two bodies are in equilibrium with a third they are in equilibrium with each other. 

A fundamental attribute of temperature is that for any body in a state of equilibrium the temperature may be expressed by a number on a temperature scale, defined without particular reference to that body. The applicability of a universal temperature scale to all physical bodies at equilibrium is a consequence of an empirical law (sometimes called the zeroth law of thermodynamics ), which states that if a body is in thermal equilibrium separately with each of two other bodies these two will be also in thermal equilibrium with each other. 

At equilibrium the two gases are at the same temperature (the zeroth law of thermodynamics), so we could, if we wanted, define the temperature to be T = (3/2). However, because the scale of temperature we use is degrees Kelvin, we need a constant of proportionality. This is k (Boltzmann s constant) when n is the number of molecules [JcT = (3 2)) and R, the gas constant, when n is the number of moles (RT - (3 2)]. If you stop and think for a moment, this result immediately gives a physical meaning to the absolute temperature as the point where molecular motion ceases. 

The zeroth law of thermodynamics states that there exists an additional intensive variable, temperature T = T p,V, N ), which has the same value for all systems in equilibrium with each other. 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

The zeroth law of thermodynamics states "Two systems in thermal equilibrium with a third system are in thermal equilibrium with each other." The zeroth law declares that two bodies in thermal equilibrium share a thermodynamic property, and that this thermodynamic property must be a state function. The thermodynamic property described by the zeroth law is temperature. 

The theory of heat has not been reduced to statistical mechanics "How can the zeroth law of thermodynamics be derived from statistical mechanics "... 

From the zeroth law of thermodynamics, we know that two systems that are in thermal equilibrium with a third system are in thermal equilibrium with one another and, by definition, have the same temperature. The zeroth law is not only important in defining systems that have the same temperature, but it also provides the basic principle behind thermometry one measures temperatures of different systems by thermometers that are, in turn, compared to some standard temperature systems or standard thermometers. 

The concept of temperature derives from a fact of common experience, sometimes called the zeroth law of thermodynamics , namely, if two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. To clarify this point, consider the three systems shown schematically in figure A2.1.1, in which there are diathermic walls between systems a and y and between systems p and y, but an adiabatic wall between systems a and p. 

---