Basic Theory and Techniques

2.1. INTRODUCTION

This chapter summarizes theory used in the discussion of thermodynamic prop-erties, giving references to where more detailed discussion may be found, and elab-orating more fully some concepts which are frequently used. The general principles are illustrated first for isotropic behavior. Of these principles, thermodynamics is essential to everyone in that it provides a systematic method of describing, relating and analyzing the bulk properties which form the subject of this monograph. Sta-tistical mechanics relates these properties to atomic and molecular structure, and so forms the basis for their theoretical explanation and prediction. Computational tools developed for this purpose are briefly mentioned, both for the statistical mechanics and for the underlying quantum theory of bonding and cohesion. Some of the appli-cations described are simple, such as ideal gases and Schottky systems; but two others of great general importance, viz. cooperative order-disorder effects and vibrational contributions, require longer discussion. So also does the extension of the theory to anisotropic behavior and elasticity.

2.2. THERMODYNAMICS

Like Section 1.3, this section deals with processes that are functions of volume and temperature. The thermodynamics of more general strain (including anisotropic expansion and elasticity) is treated in Section 2.8.

2.2.1. Definitions

Heat Capacity. Heat capacity is defined as the limit of the ratio

aQ/ aT

as

aQ

-+ 0, where

aT

is the rise of temperature resulting from an input of heat

aQ

under specified conditions. The heat capacities Cp and Cv, already defined in 27

28 Chapter 2

Section 1.3.1, can also be expressed as derivatives of the entropy: thus

cp=(aH) =T(as) _{aT p aT p} =_T(a2~) _{aT p}

^(2.1)

Cv= (~~)v ^=T(::)v =-T(:~)v

^(2.2)

where U and H are the energy and enthalpy, and F and G the Helmholtz and Gibbs free energies. The estimation of

Cp

from measurements involving finite intervals is discussed in Sections 1.2 and 3.2.

Cv

is usually not measured directly, but obtained from

Cp

by a thermodynamic relation (Eq. 2.10).

The expressions given above can apply to macroscopic systems of any size, but are often taken to refer to molar quantities. We reserve the term

specific

heat for the heat capacity per unit mass or per unit volume, both of which we denote by lower case:

cp=Cp/M, cv=Cv/M;

^OR

cp=Cp/V, cv=Cv/V

^(2.3)

Thermal Expansion. The coefficient of volumetric expansion, already defined in Eq. (1.3), can be expressed in any of the forms

~ = (alnV) aT p = _

^{(aln p )}

aT p =.!.. v apaT a2G

^(2.4)

This coefficient is often also denoted by a, but that symbol is more usefully reserved for coefficients of linear expansion, defined by

a

=

(alnl)

aT plaT p =! (!!...)

^(2.5)

When the expansion is isotropic, ~ = 3a.

Data on thermal expansion can be presented in different ways - for example as molar volumes, as dilations AV

/Vo,

or as expansion coefficients. For high precision it may be necessary to make fine distinctions, as for example between ~ and a as defined thermodynamically above and the quantities

W

and a* often used as practical definitions:

~* = ~ _Vo (av) , _{aT p}

_a ^* ₌

_t;;

¹

^(al) _aT

_p ^(2.6)

where Vo and 10 are usually taken to be the room temperature values of V and I.

Details of the treatment of primary dilatometric data are discussed in Section 3.3.1 and in several chapters of [H098].

Basic Theory and Tedmlques 29 Thermal expansion data are also obtained from the change in crystal lattice parameters measured by Bragg reflection in X-ray or neutron diffraction. Strictly such data are not equivalent to dilatometric data because of crystal imperfections.

For example, differences between the volumetric expansion coefficients of the crys-tallographic unit cell and of the bulk crystal are used to estimate the formation of vacancies (see Section 3.3.2). However, vacancies have a significant effect only near the melting point; at other temperatures the unit cell dimensions change proportion-ately to those of the macroscopic crystal, and so are equivalent to dilatometric data.

Conventions for the nomenclature of crystal axes, and the relation of the change of crystallographic parameters to bulk expansion, are discussed in Appendix A. Anal-ysis of intensities can give also the changing relative positions of atoms within the unit cell (known as internal expansion), but usually with insufficient precision to show perceptible change at low temperatures.

2.2.2. Units and Conversion Factors

Conversion between different energy scales is important for the comparison and interpretation of thermodynamic data. International convention now generally requires the use of SI units, together with allowed related units [CohS7, NeI9S]; but cgs and obsolete "practical" units are sometimes found, especially in the older lit-erature. Temperature scales, thermodynamic and practical, are discussed in Section 3.1.

The old unit of heat, the calorie, was used in the past in much good calorimetric work. It was defined originally so as to make the specific heat of water at 15°

Centigrade equal to 1 cal.g-1·deg-1, but later a thermochemical calorie (calth) was fixed as precisely 4.184 J. Heat capacities are now usually given in molar units of J·mol-I·K-1, or as specific heats in units of J.g-1·K-¹ or J·cm-³·K-1•

At the atomic level the electron volt (eV) is often used as a unit of energy, although spectroscopists may also refer to energy differences in terms of the fre-quency (in Hz) or inverse wave-length (in cm-l ) of the equivalent photon. In statistical mechanics we also need to know the temperature range in which the higher energy level becomes appreciably occupied. Table 2.1 gives equivalence factors relating these different energy scales. Thus we can see, for example, that rotational energy levels of molecules, which have microwave spectroscopic tran-sitions of a few cm-1, will contribute to heat capacities at temperatures of a few kelvin and upwards; whereas electronic levels, typically of the order of eV, will usually not contribute at all at low temperatures.

The bulk modulus and other elastic stiffnesses have the dimensions of pressure, for which the SI unit is the pascal:

1 Pa = 1 N ·m-² = 10 dyn.cm-² (2.7) The unit dyn.cm-² is now wholly obsolete, but one pre-SI practical unit, the bar, is still acceptable and widely used; 1 bar =

lOS

Pa, introduced so that the atmospheric

30 Chapter 2

Table 2.1. Equivalence factors for different energy scales

J eV hxl THz hexl em-^I kxl K

1 eV

=

1.602 x 10-¹⁹ 1 241.8 8065 11604

hxl THz

=

6.626 x 10-²² 4.136 x 10-3 1 33.36 47.99 hexl em-^I

=

1.986 x 10-²³ 1.240 x 10-⁴ 0.02998 1 1.439 kxl K

=

1.381 x 10-23 8.617 x 10-5 0.02084 0.6950 1 Also: I eV·molecule-^J = 96.49 kJ·mol-^J = 23.06 kcalll,moJ- 1

pressure is approximately 1 bar. Elastic stiffnesses of solids are typically of the order of 10 to 100 GPa, i.e., 0.1 to 1 Mbar.

2.2.3. Thermodynamic Relations

Methods used for obtaining relationships between thermodynamic quantities are summarized in Appendix B. Here we quote some results widely used in the analysis of thermodynamic data.

The ratio of

Cp

^to

Cv

is the same as that for

Bs

to

BT:

Cp

⁼

Bs

⁼

XT

^{= 1}

+

~yT

Cv BT xs

where "I is the Griineisen function defined in Eq. (1.5):

Thus

~V ~V

" 1 = =

In document at Low Temperatures (pagina 35-38)