Specific heat capacity

5.3.2 Specific heat capacity

The specific heat is another thermal property important in the processing operations of casting or heat treatment, since it determines the amount of heat required in the process. Thus, the specific heat (denoted by C p , when dealing with the specific heat at constant pressure) controls the increase in temperature, dT , produced by the addition of a given quantity of heat, dQ, to one gram of matter, so that dQ =C p dT .

The specific heat of a metal is due almost entirely to the vibrational motion of the ions. However,

a small part of the specific heat is due to the motion of the free electrons, which becomes important at high temperatures, especially in transition metals with electrons in incomplete shells.

The classical theory of specific heat assumes that an atom can oscillate in any one of three direc- tions, and hence a crystal of N atoms can vibrate in 3N independent normal modes, each with its characteristic frequency. Furthermore, the mean energy of each normal mode will be kT , so that the total vibrational thermal energy of the metal is E = 3N kT . In solid and liquid metals, the vol- ume changes on heating are very small and, consequently, it is customary to consider the specific heat at constant volume. If N, the number of atoms in the crystal, is equal to the number of atoms in

a gram-atom (i.e. Avogadro number), the heat capacity per gram-atom, i.e. the atomic heat, at constant volume is given by

dE

dQ

= = 3N k = 24.95 J K −1 dT .

dT

In practice, of course, when the specific heat is experimentally determined, it is the specific heat at constant pressure, C p , which is measured, not C v , and this is given by

where H = E + PV is known as the heat content or enthalpy, C p is greater than C v by a few percent because some work is done against interatomic forces when the crystal expands, and it can be shown that

C p −C 2 v = 9α VT /β, where α is the coefficient of linear thermal expansion, V is the volume per gram-atom and β is the

compressibility. Dulong and Petit were the first to point out that the specific heat of most materials, when deter- mined at sufficiently high temperatures and corrected to apply to constant volume, is approximately equal to 3R, where R is the gas constant. However, deviations from the ‘classical’ value of the atomic heat occur at low temperatures, as shown in Figure 5.2a. This deviation is readily accounted for by the quantum theory, since the vibrational energy must then be quantized in multiples of hν, where h is Planck’s constant and ν is the characteristic frequency of the normal mode of vibration.

According to the quantum theory, the mean energy of a normal mode of the crystal is

E(ν) = 1

2 hv + {hν/ exp(hν/kT ) − 1}, where 1 2 hν represents the energy a vibrator will have at the absolute zero of temperature, i.e. the

zero-point energy. Using the assumption made by Einstein (1907) that all vibrations have the same

Physical properties 243 Pb

1 Atomic heat

(u ⫽ 2000 K)

(b) Figure 5.2 The variation of atomic heat with temperature.

(a)

frequency (i.e. all atoms vibrate independently), the heat capacity is

C v = (dE/dT ) v

2 = 3N k(hν/kT ) 2 [ exp(hν/kT )/ {exp(hν/kT ) − 1} ] This equation is rarely written in such a form because most materials have different values of ν. It is

E E is known as the Einstein characteristic temperature. Consequently, when C v

E , the specific heat curves of all pure metals coincide and the value approaches zero at very low temperatures and rises to the classical value of 3N k = 3R ∼

= 25.2 J g −1 at high temperatures. Einstein’s formula for the specific heat is in good agreement with experiment for T

E , but is poor for low temperatures where the practical curve falls off less rapidly than that given by the

Einstein relationship. However, the discrepancy can be accounted for, as shown by Debye, by taking account of the fact that the atomic vibrations are not independent of each other. This modification to

D , which is defined by

D = hν D , where ν D is Debye’s maximum frequency. Figure 5.2b shows the atomic heat curves of Figure 5.2a

D D ≪ 1) a T 3 law is obeyed, but at high temperatures the free electrons make a contribution to the atomic heat which is proportional to

T and this causes a rise of C above the classical value.