Mathematical definition

Thermodynamiquement one can as consider as the specific heat is the derivative partial of a Fonction of state of a body compared to the Température.

The function of state is the energy interns mass U , or the mass Enthalpie H :

• mass heat capacity with constant volume : $C\_V\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_V$;

• mass heat capacity with constant pressure : $C\_p\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; H\}\; \{\backslash \; partial\; T\}\; \backslash \; right)\; \_p$.

Mass heat capacity of the solids and the liquids

The dilation coefficients of the solid body S and Liquide S are generally sufficiently low so that one neglects the difference between C_p and C_V for the majority of the applications. According to the theory of Debye, the molar heat capacity of a solid element can be given by means of the formula:

$C\_V\; (T)\; =\; 3R\; (4D\; (U)\; -\; \backslash \; frac\; \{3u\}\; \{\backslash \; exp\; \{U\}\; -\; 1\})$

with $u=\; \backslash \; frac\; \{\backslash \; Theta\}\; \{T\}$,

$\backslash \; Theta$ is the temperature of Debye, which is a characteristic of each substance,

R is the constant molar of the gases ,

and $D\; (U)\; =\; \backslash \; frac\; \{3\}\; \{u^3\}\; \backslash \; int\_\; \{0\}\; ^\; \{U\}\; (\backslash \; frac\; \{X\}\; \{2\}\; +\; \backslash \; frac\; \{X\}\; \{\backslash \; exp\; \{(X)\}-\; 1\})\; x^2dx$.

This formula is simplified at low temperature, like at high temperature; in this last case, we find the law of Dulong and Petit:

The theory is not valid any more for the made up bodies.

Measure mass heat capacity of a solid

The mass heat capacity of a solid can be measured by using an apparatus of the type DTA (thermodifférentielle Analyze, or DSC for differential scanning calorimetry ). It can be defined in the following way: when a system passes from the temperature T at a T+dT temperature, variation of internal energy of the system of the east related to the quantity of heat exchanged δQ according to:

$dU=\; \backslash \; delta\; Q\; -\; p\_edV\; \backslash ,$

with p_e the external pressure to which is subjected the system and FD the variation of volume. If V=Cte:

$dU=\; \backslash \; Q\_v\; delta\; =\; C\_v\; dT\; \backslash ,$

On the other hand, if the transformation is isobar (constant pressure), one obtains by using the function Enthalpie system, the relation:

$dH\; =\; \backslash \; delta\; Q\; +\; Vdp\; \backslash ,$

If P= cte

$dH\; =\; \backslash \; Q\_p\; delta\; =\; C\_pdT\; \backslash ,$

with C_p capacity with constant pressure. Measurement thus consists in measuring the difference in temperature created by a given heat exchange, or the flow of energy results in a difference in temperature.

The following diagram illustrates the instrumental technique used in the case of the first method (measurement of the difference in temperature).

The apparatus consists of two independent “studs” and a furnace. Thermocouple S make it possible to measure the temperature of the higher face of the studs in contact with the sample, as well as the temperature of the furnace. This one corresponds to the temperature of measurement. All measurements are taken by using an empty aluminum carry-sample on one of the studs. The first measurement of another empty aluminum carry-sample makes it possible to obtain a base line (depend on the temperature measurement by the thermocouples). Then a measurement of a reference sample of known specific heat makes it possible to calibrate the apparatus. Lastly, the sample in the form of powder is measured and its specific heat is obtained by comparison with that of the reference sample. To improve measuring accuracies, it is advisable to take into account if necessary the difference in mass between the two door samples (the correction is carried out by using the specific heat of aluminum). The source of principal error comes from the quality of the thermal contact between the stud and the carry-sample.

Mass heat capacity of gases

The mass capacity with constant volume is thus of:

• $C\_V\; =\; \backslash \; frac\; \{3R\}\; \{2M\}$ for a monoatomic perfect gas;

• $C\_V\; =\; \backslash \; frac\; \{5R\}\; \{2M\}$ for a diatomic perfect gas.

The mass capacity with constant pressure of a perfect gas can be given starting from the mass capacity with constant volume, since the equation of perfect gases expresses that:

$pv\; =\; \backslash \; frac\; \{RT\}\; \{M\}$, and thus: $\backslash \; frac\; \{\backslash \; partial\; (pv)\}\; \{\backslash \; partial\; T\}\; =\; \backslash \; frac\; \{R\}\; \{M\}$

p being pressure, v the mass volume , R the constant molar of the gases , under atmosphere normale