## Info

Upper path: isobaric heating at 0.1 MPa from 20°C to 30°C; isothermal compression to 100 MPa.

Lower path: isothermal compression at 20°C from 0.1 to 100 MPa; isobaric heating to 30°C.

The temperature-averaged thermal expansion coefficients ( a ) between 20°C and 30°C are shown on the drawing for the initial and final pressures; they differ by about 40%. The pressure-averaged compressibilities (J3 ) at the two temperatures differ by only 2.5%. For either path, the fractional volume change is given by:

For the upper path, the above equation yields:

— I = 2.6 x 10^ (30 ~ 20) - 3.9 x 10"* (100 - 0.1) = -0.0364

'upper

For the lower path, the fractional volume change is:

As required, the 3.6% volume decrease is calculated from either path. Integration along any p-T curve connecting the end states would give the same result. The path independence of Av/v0 implies that the functions a(T,p\ and 0(7",p) cannot be totally independent of each other. Indeed they are not—see Problem 2.16.

### 2.5.2 Thermal Equations of State

The thermal equations of state of condensed phases are dependent chiefly on temperature, with pressure and volume changes having rather small effects on u or h. Consequently, Equations (2.13) apply reasonably well to condensed phases as well as to gases. Thus, the thermal properties of solids and liquids are determined by their specific heats. For solids the Cy-T relationship is typified by the lower curve in Figure 2.7, which is for copper.

As T approaches zero, so does Cy, which is a consequence of quantization of the vibrations of the atoms in a solid. At the other extreme of high temperatures, Cv remains at a plateau value of 3R. This is the classical physics result (called the law of DuLong and Petit) for a collection of vibrating particles. At very high temperatures, Cv begins to increase above the value of 3R because of a combination of electronic excitation and creation of point defects in the crystal structure. Both of these processes absorb energy and so appear in the specific heat.

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