The table shows that thermal equilibration in an isolated system results in an increase in entropy, as indicated in Fig. 1.18.

3.7 Effect of phase changes on the thermal properties of condensed phases

This topic, which properly belongs in Chap. 2, has been postponed until this point in order to complete the account of the effect of temperature on the properties of solids and liquids.

3.7.1 Heat Capacities

The discussion of Sect. 2.5.2 alluded to the small effect of pressure on the thermal properties of solids and liquids. One consequence of this fact is the close proximity of CP and Cv, which permits dealing with CP only. Another feature of the heat capacities of condensed phases is their relative insensitivity to temperature as well (at least at room temperature and above, see Fig. 2.6). For the present purposes, specific heat will be taken to mean CP and for most applications will be assumed to be independent of both p and T.

However, the difference between the heat capacities of the liquid (CPL) and that of the solid (CPs) is significant; for all substances, CPL is greater than CPs The right-hand plot in Fig. 3.5 shows this behavior schematically. In accord with the approximations stated above, ACP is taken to be independent of both temperature and pressure.

n re



Or aOp



Fig. 3.6 Effect of temperature and melting on enthalpy and specific heat 3.7.2 Enthalpy

The left-hand plot of Fig. 3.5 shows the temperature effect on the enthalpies of the individual phases and on the enthalpy of fusion. Because of the underlying assumption of constant CPs and Cpl, these enthalpy variations are linear in temperature. The dashed lines are extrapolations of hL into the subcooled-liquid region (T < TM) and of hS into the superheated-solid region (T > TM). Even though the dashed lines do not represent thermodynamically stable states, some phase equilibrium and chemical equilibrum calculations are based on estimates of the heat of fusion, AhM, at temperatures other than the melting point.

As expressed by Eq (2.14), the enthalpy of a substance is the temperature integral of its heat capacity. To perform the integration, a reference temperature Tref and a reference enthalpy must be selected. This choice is arbitrary, but usually Tref is taken as 298 K (room temperature) and the enthalpy of the solid at Tref is specified as href,s, which is usually set equal to zero. The enthalpy of the solid phase at other temperatures (including T > Tm) is:

Tref f)

The enthalpy of the liquid must reflect the same reference state as that of the solid, so that hL is given by:

hL(T) = hrefs + Cps(Tm - Tref) + AhM^) + Cpl(T - Tm) (3.20)

AhM(TM) accounts for the enthalpy increase on melting at the normal melting point, Tm. The heat of fusion changes with temperature because the slopes of the hL vs T and hS vs T differ. Even though melting physically occurs only at TM, later analyses of phase and chemical equilibria will require estimates of AhM at T ^ TM. This is obtained by subtracting Eq (3.19) from Eq(3.20). In so doing, Eq (3.19) is written as: hs(T) = hrefs + CPs(TM - Tref) + CPs(T - Tm), whether T is larger or smaller than TM. The result is:

Getting Started With Solar

Getting Started With Solar

Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.

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