Entropy phase changes

A lustrous metal has the heat capacities as a function of temperature shown in Table 1-4 where the integers are temperatures and the floating point numbers (numbers with decimal points) are heat capacities. Print the curve of Cp vs. T and Cp/T vs. T and determine the entropy of the metal at 298 K assuming no phase changes over the interval [0, 298]. Use as many of the methods described above as feasible. If you do not have a plotting program, draw the curves by hand. Scan a table of standard entropy values and decide what the metal might he. 

Molecular Nature of Steam. The molecular stmcture of steam is not as weU known as that of ice or water. During the water—steam phase change, rotation of molecules and vibration of atoms within the water molecules do not change considerably, but translation movement increases, accounting for the volume increase when water is evaporated at subcritical pressures. There are indications that even in the steam phase some H2O molecules are associated in small clusters of two or more molecules (4). Values for the dimerization enthalpy and entropy of water have been deterrnined from measurements of the pressure dependence of the thermal conductivity of water vapor at 358—386 K (85—112°C) and 13.3—133.3 kPa (100—1000 torr). These measurements yield the estimated upper limits of equiUbrium constants, for cluster formation in steam, where n is the number of molecules in a cluster. 

It should be noted that the methodology for a and b results in a ASf s associated with the phase change from a sohd at 0 K to the liquid at Tmit No entropy changes resulting from solid transitions are taken into account, and ASfus for a substance that undergoes such a transition will be overestimated by this technique. 

Notice from the figure that the effect of temperature on entropy is due almost entirely to phase changes. The slope of the curve is small in regions where only one phase is present. In contrast, there is a large jump in entropy when the solid melts and an even larger one when the liquid vaporizes. This behavior is typical of all substances melting and vaporization are accompanied by relatively large increases in entropy. 

Phase changes are examples of (c). In order to melt a mole of ice in contact with air at 298.15 K, heat must flow into the system from the air. The increase in entropy of the system is 22.00 J K-1 - mol-1. The heat leaving the air decreases its entropy by 20.15 J K-1 mol-1. The net change in the universe is once again positive ... 

Experience indicates that the Third Law of Thermodynamics not only predicts that So — 0, but produces a potential to drive a substance to zero entropy at 0 Kelvin. Cooling a gas causes it to successively become more ordered. Phase changes to liquid and solid increase the order. Cooling through equilibrium solid phase transitions invariably results in evolution of heat and a decrease in entropy. A number of solids are disordered at higher temperatures, but the disorder decreases with cooling until perfect order is obtained. Exceptions are... 

A cryogenic calorimeter measures Cp,m as a function of temperature. We have seen that with the aid of the Third Law, the Cp,m data (along with AHm for phase changes) can be integrated to give the absolute entropy... 

It follows that we can replace qm, in the expression for the entropy change by AH for the phase change. 

Calculate the standard entropy of a phase change (Example 7.6). 

C(s, diamond) < Ff20(s) < H20(1) < H20(g). Diamond, being a covalently bound monatomic solid, has less disorder than H20(s), which is held together by weaker interniolecular forces. The three phases of water increase in entropy in changing from a solid to a liquid to a gas. 

Phase changes, which convert a substance from one phase to another, have characteristic thermodynamic properties Any change from a more constrained phase to a less constrained phase increases both the enthalpy and the entropy of the substance. Recall from our description of phase changes in Chapter 11 that enthalpy increases because energy must be provided to overcome the intermolecular forces that hold the molecules in the more constrained phase. Entropy increases because the molecules are more dispersed in the less constrained phase. Thus, when a solid melts or sublimes or a liquid vaporizes, both A H and A S are positive. Figure 14-18 summarizes these features. 

Schematic view of the three phase changes leading from more constrained to less constrained phases, illustrated by the phase changes for water. Each is accompanied by positive enthalpy and entropy changes for the substance.

As described in Section 14-1. when AR and ZlS have the same sign, the spontaneous direction of a process depends on T. For a phase change, enthalpy dominates AG at low temperature, and the formation of the more constrained phase is spontaneous, hi contrast, entropy dominates AG at high temperature, and the formation of the less constrained phase is spontaneous. At one characteristic temperature, A G = 0, and the phase change proceeds in both directions at the same rate. The two phases coexist, and the system is in a state of d Tiamic equilibrium. 

The spontaneous direction of a phase change also depends on pressure, primarily because the molar entropy of a... 

The quantity (Sp - Sa) is the entropy change in the phase change and hence (Sp - Sa) = AS = L/T, where L is the latent heat per mole associated with the phase change at temperature r. Taking this into account, the above relationship can be expressed as... 

From Eq. (6) it clearly can be seen that pressure and temperature are related by the changes in entropy and volume for the phase change. This equation also shows that once temperature is chosen, pressure must be fixed accordingly. It is easy to show that the change in entropy for the phase transition is simply the... 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

Even if one of the processes is not chemical but is categorized as a phase change, for example, evaporation, the extended De Donder s equation (Equation 13.17), is known to be valid. Any large magnitude of entropy production rate (diS/dt) due to evaporation might give a correlation such as... 

Chickos, J.S., Acree, W.E., Jr., Liebman, J.F. (1999) Estimating solid-liquid phase change enthalpies and entropies. J. Phys. Chem. Ref. Data 28, 1535-1673. 

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