Negative Heat Capacities

If //is 00 (very large) or T is zero, tire system is in the lowest possible and a non-degenerate energy state and U = -N xH. If eitiier // or (3 is zero, then U= 0, corresponding to an equal number of spins up and down. There is a synnnetry between the positive and negative values of Pp//, but negative p values do not correspond to thennodynamic equilibrium states. The heat capacity is... 

Gas flow in these rotary dryers may be cocurrent or countercurrent. Cocurrent operation is preferred for heat-sensitive materials because gas and product leave at the same temperature. Countercurrent operation allows a product temperature higher than the exit gas temperature and dryer efficiency may be as high as 70%. Some dryers have enlarged cylinder sections at the material exit end to increase material holdup, reduce gas velocity, and minimize dusting. Indirectly heated tubes are installed in some dryers for additional heating capacity. To prevent dust and vapor escape at the cylinder seals, most rotary dryers operate at a negative internal pressure of 50—100 Pa (0.5—1.0 cm of water). 

We shall assume that Tx is always positive, i.e., if the temperature of a system is raised at constant configuration, heat is absorbed. As Duhem (Mecaniquc eliimique, I., 164) points out, this is by no means self-evident, because there are some heat capacities which are negative the heat capacity at constant configuration, Fx, appears, however, always to be positive. 

As an example of a negative heat capacity we have the specific heat of saturated steam. If unit mass of steam in the condition of saturation is raised one degree in temperature, and at the same time compressed so as to keep it just saturated at each temperature, it is found that heat is evolved, not absorbed, because the work spent in the compression exceeds the increase of intrinsic energy. 

Note that the m(dstrong electrolytes in dilute solution. It results because the charged ions break up the hydrogen bonded structure of the water and decrease the heat capacity of the solution over that of pure water. Thus, the contribution of Cp. 2 to Cp m is negative. 

FIGURE 6J5 If the heat capacity ot the reactants is larger than that of the products, the enthalpy of the reactants will increase more sharply with increasing temperature. If the reaction is exothermic, the reaction enthalpy will become more negative, as shown here. If the reaction is endothermic, the reaction enthalpy will become less positive and may even become negative. 

We note several things about this example. First, the number of moles of water and the molar heat capacity were used. Second, since the heat was removed, the value used in the equation was negative. The final temperature is obviously lower than the initial temperature, since heat was removed. 

For cases where AH0 is essentially independent of temperature, plots of in Ka versus 1/T are linear with slope —(AH°/R). For cases where the heat capacity term in equation 2.2.7 is appreciable, this equation must be substituted in either equation 2.5.2 or equation 2.5.3 in order to determine the temperature dependence of the equilibrium constant. For exothermic reactions (AH0 negative) the equilibrium constant decreases with increasing temperature, while for endothermic reactions the equilibrium constant increases with increasing temperature. 

Due also to their (amorphous) composition, the heat capacity of a ruthenium oxide resistor is much higher than that of a Ge thermistor of equal mass [61]. This negative property prevents the use of Ru02 resistors as detector sensors (see Chapter 15). 

To determine the heat capacity of the calorimeter, recognize that the heat evolved by the reaction is the negative of the heat of combustion. 

As shown by equation 2.11, known as the Kirchhofif equation, the standard reaction heat capacity (ArC°) is the difference between the standard heat capacities of the products and reactants (recall that v are the stoichiometry coefficients—negative for the reactants and positive for the products) ... 

It can be concluded from equations 12.11 and 12.12 that the small deviation of the zero line relative to the isothermal baseline under the same scanning conditions is proportional to the heating rate and the difference in heat capacities of the two empty crucibles. This deviation can be positive (as in figure 12.4) or negative, depending on the magnitude of the intrinsic thermal asymmetry of the system under scanning conditions and the relative masses of the two crucibles. When the sample is introduced in the sample crucible,... 

We can calculate AH from thermal data alone, that is, from calorimetric measurements of enthalpies of reaction and heat capacities. It would be advantageous if we could also compute AS from thermal data alone, for then we could calculate AG or Ay without using equilibrium data. The requirement of measurements for an equilibrium state or the need for a reversible reaction thus could be avoided. The thermal-data method would be of particular advantage for reactions for which AG or AT is very large (either positive or negative) because equilibrium measurements are most difficult in these cases. 

The heat capacities of series of ammonium bromides ([(Ci)2C3HOC2N]Br, [(Cj)2C4HOC2N]Br, and [(Q)2QHOC2N]Br) were measured for the solid and liquid phase, and the difference in the solute heat capacity between the liquid and solid phase at the melting temperature, j has been determined by the DSC analysis [79]. Negative values of were observed for these... 

The minns sign in Eq. (4.2) arises dne to the fact that in order for there to be heat flow in the +y direction, the temperatnre gradient in that direction must be negative—that is, lower temperature in the direction of heat flow. If the temperature gradient is expressed in units of K/m, and the heat flux is in J/m - s, then the thermal conductivity has units of J/K m s, or W/m K. A related quantity is the thermal diffusivity, which is often represented by the lowercase Greek letter alpha, a. Thermal diffusivity is defined as k/pCp, where k is the thermal conductivity, p is the density, and Cp is the heat capacity at constant pressure per unit mass. We will see in a moment why the term diffusivity is used to describe this parameter. We will generally confine our descriptions in this chapter to thermal conductivity. 

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