Difference Between Cp and

The differential relationships can be used to derive a general relationship for the difference between Cp and Cy. 

Substitution of the second equation into the first and cancelling terms gives 

For ideal gases, we have shown earlier that 

Solids and liquids have smaller a, k, and Vm than do gases, and the difference between Cp m and CV.m is smaller than for a gas. 

In an adiabatic expansion or compression, the system is thermally isolated from the surroundings so that q = 0. If the change is reversible, we can derive a general relationship between p, V, and T, that can then be applied to a fluid (such as an ideal gas) by knowing the equation of state relating p, V, and T. 

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Thermodynamics provides a way to calculate the difference between Cp and Cr- The procedure will be described later in this chapter and more generally in Chapter 3. 

What is the difference between CP and CV1 We first express this difference in terms of a standard thermodynamic identity for 1 mole of a simple gas ... 

There is another, nonmathematical, way to think about the difference between Cp and C,. To extract work from any process requires an energy input, like burning wax. Some of this energy will be extracted as work (Cp - Cv). But most of this energy input will wind up as heat (Cv). 

In Chapter 3, we will see how the difference in CP and Cv can generally be obtained from the equation of state. For condensed phases, (3V/3T)P is very small, but (3U/3V)T is very large, and substantial differences between CP and Cr can result. 

For example, the differences between CP and Br in complexes of the type [Ru (r 6-arene)(en)Z]+, where arene is biphenyl, indane or benzene, is not dramatic, however, when Z = I the hydrolysis is slower (3- to 7-fold). Other leaving groups such as pyridine or pyridine derivatives can slow down the hydrolysis further and even block it almost completely on biologically-relevant time scales. Such complexes are not cytotoxic towards cancer cells within 24-h drug exposures. There are a few exceptions such as [Ru(ri6-hexamethylbenzene)(en)(SPh)]PF6, where the monodentate ligand is thiophenolate. This complex does not hydrolyse, but intrigu-ingly is active. The mechanism of activation of this complex is different (vide infra). 

N is Avogadro s number and is the Debye temperature) and whose intercept at T = 0 is y. In the Debye model, the slope has a dependency (actually, r" for an n-dimen-sional solid) owing to the lattice or phonon contribution to the heat capacity. Of course, the heat capacity normally measured is Cp, the heat capacity at constant pressure. However, for solids the difference between Cp and Cy is typically less than 1 percent at low temperatures and thus can be neglected. 

For some metals Cps is greater than CpiJ From older values, Skaupy concluded that for metals Cp/c l-Scps, but there are exceptions. There are usually large differences between Cps and Cpi for organic compounds in some cases (undecoic acid, 10 carbon tetrachloriden) Cps is reported to be greater than Cpi. Mache ... 

The mean Al/Si ratio relative to Cl varies among IDPs by an amount similar to that seen for Mg/Si. Again there is a systematic difference between CP and CSIDPs with the latter being... 

The difference between Cp and Cy is of importance only when the volume of the system changes significantly— that is, when different numbers of moles of gases appear on either side of the chemical equation. For reactions involving only liquids and solids, Cp and Cv are for all practical purposes identical. 

Further differences between CP and CD relate to their binding sites on the actin molecule CP binds to the C terminus and CD to the N terminus of actin (Mezgueldi et al., 1992 Miki et al., 1992), consistent with different mechanisms of their action at the actin-myosin interface (Miki et al., 1992). Noda et al. (1992) have observed the induction of structural changes in actin upon binding to CP. 

For condensed phases the difference between Cp and Cy is determined by the constants a and k and, thus varies from substance to substance. 

Illustrative Problem. Derive an expression for the difference between Cp and Cy when gases do not behave ideally. At constant pressure, Cp = dH/df )p n and the specific heat at constant volume is Cy = (9f//9r)y aii Ni-... 

Figure 3.15 Difference between Cp and c for R134a (1,1,1,2-tetrafluoroethane). (Data from [63].)...

For ideal gases, the difference between Cp and Cy can be determined. From the definition of enthalpy (Equation 7.10) we have... 

The difference between Cp and Cg depends on pressure, temperature, and composition and may vary from 10 to several hundred percent in the single-phase state. 

The heat capacity at constant pressure is the most commonly measured heat capacity for solids and liquids. We now obtain an expression for the difference between Cp and Cy. We begin with... 

The physical explanation for this difference between Cp and Cy is that heating an ideal gas at constant volume does not work on the surroundings. In heating at constant pressure some of the heat is turned into work against the external pressure as the gas expands. A larger amount of heat is therefore required for a given change in the temperature than for a constant-volume process. In Chapter 4 we will be able to show that Cp cannot be smaller than Cy for any system. 

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