Heat capacity variation with volume

It is thus seen that heat capacity at constant volume is the rate of change of internal energy with temperature, while heat capacity at constant pressure is the rate of change of enthalpy with temperature. Like internal energy, enthalpy and heat capacity are also extensive properties. The heat capacity values of substances are usually expressed per unit mass or mole. For instance, the specific heat which is the heat capacity per gram of the substance or the molar heat, which is the heat capacity per mole of the substance, are generally considered. The heat capacity of a substance increases with increase in temperature. This variation is usually represented by an empirical relationship such as... 

The enthalpies and internal energies of steam and water also converge at the critical point. Tlie heat capacity at constant pressure. C,. is defined as the derivative of enthalpy with respect to temperature. The value of Cr, becomes very large in the vicinity of the critical point. The variation is much smaller for tlie heat capacity at constant volume, C,.. 

This relation constitutes what we call the Kirchhoff relation. An equivalent relation would give the variation of the heat of transformation at constant volume with the temperature as a function of the molar specific heat capacity at constant volume associated with the transformation. 

If we sum the three contributions to calculate the value for the heat capacity at constant volume, as the characteristic temperatures of rotation are often lower than the Einstein temperature (see Table 7.3), the variation in the heat capacity, for example of a diatomic molecule, with temperature takes the form of the curve in Figure 7.12(c). At low temperatures, the only contribution is that of translation, given by 3R/2. Then if the temperature increases, the contribution of rotation, is added according to the curve in Figure 7.12(a) until the limiting value of this contribution is reached, then the vibrational contribution is involved until the molecule dissociates which makes the heat capacity become double that of the translational contribution of monoatomic molecules. The limiting value of the vibrational contribution is sometimes never reached. This explains why the values calculated in Table 7.2 are too low if we do not take into account the vibration and too high in the opposite case. 

In terms of comparison with experimental values, we shall give the example of the variation in heat capacity at constant volume as a function of the temperature calculated by Eyring s semi-microscopic method. Remember that it is a cellular model including vacancies and a degeneration coefficient (see section 1.6). Figure 1.9 illustrates such a comparison and exhibits good accordance between the results obtained by the model and the experimental results. 

The variation with temperature of the vibrational contribution to the heat capacity at constant volume for many relatively simple crystalline solids is shown in Figure 19.2. The C is zero at 0 K, but it rises rapidly with temperature this corresponds to an increased ability of the lattice waves to enhance their average energy with increasing temperature. At low temperatures, the relationship between C and the absolute temperature T is... 

By differentiating Eq.(6) with respect to time, considering that the variations of the volume and total pressure are negligible, and using the enthalpy definition. Hi = CpiT, where Cp is the heat capacity of i-reactant (kJ/mol °C), Eq.(5) can be written as follows ... 

The constant-volume heat capacity of the substance ( Cy) represents the variation of total energy U with T in the harmonic approximation (X is constant over T), and the integration of Cy over T gives the (harmonic) entropy of the substance ... 

Fig. 17. Temperature variation of internal energy. Et (kj mole"1), coordination number (CN), and gmi /gm.x ratios of water showing the occurrence of the glass transition in the 200-240 K range. Volume also shows a similar change, but a slightly lower temperature. In the inset, variation of the configurational heat capacity, Cp (J deg 1 mole-1), with temperature is shown. (From Chandrasekhar and Rao (73).)...

However, there are some situations where the one-dimensional descriptions do not work well. For example, with highly exothermic reactions, a fixed-bed reactor may contain several thousand tubes packed with catalyst particles such that djdp 5 in order to provide a high surface area per reaction volume for heat transfer. Since the heat capacities of gases are small, radial temperature gradients can still exist for highly exothermic gas-phase reactions, and these radial variations in temperature produce large changes in reaction rate across the radius of the tube. In this case, a two-dimensional reactor model is required. 

Heat capacities are also useful in determining the variation of temperature with pressure or volume in isentropic processes. For this purpose introduce Eq. (1.3.8) to write... 

The magnitude of the variation of with pressure is thus determined by the variation of a with temperature. Since docjdT is always positive, the heat capacity at constant pressure decreases with increased pressure, but the effect becomes small at high temperatures since docjdT falls off more rapidly than T increases (c/. fig. 12.1). A similar formula is readily obtained for the effect of volume on c. ... 

It is of interest to note that (d P/dT )v is zero for a van der Waals gas, as well as for an ideal gas hence, Cv should also be independent of the volume (or pressure) in the former case. In this event, the effect of pressure on Cp is equal to the variation of Cp — Cv with pressure. Comparison of equations (21.4) and (21.13), both of which are based on the van der Waals equation, shows this to be true. For a gas obeying the Berthelot equation or the Beattie-Bridgeman equation (d P/dT )v would not be zero, and hence some variation (f Cv with pressure is to be expected. It is probable, however, that this variation is small, and so for most purposes the heat capacity of any gas at constant volume may be regarded as being independent of the volume or pressure. The maximum in the ratio y of the heat capacities at constant pressure and volume, respectively, i.e., Cp/Cv, referred to earlier ( lOe), should thus occur at about the same pressure as that for Cp, at any temperature. 

Figure 2.23 Idealized variations in volume (V) and enthalpy (H). Also shown are a, the volume coefficient of expansion and Cp, the heat capacity, which are, respectively, the first derivatives of V and H with respect to temperature (T).

Since no work is performed in these constant volume chambers, the heat measured equals the change in internal energy U of the system. With known temperature change, the heat capacity Cy at constant volume V can be derived under the assumption that Cy is constant for the small temperature variation measured ... 

Because it applies mostly to electrolytes, it is discussed in Chapter 15. Briefly, Helgeson models the behavior of solutes by developing equations for the standard state partial molar volume (Helgeson and Kirkham 1976) and standard state partial molar heat capacity (Helgeson et al. 1981) as a function of P and T, with adjustable constants such that they can be applied to a wide variety of solutes. If you know these quantities (V°, C°p), you can calculate the variation of the standard state Gibbs energy, and that leads through fundamental relationships to equilibrium constants, enthalpies, and entropies. 

The extreme case of complete heat transfer control for COj-SA is illustrated in Figure 6.15. For this system diffusion is much faster and even in relatively large crystals the uptake rate is controlled by heat transfer. Uptake curves are essentially independent of crystal size but vary with sample size due to changes in the effective heat capacity and external area-to-volume ratio for the sample. Analysis of the uptake curves according to Eq. (6.70) yields consistent values for the overall heat capacity (34 mg sample C 0.32 and 12.5 mg sample 0.72 cal/g deg.). The variation of effective neat capacity with sample size arises from the increasing importance of the heat capacity of the containing pan when the adsorbent weight is small. 

Variations in heat capacities with pressure or volume From the symmetry of matrix u(P,T), it is easy to show that ... 

---