Heat capacity under constant volume

Here, we employ Fourier s law which relates the temperature gradient to the heat flux. It should be noted that the temperature plays a role of potential, and Cv is the heat capacity under constant volume, calculated as follows ... 

The heat capacity at constant volume is the derivative of the energy with respect to temperature at constant volume (eq. (16.1). There are several ways of calculating such response properties. The most accurate is to perform a series of simulations under NVT conditions, and thereby determine the behaviour of (f/) as a function of T (for example by fitting to a suitable function). Subsequently this function may be differentiated to give the heat capacity. This approach has the disadvantage that several simulations at different temperatures are required. Alternatively, the heat capacity can be calculated from the fluctuation of the energy around its mean value. 

The specific heat of a substance must always be defined relatively to a particular set of conditions under which heat is imparted, and it is here that the fluid analogy is very liable to lead to error. The number of heat units required to produce unit rise of temperature in a body depends in fact on the manner in which the heat is communicated. In particular, it is different according as the volume or the pressure is kept constant during the rise of temperature, and we have to distinguish between specific heats (and also heat capacities) at constant volume and those at constant pressure, as well as other kinds to be considered later. 

In (1.18.12) the symbols Cv and CP represent the heat capacities at constant volume and constant pressure, respectively. As will be seen in Section 1.19, these quantities represent the heat absorbed by a system per unit increase in its temperature under the indicated constraints. Such a quantity may be measured experimentally by use of a calorimeter. Equations (1.18.12) thus furnish a basis for determining E, H, or S from experimental measurements. 

The heat capacity at constant volume, Cy, is the ratio dq / dT for aproeess in a closed constant-volume system with no nonexpansion work—that is, no work at all. The first law shows that under these eonditions the internal energy change equals the heat dU = dq (Eq. 5.3.9). We can replace dq by dU and write Cy as a partial derivative ... 

Consider the process of changing the temperature of a phase at constant volume. The rate of change of internal energy with T under these conditions is the heat capacity at constant volume Cv = (9t//dT)v (Eq. 7.3.1). Accordingly, an infinitesimal change of U is given by... 

The heat capacity is, as shown in Section II.l.l defined as the increase in heat content if the tanperature of the sample is increased by one degree Celsius. In the following discussion, the heat capacity is measured under the condition of constant pressure. Heat capacities at constant volume, which are rntwe amenable to theoretical interpretation are not determined directly, but are calculated using the thermo-d5mamic relationship (see Eq. 11.25). 

For purposes of this calculation, latent heats at constant volume and at constant pressure are assumed equal, heat capacities at constant pressure and at constant volume are assumed equal for solids and liquids [See also Calculation of Temperature of Detonation (and Explosion) 1 and Experimental Determination of Temperature of Detonation [and Explosion) , under Detonation (and Explosion) Temperature Developed On in Vol 4 of Encycl, pp D589 L to D601-R]... 

Under isentropic conditions and with constant heat capacities, the pressure-volume relation is... 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

Heat capacities can be defined for processes that occur under conditions other than constant volume or constant temperature. For example, we could define a heat capacity at constant length of a sample. However, regardless of the nature of the process, the heat capacity will always be positive. This is ensured by the zeroth law of thermodynamics, which requires that as positive heat is transferred from a heat reservoir to a colder body, the temperature of the body will rise toward that of the reservoir in approaching the state of thermal equilibrium, regardless of the constraints of the heat-transfer process. 

When a system is heated, its temperature generally increases. This increase in temperature is dependent on the heat capacity of the system under constant volume or constant pressure. Therefore, the heat capacity is defined as the ratio of heat added to a system to its corresponding temperature change. If the system is under constant volume, the molar heat capacity is Cv, whereas the molar capacity is Cp for a system under constant pressure. Then,... 

Under constant-volume conditions, the heat of combustion of glucose (C5H12O6) is 15.57 kj/g. A 3.5(X)-g sample of glucose is burned in a bomb calorimeter. The temperature of the calorimeter increased from 20.94 C to 24.72 "C. (a) What is the total heat capacity of the calorimeter (b) If the size of the glucose sample had been exactly twice as large, what would the temperature change of the calorimeter have been ... 

The subscript p on Cp indicates that this is the heat capacity at constant pressure. Under other conditions, such as constant volume, the value of the heat capacity may differ slightly.)... 

The vaiiabiUty of circumstances, under which the heat capacity can be measured, extends as soon as the thermodynamical states of a system are no longer unambiguously characterized by means of the internal energy, the volume and the numbers of moles alone (or other variables which can be gained from these variables by Legendre-transformations) (Section II. 1.5). Thus the variaUes p and V, for example, are not sufficient to describe in an unique way the mechanical state of an elastic solid under the influence of external forces (Section II.1.5.1). On the contrary, the pressure has to be replaced by the six components of the stress tensor and the volume by the six conqxments j/fp of the strain tensor (multiplied by a suitable factor which makes these variables extensive variables). The analc e of Cp is then the heat capacity at constant strain [Eq.(37)], the analogue of Cp tiie heat capacity at constant stress [Eq.(38)]. Eq.(39) takes the place of the relation (25a). Butaccord-ing to the particular physical situation heat capacities can also be measured which correspond to a mixed set of variables, like constant y . t , y , t , y . 

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