Heat capacity constant-volume

Specific Heat and Heat Capacity Constant-Volume Calorimetry Constant-Pressure Calorimetry... 

Cv isochoric specific heat capacity (constant volume) J/(kg-K) Btu/(lbm-°R)... 

Cv isochoric specific heat capacity (constant volume)... 

The measurement of the specific heat at constant volume is attended with considerable difficulty, because the thermal capacity of a vessel strong enough to contain the gas after heating has a value much greater than that of the thermal capacity of the enclosed gas. 

STRATEGY We expect the temperature to rise more as a result of heating at constant volume than at constant pressure because at constant pressure some of the energy is used to expand the system. Oxygen is a linear molecule and its heat capacities can be... 

The heat capacity of a substance can differ, depending on which are the variables held constant, with the quantity being held constant usually being denoted with a subscript. For example, the specific heat at constant pressure is commonly denoted cP, while the specific heat at constant volume is commonly denoted cv ... 

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]... 

The heat capacity per unit mass, cr, is called the specific heat at constant volume. The heat capacity per mole is called the molar heat capacity at constant volume, CVm. For homogeneous systems, the system heat capacity can be calculated as... 

The constant-volume heat capacity appears in this expression because when a gas is heated at constant volume, all the input energy (heat) goes toward increasing E (no heat is needed to do work). 

If we are speaking of molar heat capacities, the volume in the derivative is the molar volume from the equation of state, V= RT/p. Differentiating with respect to temperature, keeping the pressure constant, yields dVIdT) = R/p. Putting this value in Eq. (7.41) reduces it to the simple result... 

As shown already, specific heat is the quantity of heat required to raise the unit mass of the material through 1°C, that is, the heat capacity of unit mass. There is the specific heat at constant volume which is virtually impossible to measure, and the specific heat at constant pressure, which is the quantity normally measured. The difference between the two specific heats is usually small enough to be ignored. In fact, the heat capacity at constant volume per unit volume is given by p C, so that it is found ... 

The practical utility of the heat capacities is twofold. First, they allow us to calculate heat in constant-volume and constant-pressure processes. This is useful in energy balances. Second, they allow us to calculate changes in internal energy and enthalpy. This allows us to calculate these properties using equations rather than tables, or to obtain their values in states that are not found in tables. There is a limitation, however. Equation (. 17) maybe used only between two states of the same volume, and eg. f. iQ ) only between two states of the same pressure. The general calculation of properties between any two states will be discussed in Chanter r. 

When a gas is heated at constant volume, AV is zero i.e., no work is done. All the heat absorbed by the system is used to increase the internal energy of the system. Consider 1 mole of an ideal gas whose temperature is raised by 1° at constant volume. The increase in internal energy itself gives the molar heat capacity at constant volume,... 

Table 7.2. Heat capacities at volume constant in the classical theorem of equipartition of energy...

There is no difficulty in keeping the number of moles constant during the measurement of heat, but to keep the volume constant is only easy with gases. Solids and liquids develop enormous pressures when heated at constant volume. To allow easier experimentation, it would be better to use the variables pressure, p, temperature, T, and number of moles, n, instead of volume, temperature, and number of moles. With these new variables, Eq. (2) takes on the form given by Eq. (5). To combine Eq. (5) with Eq. (1), one needs to express volume, which is also an extensive function of state, in the same variables as U. This is done in Eq. (6). When one now inserts Eqs. (5) and (6) into Eq. (1), to come up with an expression for dQ. one gets to the somewhat unhandy expression of Eq. (7), which is to be compared to Eq. (3). All three terms in Eq. (7) are made up of two components. It is just not convenient to define a heat capacity at constant pressure. 

Thus, by using thermodynamic relations, in the same way as in the case of volumetric and thermal properties of solntions (see Eq. (2.57)), it is possible to correlate the compressibility and thermal properties. By differentiation of the isochoric thermal pressure coefficient y T m) with regard to T, the change of isochoric heat capacity with volume at constant temperature can be evaluated. Its value for pure water and citric acid solutions increases with increasing volume because the second derivative of the pressure with respect to temperature is positive, g T m) = T d P / dT )y > 0. 

As one raises the temperature of the system along a particular path, one may define a heat capacity C = D p th/dT. (The tenn heat capacity is almost as unfortunate a name as the obsolescent heat content for// alas, no alternative exists.) However several such paths define state functions, e.g. equation (A2.1.28) and equation (A2.1.29). Thus we can define the heat capacity at constant volume Cy and the heat capacity at constant pressure as... 

In typical metals, both electrons and phonons contribute to the heat capacity at constant volume. The temperaPire-dependent expression... 

Differentiating this with respect to J = - TJ yields a heat capacity at constant volume,... 

However, the discovery in 1962 by Voronel and coworkers [H] that the constant-volume heat capacity of argon showed a weak divergence at the critical point, had a major impact on uniting fluid criticality widi that of other systems. They thought the divergence was logaritlnnic, but it is not quite that weak, satisfying equation (A2.5.21) with an exponent a now known to be about 0.11. The equation applies both above and... 

The heat capacity at constant volume Cj is defined from the relations... 

Except for gases, it is very difficult to detennine Cy. For a solid or liquid the pressure developed in keeping the volume constant when the temperature is changed by a significant amount would require a vessel so massive that most of the total heat capacity would be that of the container. It is much easier to measure the difference... 

Magee J W, Blanco J C and Deal R J 1998 High-temperature adiabatic calorimeter for constant-volume heat capacity of compressed gases and liquids J. Res. Natl Inst. Stand. Technol. 103 63... 

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