Heat capacity at constant volume

In Section 3.8, we saw that the heat capacity (at constant volume) is expressible in terms of ordinary MDF s of up to order four. The order is reduced to two by using GMDF s. This is demonstrated below. Differentiating (5.45) with respect to temperature gives 

In this section, we assume that N b ) has been defined in the T, V, N ensemble, and all the derivatives that follow are at constant N and V. 

All of the averages in (5.53) can be now rewritten as integrals over the singlet and the pair GMDF s based on BE. To do this, we use the relations 

Relations (5.54)-(5.56) can be easily verified using arguments similar to those employed in (5.35). Substituting these in (5.53), we obtain the final expression for the heat capacity in the T, F, N ensemble  

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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,... 

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

Note This method of temperature regulation does not give all properties of the canonical ensemble. In particular, you cannot calculate Cy, heat capacity at constant volume. 

The procedure now is analogous to that of the preceding sec tion. Define the heat capacity at constant volume by... 

Cp = Heat capacity at constant pressure. Btu/lb F = Heat capacity at constant volume, Btu/lb°F... 

Cv specific molar heat capacity or heat capacity at constant volume, J/mol K... 

Macroscopic observables, such as pressme P or heat capacity at constant volume C v, may be calculated as derivatives of thermodynamic functions. 

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. 

A = work function (Helmholtz free energy), Btu/lb or Btu C = heat capacity, Btu/lb °R Cp = heat capacity at constant pressure = heat capacity at constant volume F= (Gibbs) free energy, Btu/lb or Btu g = acceleration due to gravity = 32.174 ft/s ... 

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. 

Thus, for the ideal gas the molar heat capacity at constant pressure is greater than the molar heat capacity at constant volume by the gas constant R. In Chapter 3 we will derive a more general relationship between Cp m and CV m that applies to all gases, liquids, and solids. 

Heat Capacities at Constant Volume and Constant Pressure... 

Because heat transferred at constant volume can be identified with the change in internal energy, AU, we can combine Eq. 8 with C = q/AT and define the heat capacity at constant volume, Cv, as... 

We can see how the values of heat capacities depend on molecular properties by using the relations in Section 6.7. We start with a simple system, a monatomic ideal gas such as argon. We saw in Section 6.7 that the molar internal energy of a monatomic ideal gas at a temperature T is RT and that the change in molar internal energy when the temperature is changed by AT is A(Jm = jRAT. It follows from Eq. 12a that the molar heat capacity at constant volume is... 

The high-temperature contribution of vibrational modes to the molar heat capacity of a solid at constant volume is R for each mode of vibrational motion. Hence, for an atomic solid, the molar heat capacity at constant volume is approximately 3/. (a) The specific heat capacity of a certain atomic solid is 0.392 J-K 1 -g. The chloride of this element (XC12) is 52.7% chlorine by mass. Identify the element, (b) This element crystallizes in a face-centered cubic unit cell and its atomic radius is 128 pm. What is the density of this atomic solid ... 

Estimate the molar heat capacity (at constant volume) of sulfur dioxide gas. In addition to translational and rotational motion, there is vibrational motion. Each vibrational degree of freedom contributes R to the molar heat capacity. The temperature needed for the vibrational modes to be accessible can be approximated by 6 = />vvih/, where k is Boltzmann s constant. The vibrational modes have frequencies 3.5 X... 

Hz, 4.1 X 1013 Hz, and 1.6 X 1013 Elz. (a) What is the high-temperature limit of the molar heat capacity at constant volume (b) What is the molar heat capacity at constant volume at 1000. K (c) What is the molar heat capacity at constant volume at room temperature ... 

STRATEGY We expect a positive entropy change because the thermal disorder in a system increases as the temperature is raised. We use Eq. 2, with the heat capacity at constant volume, Cv = nCV m. Find the amount (in moles) of gas molecules by using the ideal gas law, PV = nRT, and the initial conditions remember to express temperature in kelvins. Because the data are liters and kilopascals, use R expressed in those units. As always, avoid rounding errors by delaying the numerical calculation to the last possible stage. 

Thus, even at temperatures well above absolute zero, the electrons are essentially all in the lowest possible energy states. As a result, the electronic heat capacity at constant volume, which equals d tot/dr, is small at ordinary temperatures and approaches zero at low temperatures. 

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... 

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