Heat capacity relationship between constant volume

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. 

In Chapter 4 the following expression [Equation (4.61)] was shown to be a general relationship between the heat capacity at constant pressure and the heat capacity at constant volume ... 

Recall (see any standard physical chemistry text) that for adiabatic expansions or compressions of an ideal gas, there are several relationships between P, V, and T that hold e.g., PVy = constant, where y is the ratio of the heat capacities at constant pressure and volume, i.e., y = cp/cv. Most useful in the context of potential temperature is TPy/y 1 = constant. Applying this latter relationship,... 

This equation relates temperature and volume for a mechanically reversible adiabatic process involving an ideal gas with constant heat capacities. The analogous relationships between temperature and pressure and between pressure and volume can be obtained from Eq. (3.22) and the ideal-gas equation. Since P, V,/ T, = P2V2j T2, we may eliminate V,/ V2 from Eq. (3.22), obtaining ... 

For example, it a gas expands under adiabatic conditions, its temperature falls (work is done against the retreating walls of the container). The adiabatic equation describes the relationship between the pressure ip) of an ideal gas and its volume (V), i.e. pV = K, where y is the ratio of the principal specific heat capacities of the gas and K is a constant. 

All experimental techniques lead to heat capacities at constant pressure, Cp. In terms of microscopic quantities, however, heat capacity at constant volume, C, is the more accessible quantity. The relationship between Cp and is listed as Eq. (4) in Fig. 2.31 in continuation of Fig. 2.22. It involves several correlations, easily (but tediously) derivable from the first and second law expressions as will be shown next. To simplify the derivation, one starts assuming constant composition for the to be derived equation (no latent heats, dn = 0). From the first law, as given by Eq. (3) of Fig. 2.10, one differentiates dQ at constant pressure. Since (5Q/3T)p = Cp and (8U/8T)v = C, this differentiation gives Eq. (12) of Fig. 2.10 ... 

For a perfect gas at constant pressure, P = const., Eq. 2.1 gives the relationship between the heat capacities at constant pressure and at constant volume, viz., Cp — Cv = R (R is the gas constant). Similarly, at T = const, Eq. 2.1 predicts that the external work can only be performed at a cost of the internal energy PdV = —dU. 

Recall that we defined two different heat capacities, one for a change in a system kept at constant volume, and one for a change in a system kept at constant pressure. We labeled them C and Cp. What is the relationship between the two ... 

Starting from the relationship between temperature and kinetic energy for an ideal gas, find the value of the molar heat capacity of an ideal gas when its temperature is changed at constant volume. Find its molar heat capacity when its temperature is changed at constant pressure. 

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... 

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