Heat capacity at constant pressure

In this and in the following examples, we assume that the GMDF s have been defined in the T, P, N ensemble, using the basic distribution function 

The heat capacity at constant pressure is obtained from the temperature derivative of the enthalpy 

The term in the square brackets under the integral sign can be viewed as the enthalpy of the system for a given volume V and configuration R.  

By direct differentiation of (5.61) with respect to temperature, and by using similar arguments (we omit the details) to those employed in the derivation of (5.53) and (5.57), we get for Cp 

Relations (5.62) and (5.63) should be compared with (5.53) and (5.57), respectively. Here, the expression is more complicated due to the appearance of cross fluctuation between BE and the volume of the VP for a single particle, as well as between a pair of particles. Nevertheless, we note that the order of the GMDF s needed does not exceed two. Again, we recall the expression for Cp cited in Section 1.4. There, we were concerned with 

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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... 

Figure A2.5.2. Schematic representation of the behaviour of several thennodynamic fiinctions as a fiinction of temperature T at constant pressure for the one-component substance shown in figure A2.5.1. (The constant-pressure path is shown as a dotted line in figure A2.5.1.) (a) The molar Gibbs free energy Ci, (b) the molar enthalpy n, and (c) the molar heat capacity at constant pressure The fimctions shown are dimensionless...

For the first pari of this project, we suppose that we are presented with the following experimental data on the heat capacity at constant pressure Cp of solid lead at various temperatures up to and including 298 K (Table 1-2). 

Table 1-2 Experimental Heat Capacities at Constant Pressure for Lead...

Extensive tables and equations are given in ref. 1 for enthalpy of vaporization and heat capacity at constant pressure. 

Substance Statef Heat capacity at constant pressure (T=K 0°C = 273.1 K), cahdeg mol Range of temperature, K Uncertainty, %... 

Here, erfcjc is the eiTor function complement of jc and ierfc is its inverse. The physical properties are represented by a, the thermal dijfusivity, which is equal to lejpCp, where k is the drermal conductivity, p is the density and Cp, the specific heat capacity at constant pressure. The surface temperature during this iiTadiation, Tg, at jc = 0, is therefore... 

From a survey of experimental data for a large number of compounds, Kelley concluded that the heat capacity at constant pressure above room temperature could usefully be represented by an empirical equation using only the temperature as variable ... 

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

Cp specific molar heat capacity or heat capacity at constant pressure, J/mol K... 

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 ratio of the heat capacity at constant pressure to that at constant volume is... 

When a substance is heated at constant pressure without change of phase through a temperature rise dr the heat absorbed is Cp dr, where Cp is the molar heat capacity at constant pressure, and the entropy increase is... 

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 capacity at constant pressure is then easily obtained for the ideal gas from... 

Similarly, because heat transferred at constant pressure can be identified with the change in enthalpy, AH, we can define the heat capacity at constant pressure, Cr, as... 

J-K 1 mol 1 =21.1 J-K -mol-1, a difference of 65%. The heat capacity at constant pressure is greater than that at constant volume because at constant pressure not all the heat supplied is used to raise the temperature some returns to the surroundings as expansion work and C = q/AT is larger (because AT is smaller) than at constant volume (when all the energy remains inside the system). 

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