Heat Capacity c and cP

the slope of the curve gives us the heat capacity at any temperature. In these data, the heat capacity at Ti is less than that at T - Typically, heat capacity changes with T. By taking the slope of this curve as a function of temperature, we can get  

Parameters a, B, C, D, and E are then tabulated and can be used any time we want to know how the internal energy of species A changes with temperature at constant volume. We can then find A by integration  

Heat capacity should only be used for temperature changes between the same phase. When a phase change occurs, the latent heat must also be considered, as discussed shortly. 

Since the system is now doing Pv work as it expands, the energy balance contains a term for work  

in this case, an energy balance tells us the heat supplied at constant pressure is just equal to the change in the thermodynamic property, enthalpy. Therefore, we define the heat capacity at constant pressure as  

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No phase changes or chemical reactions take place within the system 0 and H are independent of pressure and the mean heat capacities C and Cp of the system contents (and of the inlet and outlet streams) are independent of composition and temperature, and hence unchanging with time. Then if Tr is a reference temperature at which is defined to be zero and M is the mass (or number of moles) of the system contents,... 

To summarize, the conditions under which Equations 11.3-12 and 11.3-13 are valid are (a) negligible kinetic and potential energy changes, (b) no accumulation of mass in the system, (c) pressure independence of U and / , (d) no phase changes or chemical reactions, and (e) a spatially uniform system temperature. Any or all of the variables T, T, , Q and (or IV) may vary with time, but the system mass, M, the mass throughput rate, m, and the heat capacities, C and Cp, must be constants. 

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