Specific Heat: cp and cv

Earlier, we looked at the specific heat ratio, and we also looked at the variation in specific heat with temperature here and here. Today we’ll look in a little more detail at the properties constant volume and constant pressure specific heat.
The definitions of these material properties are:
constant pressure specific heat: c_p=(∂h/∂T)_p
constant volume specific heat:  c_v=(∂u/∂T)_v

From a practical standpoint, it is sometimes convenient to think of the specific heat as the amount of heat required to raise the temperature of a material. 

Note that while the definition is expressed in terms of constant pressure or constant volume processes, once defined, c_p and c_v, are properties (so the change between states is independent of the path) and can be applied in any kind of process.

Another important point is that these property definitions are general for any material.  For the special case of ideal gases, internal energy, u, and enthalpy, h, depend only on temperature, so the partial derivative notation can be dropped and:

c_p (ideal gas)=(dh/dT)  and  c_v (ideal gas)=(du/dT)

Still considering the special case of an ideal gas, the difference between c_p and c_v is R, the gas constant for a particular gas.  The gas constant for a particular gas, R, is related to the universal gas constant (here we’ll call it R) by the molar mass for the particular gas: R=R/MM

R=8.314 kJ/(kmole*K) or 1545 ft*lbf/(lbmole*°R)

This table demonstrates values of c_p, c_v , and R for several gases: 

 

For incompressible things (many liquids and solids can be approximated as being incompressible) c_p≈c_v , and sometimes is just represented by c.  This table shows values of c for some liquids and solids.