Relation between specific heats  1For thermoelastic materials, show that the specific heats are related by the relation 
Proof:
Recall that
and
Therefore,
Also recall that
Therefore, keeping constant while differentiating, we have
Noting that , and plugging back into the equation for the difference between the two specific heats, we have
Recalling that
we get
Relation between specific heats  2For thermoelastic materials, show that the specific heats can also be related by the equations 
Proof:
Recall that
Recall the chain rule which states that if
then, if we keep u fixed, the partial derivative of g with respect to t is given by
In our case,
Hence, we have
Taking the derivative with respect to T keeping constant, we have
or,
Now,
Therefore,
Again recall that,
Plugging into the above, we get
Therefore, we get the following relation for :
Recall that
Plugging in the expressions for we get:
Therefore,
Using the identity , we have
Specific heats of SaintVenant–Kirchhoff materialConsider an isotropic thermoelastic material that has a constant coefficient of thermal expansion and which follows the SaintVenant–Kirchhoff model, i.e, where α is the coefficient of thermal expansion and where K,μ are the bulk and shear moduli, respectively. Show that the specific heats related by the equation 
Proof:
Recall that,
Plugging the expressions of and into the above equation, we have
Therefore,
