The specific heat involves the consideration of a homogeneous phase of a substance of constant composition. This phase may be a solid, a liquid, or a gas, but no change of phase will occur. A variable termed the specific heat is then defined, which is the amount of heat required per unit mass to raise the temperature by one degree. Since it would be of interest to examine the relation between the specific heat and other thermodynamic variables, it is first noted that the heat transfer is given by Eq6 for the first law. Neglecting changes in kinetic and potential energies, and assuming a simple compressible substance and quasi-equilibrium process, for which the work in Eq6 for the first law is given by:
δQ = dU + δW = dU + P dV
This expression can be evaluated for two separate special cases:
1. Constant volume, for which the work term (P dV) is zero, so that the specific heat (at constant volume) is:
2. Constant pressure, for which the work term can be integrated and the resulting PV terms at the initial and final states can be associated with the internal energy terms, as in the lesson Enthalpy, thereby leading to the conclusion that the heat transfer can be expressed in terms of the enthalpy change. The corresponding specific heat (at constant pressure) is:
Note that in each of these special cases, the resulting expression, Eq1 or Eq2, contains only thermodynamic properties, from which it is concluded that the constant-volume and contant-pressure specific heats must themselves be thermodyanamic properties. This means that, although the discussion began by considering the amount of heat transfer required to cause a unit temperature change and then proceeded through a very specific development leading to Eq1 or Eq2, the result ultimately expresses a relation among a set of thermodynamic proeprties and therefore consitutes a definition that is independent of the particular process leading to it (in the same sense that the definition of enthalpy in the previous section is independent of the process used to illustrate one situation in which the property is useful in a thermodynamic analysis). As an example, consider the two identical fluid masses as shown:
In the first system 100 kJ of heat is transferred to it, and in the second system 100 kJ of work is done on it. Thus, the change of internal energy is the same for each, and therefore the final state and the final temperature are the same in each. In accordance with Eq1, therefore, exactly the same value for the average constant-volume specific heat would be found for this substance for the two processes, even though the two processes are very different as far as heat transfer is concerned.
Solids and Liquids
As a special case, consider either a solid or a liquid. Since both of these phases are nearly incompressible:
dh = du + d(Pv) ≈ du + v dP
Also, for both of these phases, the specific volume is very small, such that in many cases:
dh ≈ du ≈ C dt
where C is either the constant-volume or the constant-pressure specific heat, as the two would be nearly the same. In many processes involving a solid or a liquid, it might be further assumed that the specific heat in Eq3 is constant (unless the process occurs at low temperature or over a wide range of temperatures). Eq3 can then be integrated to:
h2 − h1 ≃ u2 − u1 ≃ C(T2 − T1)
In other processes for which it is not possible to assume constant specific heat, there may be a known relation for C as a function of temperature. Eq3 could then also be integrated.