# Thermodynamic property relations

For a single-component closed system (fixed mass), the first law of thermodynamics gives us:

$d\hat E = \delta Q - \delta W\qquad \qquad(1)$

where $\hat E$ is the total energy of the closed system, δQ is heat transferred to a system and δW is the work done by the system to the surroundings. The contribution to the total energy is due to internal (E), kinetic, potential, electromagnetic, surface tension or other form of energies. If change of all other forms of energy can be neglected, then $\hat E = E$. Heat transfer to a system is positive (system receives heat), whereas heat transfer from the system is negative (system loses heat). In contrast, work done by a system is positive (system loses work), and the work done to the system is negative (system receives work). The mechanical work for a closed system is usually expressed as δW = pdV, where p is the pressure and V is the volume of the system – both are thermodynamic properties of the system. Change of a thermodynamic property depends on initial and final states only and does not depend on the path by which the change occurred. Therefore, thermodynamic properties are path-independent and its infinitesimal change is represented by exact differential d (such as dE or dV). The heat transfer, Q, and work, W, on the other hand, are path-dependent functions. Infinitesimal heat transfer and work are represented by δQ and δW, respectively, in order to distinguish them from the change of a path-independent function. If it is assumed that the only work done is by volume change, and that potential and kinetic energies are negligible,

$dE = dQ - pdV\qquad \qquad(2)$

The second law of thermodynamics for the single-component closed system can be described by the Clausius inequality, i.e.,

$dS \ge \frac{{\delta Q}}{T}\qquad \qquad(3)$

where dS is the change of entropy of the closed system. The equal sign designates a reversible process, which is defined as an ideal process that after taking place can be reversed without leaving any change to either system or surroundings. The greater-than sign denotes an irreversible process. Combining these general forms of the first two laws of thermodynamics results in an expression that is very useful for determining the conditions for equilibrium and stability of systems, namely, the fundamental relation of thermodynamics:

$dE \le TdS - \delta W\qquad \qquad(4)$

where the inequality is used for irreversible processes and the equality for reversible processes. For a finite change in a system, the fundamental thermodynamic relationship becomes

$\Delta E \le T\Delta S - W\qquad \qquad(5)$

where $W = \int_{{V_1}}^{{V_2}} {pdV}$ is the work done by the system to the surroundings. It is desirable to have a property that allows us to compare the energy storage capabilities of various substances under various processes such as constant volume or constant pressure. This property is specific heat. Two kinds of specific heat are used: specific heat at constant volume, cv, and specific heat at constant pressure, cp. For a closed system undergoing a constant volume process, consider the first and second laws of thermodynamics for a reversible process, and using the definition of specific heat,

${\left( {\delta Q} \right)_{rev}} = de = {c_v}dT \qquad \qquad (6)$

${c_v} = {\left( {\frac{{\delta {Q_{rev}}}}{{mdT}}} \right)_v} = {\left( {\frac{{\partial e}}{{\partial T}}} \right)_v} = T{\left( {\frac{{\partial s}}{{\partial T}}} \right)_v} \qquad \qquad (7)$

Similarly, the expression for specific heat at constant pressure cp can be obtained for a constant pressure process,

${c_p} = {\left( {\frac{{\delta {Q_{rev}}}}{{mdT}}} \right)_p} = {\left( {\frac{{\partial h}}{{\partial T}}} \right)_p} = T{\left( {\frac{{\partial s}}{{\partial T}}} \right)_p} \qquad \qquad (8)$

where h = e + pv is the specific enthalpy and s is the specific entropy. Another important property that is often used for constant pressure processes is the volumetric coefficient of thermal expansion in terms of specific volume (v),

$\beta = \frac{1}{v}{\left( {\frac{{\partial v}}{{\partial T}}} \right)_p} \qquad \qquad (9)$

The coefficient of thermal expansion is often used in natural convective heat and mass transfer in terms of density,

$\beta = - \frac{1}{\rho }{\left( {\frac{{\partial \rho }}{{\partial T}}} \right)_p} \qquad \qquad (10)$

The first and second laws of thermodynamics for open systems will be discussed in Chapter 3.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.