Measurement of Properties
Only few thermodynamic properties can be measured easily, namely temperature T , pressure p, and volume v. These are related by the thermal equation of state p (T, v) which is therefore relatively easy to measure.
The specific heats (16.20) can be measured in careful measurements where, because of (16.26), it suffices to measure either cv or cp. These calorimetric measurements employ the first law, where the change in temperature in response to the heat added to the system is measured.
Other important quantities, however, e.g., u, h, f, g, s, cannot be measured directly. In the following we shall study how they can be related to measurable quantities, i.e., T , p, v, and cv by means of the Gibbs equation and the differential relations derived above.
We first consider the measurement of internal energy. The differential of u (T, v) is
Thus, only the specific heat cv (T, v) must be measured when the thermal equation of state p (T, v) is already known.
To determine what measurements must be taken to determine the specific heat cv (T, v) = ( ∂u ) , we consider its differential,
Thus, the volume dependence of cv follows from measurement of the thermal equation of state.
Accordingly, in order to determine the specific heat cv (T, v) for all T and v it is sufficient to measure the thermal equation of state p (T, v) for all (T, v) and the specific heat cv (T, v0) for all temperatures T but only one volume v0. Then, cv (T, v) follows from integration of (16.30). Finally, integration of (16.28) gives the internal energy u (T, v).
Integration is performed from a reference state (T0, v0) to the actual state (T, v). Since internal energy is a point function, its differential is exact, and the integration is independent of the path chosen. The easiest integration is in two steps, first at constant volume v0 from (T0, v0) to (T, v0), then at constant temperature T from (T, v0) to (T, v). The integration results in
The internal energy can only be determined apart from a reference value u (T0, v0). As long as no chemical reactions occur, the energy constant u (T0, v0) can be arbitrarily chosen; see Chapter 23 on chemical reactions for additional discussion. When phase changes are involved, the respective energies Δui have to be added.
Enthalpy can be obtained from integration of its differential
where (16.19) was used. Here, integration is performed from (T0, p0) to (T, p). When we integrate in two steps, first at constant pressure p0 from (T0, p0) to (T, p0), then at constant temperature T from (T, p0) to (T, p), the integration gives
Here we have explicitly introduced the heats of phase change Δhi which must be added whenever the line of integration crosses a saturation curve in the p-T-diagram. The reference enthalpy h (T0, p0) can be chosen arbitrarily as long as no chemical reactions occur. In case of chemical reactions, it should be chosen as the enthalpy of formation, see the discussion in the chapter on chemical reactions.
Entropy s (T, p) follows by integration of the Gibbs equation, e.g., in the form
Also entropy can be determined only apart from a reference value s (T0, v0) which only plays a role when chemical reactions occur; see Chapter 23. When the line of integration crosses saturation lines in the p-T-diagram, the corresponding entropy changes Δsi = Δhi must be included. This can be seen as follows: At an equilibrium phase interface, temperature T and (saturation) pressure psat (T ) are continuous. Integration of the Gibbs equation T ds = dh − vdp across the phase interface yields T Δs = Δh.
For the ideal gas, where ( ∂v ) = R and the specific heat depends on T only, enthalpy and entropy assume the familiar forms (with suitable choice of integration constants)
After u, h and s are determined, Helmholtz free energy f and Gibbs free energy g simply follow by means of their definitions (16.10, 16.12). Thus the measurement of all thermodynamic quantities requires only the measurement of the thermal equation of state p (T, v) for all (T, v) and the measurement of the specific heat at constant volume cv (T, v0) for all temperatures, but only one volume, e.g., in a constant volume calorimeter.2 All other quantities follow from differential relations that are based on the Gibbs equation, and integration.
Above we have outlined the necessary measurements to fully determine all relevant thermodynamic properties. We close this section by pointing out that all properties can be determined if just one of the thermodynamic potentials is known, this is shown in the next example. Since all properties can be derived from the potential, the expression for the potential is sometimes called the fundamental relation.
Example: Gibbs Free Energy as Potential
In this example we consider a particular function for the Gibbs free energy g (T, p), to show that knowledge of one potential allows to determine all relevant property relations, including all other potentials.
We consider the fundamental relation (A is a constant with the appropriate dimensions)
