Variable of state

As the pure substance, hydrogen H2 may exist in various physical phases as vapor, liquid and solid. As with any pure substance, the thermodynamic state of hydrogen is completely defined by specifying two independent intensive state variables. All other variables of state can then be determined by using one of the three relationships of state. 

The equation of state is the mathematical relationship between the following three intensive thermodynamic properties pressure p, temperature T, and specific volume V (or density q = 1/v) 

The calorific equation of state relates the internal energy w to Tand v or enthalpy h to T and p 

The third state relationship relates entropy s to T and v or Tand p 

These relationships of state have to be determined experimentally and are available in the form of equations, tables or diagrams [21]. 

---

One must now examine the integrability of the differentials hi equation (A2.1.121 and equation (A2.1.13), which are examples of what matliematicians call Pfoff differential equations. If the equation is integrable, one can find an integrating denominator k, a fiiiictioii of the variables of state, such that = d( ) where d( ) is... 

F is the number of degrees of freedom, i.e. the number of variables of state such as temperature and pressure that can be varied independently, P is the number of phases and C is the number of components. Components are to be understood as independent, pure substances (elements or compounds), from which the other compounds that eventually occur in the system can be formed. For example ... 

In phase diagrams for two-component systems the composition is plotted vs. one of the variables of state (pressure or temperature), the other one having a constant value. Most common are plots of the composition vs. temperature at ambient pressure. Such phase diagrams differ depending on whether the components form solid solutions with each other or not or whether they combine to form compounds. 

It follows that in the limit of absolute zero, the entropy of a perfect crystalline substance must be independent of changes in pressure or volume (or any other variable of state except T). Thus,... 

These three equations of conservation may be looked upon as defining any three of the four variables p, p, U, u in terms of the 4th, if it is assumed that the equation of the medium, f(p,p,T)=0, as well as the dependence of internal energy of any pair of these variables of state is known. Therefore, the properties of a stationary shock wave follow from the knowledge of the velocity of the piston maintaining the wave, which is also the material (particle) velocity, u... 

The temporal change of the molar quantity of the element sulfur in the film on a fluidized-bed particle is caused by inflow and outflow of the material. This mass balance is needed only once, since the fluidized-bed particles and, thus, the liquid films on the particles are regarded as ideally mixed. The condition of the ideal mixing leads to the fact that with the mass and energy balance of the film, the average variables of state of the gas in the respective driving force approximations find the following form... 

The number of degrees of freedom is the number of variables of state that can be altered independently and arbitrarily, within limits, without resulting in the appearance of existing phases or in the generation of new ones. 

We can now proceed to calculate the number of degrees of freedom f for the assembly of phases. As discussed earlier, if the phases were all separate systems, p(c + 1) independent variables of state would have to be specified. However, as a result of equilibration among the phases one must now introduce the 2(p - 1) constraints of Eq. (2.1.6a) and (2.1.6b) to ensure uniformity in T and P. One must also take note of the c(p — 1) interrelations in Eq. (2.1.7) the equality among appropriate p s provides interrelations among the mole fractions of any given component in the various phases. The totality of constraints therefore is (c + 2)(p — 1). The number of degrees of freedom remaining is then... 

If we change any of the external variables governing the system, such as temperature, pressure, etc., then Eq. (XV.5.1) or (XV.5.2) can be used to estimate the effect of such changes on the rate constant s so long as the changes in external variables do not alter the mechanism of the reaction. But this last proviso defines a very interesting situation. Since Eq. (XV.5.2) involves only thermodynamic factors, the only external variables that need concern us are the thermodynamic variables of state, i.e., those needed to describe an equilibrium state of a system. 

Due to its low density, the storage of hydrogen at reasonable energy densities poses a technical and economic challenge. This chapter is dedicated to the storage of hydrogen in the pure form, which is defined by the thermodynamic variables of state and thus can best be analyzed on the basis of a depiction of these variables like the T-s-diagram. 

Since Jri - is dependent only on the values of the variables of state at the beginning and end of the process, it follows that the work done by the system is the same for all reversible and isothermal processes, which bring the system from state I to state 2. 

The molecular DFT approach [7, 8] places to our disposal the contributions of the free energy, the solid-fluid-interactions and the chemical potential to the grand potential functional on the basis of suitable model conceptions. The final functional expression fl[p] can be differentiated at fixed wall potential v (r,w) and variables of state T,p in order to yield an analytically given relation which enables the calculation of the equilibrated density profile p (z,oj). 

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

In the equilibrium thermodynamics, the physical properties of the system are fully identified by the fundamental thermodynamic potential / = /(oq,. .., xn) as a real-valued function of n real variables, which are called the variables of state. The macroscopic state of the system is fixed by the set of independent variables of state. x=(oq,. .., xn). Each variable of state x(, which is related to the certain thermodynamic quantity, describes some individual property of the system. The first and the second partial derivatives of the thermodynamic potential with respect to the variables of state define the thermodynamic quantities (observables) of the system, which describe other individual properties of this system. The first differential and the first partial derivatives of the fundamental thermodynamic potential with respect to the variables of state can be written as... 

If the function f(xv. .., xn) is convex (concave) on squadratic form (Eq. (2)) in s variables is positive definite (negative definite). The quadratic form (Eq. (2)) in s variables for which is positive definite (negative definite) if [18]... 

Solving this system of equations, we obtain m functions of the variables of state,... 

Then m -extensive and n-m -intensive variables of state satisfy the additivity relations [6,15]... 

Note that the zero law of thermodynamics is expressed by Eqs. (21) and (25) when the temperature T is a function or the second equation of Eq. (23) when temperature T is a variable of state. 

---