Calculating Concentrations for Liquids or Solids

Another frequent mistake among students is to try to apply the ideal gas law to calculate the concentrations of species in condensed-matter phases (e.g., liquid or solid phases). Do not make this mistake] The ideal gas law only applies to gases. To calculate concentrations for liquid or solid species, information about the density (pj) of the liquid or solid phase is required. Both mass densities and molar densities (concentrations) as well as molar and atomic volumes may be of interest. The complexity of calculating these quantities tends to increase with the complexity of the material under consideration. In this section, we will consider three levels of increasing complexity pure materials, simple compounds or dilute solutions, and more complex materials involving mixtures of multiple phases/compounds. 

Mass densities (g/cm ) are readily available for most pure materials. For example, the density of Si is psi = 2.33 g/cm while the density of Si02 is Psi02 2.65 g/cm. Calculating the molar densities (molar concentrations) and molar volumes of pure Si and pure Si02 (or any other pure material) from their mass densities and their molecular weights is quite straightforward  

Molar concentrations and molar volumes can be converted into number densities ( ) and atomic volumes Q., by multiplying or dividing by Avogadro s number, 

These quantities correspond to the number of atoms or molecules of species i per unit volume (e.g., no./cm ), and the volume associated with a single atom or molecule of a species i (e.g., cm ), respectively. Thus, the number density and atomic volume of Si02 are  

2 Calculating Densities/Concentrations in Stoichiometric Compounds or Dilute Solutions 

---

When calculating concentrations for liquid or solid species (which will be frequently encountered in materials kinetics problems), the ideal gas law DOES NOT APPLY Instead, information about the density (or atomic structure and packing) is needed. These calculations can become increasingly complicated depending on the number of phases/components involved. The last section of the chapter provides detailed examples of such calculations. Mastering these concepts will be extremely useful as we move forward in our exploration of materials kinetics, as most kinetic equations involve species concentration. 

A crucial feature of PNC experiments in atoms, molecules, liquids or solids is that for interpretation of measured data in terms of fundamental constants of the P,T-odd interactions, one must calculate those properties of the systems, which establish a connection between the measured data and studied fundamental constants (see section 4). These properties are described by operators heavily concentrated near or on heavy nuclei they cannot be measured and their theoretical study is not a trivial task. During the last several years the significance of (and requirement for) ab initio calculation of electronic structure providing a high level of reliability and accuracy in accounting for both relativistic and correlation effects has only increased (see sections 3 and 10). 

The numerical value of an electrode potential depends on the nature of the particular chemicals, the temperature, and on the concentrations of the various members of the couple. For the purposes of reference, half-cell potentials are taken at the standard states of all chemicals. Standard state is defined as 1 atm pressure of each gas (the difference between 1 bar and 1 atm is insignificant for the purposes of this chapter), the pure substance of each liquid or solid, and 1 molar concentrations for every nongaseous solute appearing in the balanced half-cell reaction. Reference potentials determined with these parameters are called standard electrode potentials and, since they are represented as reduction reactions (Table 19-1), they are more often than not referred to as standard reduction potentials (E°). E° is also used to represent the standard potential, calculated from the standard reduction potentials, for the whole cell. Some values in Table 19-1 may not be in complete agreement with some sources, but are used for the calculations in this book. 

Substituting the appropriate ideal expression for the activity of gaseous or dissolved species from Equation 14.8a or 14.8b leads to the forms of the mass action law and the equilibrium constant K already derived earlier in Section 14.3 for reactions in ideal gases or in ideal solutions. We write the mass action law for reactions involving pure solids and liquids and multiple phases by substituting unity for the activity of pure liquids or solids and the appropriate ideal expression for the activity of each gaseous or dissolved species into Equation 14.9. Once a proper reference state and concentration units have been identified for each reactant and product, we use tabulated free energies based on these reference states to calculate the equilibrium constant. 

Similar results may be obtained for convective mass transfer. If a fluid of species concentration Ci, flows over a surface at which the species concentration is maintained at some value Clilv C, transfer of the species by convection will occur. Species 1 is typically a vapor that is transferred into a gas stream by evaporation or sublimation at a liquid or solid surface, and we are interested in determining the rate at which this transfer occurs. As for the case of heat transfer, such a calculation may be based on the use of a convection coefficient [3, 5]. In particular we may relate the mass flux of species 1 to the product of a transfer coefficient and a concentration difference... 

It can be seen that it is AF which we wish to know. But it should be noted that AF does not depend solely on the nature of the chemical reaction. It also depends on the concentrations of the reactants, and the direction of the reaction will also depend on these concentrations. It would not be practicable to compile tables of AF for all concentrations, so its value is determined under defined conditions liquids or solids in the pure state, gas at a pressure of 1 atmosphere, solutions at OTM, temperature 25 . Concentrations thus defined are assigned a value of unity and, from the AF° value thus defined, it is possible to calculate values of AF for other conditions. 

Source Considerations. Many CVD sources, especially sources for or-ganometallic CVD, such as Ga(CH3)3 and Ga(C2H5)3, are liquids at near room temperatures, and they can be introduced readily into the reactor by bubbling a carrier gas through the liquid. In the absence of mass-transfer limitations, the partial pressure of the reactant in the gas stream leaving the bubbler is equal to the vapor pressure of the liquid source. Thus, liquid-vapor equilibrium calculations become necessary in estimating the inlet concentrations. For the MOCVD of compound-semiconductor alloys, the computations have also been used to establish limits on the control of bubbler temperature to maintain a constant inlet composition and, implicitly, a constant film composition (79). Similar gas-solid equilibrium considerations govern the use of solid sources such as In(CH3)3. 

For the case with porous particles, the pore fluid can be treated as a mass transfer medium rather than a separate phase thus enabling it to be combined with the bulk fluid in the overall mass balance. Under plug flow transfer conditions, at the end of each time increment, the pore fluid was assumed to remain stagnant, and only the bulk fluid was transferred to the next section. Based on these assumptions and initial conditions, the concentrations of the polypeptide or protein adsorbate in both liquid and solid phase can be calculated. The liquid phase concentration in the last section C , is the outlet concentration. The concentration-time plot, i.e., the breakthrough curve, can then be constructed. Utilizing this approach, the axial concentration profiles can also be produced for any particular time since the concentrations in each section for each complete time cycle are also derived. 

Strictly speaking, the letters in brackets represent activities, but we will usually follow the practice of substituting molar concentrations for activities in most calculations. Thus, if some participating species A is a solute, [A] is the concentration of A in moles per liter. If A is a gas, [A] in Equation 18-12 is replaced by p, the partial pressure of A in atmospheres. If A is a pure liquid, a pure solid, or the solvent, its activity is unity, and no term for A is included in the equation. The rationale for these assumptions is the same as that de.scribed in Section 9B-2, which deals with equilibrium-constant expressions. 

---