Molar transition quantities

The quantity AvapTf is the molar enthalpy change for the reversible process in which liquid changes to gas at a temperature and pressure at which the two phases coexist at equilibrium. This quantity is called the molar enthalpy of vaporization. Since the pressure is constant during the process, Ayap f is equal to the heat per amount of vaporization (Eq. 5.3.8). Hence, AyapTif is also called the molar heat of vaporization. 

The first edition of this book used the notation AyapFfm. with subscript m, in order to make it clear that it refers to a molar enthalpy of vaporization. The most recent edition of the lUPAC Green Book recommends that Ap be interpreted as an operator symbol  

Ap = 9/9 p, where p is the abbreviation for a process at constant T and p (in this case vap ) and p is its advancement. Thus AyapTf is the same as (dTf/d vaplup where vap is the amount of liquid changed to gas. 

Here is a list of symbols for the molar enthalpy changes of various equilibrium phase transitions  

AfusT/ molar enthalpy of fusion (solid liquid) 

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The subscripts in the list above are also used for other molar transition quantities. Thus, there is the molar entropy of vaporization Ayap5, the molar internal energy of sublimation Asubf. and so on. 

A molar transition quantity of a pure substance is the change of an extensive property divided by the amount transferred between the phases. For example, when an amount n in a liquid phase is allowed to vaporize to gas at constant T and p, the enthalpy change is... 

A molar property of a phase, being intensive, usually depends on two independent intensive variables such as T and p. Despite the fact that Ayap/f is the difference of the two molar properties H and H], its value depends on only one intensive variable, because the two phases are in transfer equilibrium and the system is univariant. Thus, we may treat Ayapi/ as a function of T only. The same is true of any other molar transition quantity. 

Let X represent one of the thermodynamic potentials or the entropy of a phase. The standard molar transition quantities Avap3t° = A°(g) — Xm(l) and Asub ° = m(g) — Zm(s) are functions only of T. To evaluate Avap3t° or Asub t at a given temperature, we must calculate the change of Xm for a path that connects the standard state of the liquid or solid with that of the gas. The simplest choice of path is one of constant temperature T with the following steps ... 

Partial molar thermodynamic quantities close to bulk water values. Region of rapid rise in adsorption isotherm. Transition region in heat capacity... 

These quantities (which are standard molar quantities) describe the process initial state transition state... 

The quantity of primary interest in our thermodynamic construction is the partial molar Gibbs free energy or chemical potential of the solute in solution. This chemical potential reflects the conformational degrees of freedom of the solute and the solution conditions (temperature, pressure, and solvent composition) and provides the driving force for solute conformational transitions in solution. For a simple solute with no internal structure (i.e., no intramolecular degrees of freedom), this chemical potential can be expressed as... 

One would expect the enthalpy of sublimation (d) to be the largest of the four quantities cited. Molar heat capacities are quite small, on the order of fractions of a kilojoule per mole-degree. (Remember that specific heats have values of joules per gram-degree.) All of the heats of transition (or latent heats) are positive numbers and on the order of kilojoules per mole. Since the heat of sublimation is the sum of the heat of fusion and the heat of vaporization, AHsubl must be the largest of the three. 

The standard enthalpy difference between reactant(s) of a reaction and the activated complex in the transition state at the same temperature and pressure. It is symbolized by AH and is equal to (E - RT), where E is the energy of activation, R is the molar gas constant, and T is the absolute temperature (provided that all non-first-order rate constants are expressed in temperature-independent concentration units, such as molarity, and are measured at a fixed temperature and pressure). Formally, this quantity is the enthalpy of activation at constant pressure. See Transition-State Theory (Thermodynamics) Transition-State Theory Gibbs Free Energy of Activation Entropy of Activation Volume of Activation... 

Note 3 Numerical values of the molar entropy of transition should be given as the dimensionless quantity AxyS/R where R is the gas constant. 

Figure 17. Specific volume Vt and isothermal compressibility (at the glass transition temperature Tg) calculated from the LCT as a function of the inverse number l/M of united atom groups in single chains for constant pressure (P = I atm 0.101325 MPa) F-F and F-S polymer fluids. Both quantities are normahzed by the corresponding high molar mass limits (i.e., by...

All upward radiative transitions in Figure 3.23 are absorptions which can promote a molecule from the ground state to an excited state, or from an excited state to a higher excited state. We have seen that the probability of these transitions is related ultimately to the transition moment between the two states and thereby to the Einstein coefficient A. In practice two other related quantities are used to define the intensity5 of an absorption, the oscillator strength f and the molar decadic extinction coefficient e. 

If a substance undergoes a transformation from one physical stale to another, such as a polymorphic transition, the fusion or sublimation of a solid, or the vaporization of a liquid, the heat adsorbed hy the substance during the transformation is defined as the latent heat of transformation (transition, fusion, sublimation or vaporization). It is equal in the enthalpy change of the process, which is the difference between the enthalpy of the substance in the two states at (he temperature of the transformation. For the purpose of thcrmochemical calculations, i( is usually reported as a molar quantity with die units of calories (or kilocalories) per mule (or gram formula weight). The symbol L or AH. with a subscript i.f (or in), s. and n is commonly used and the value is usually given at the equilibrium temperature of the transformation under atmospheric pressure, or at 25 C. 

The molar extinction coefficient in ND8 is about 10% larger than that previously reported for NH8 solutions (9, 10). Perhaps the density of the solutions could account for the discrepancy since the densities of NH8 and ND8 at —33.7° C. are 0.68 and 0.80 g./cc., respectively (12). Since the solutions under consideration are very dilute, their densities are approximately that of the solvent itself. However, the molar volume of the solvent is a more significant parameter since the solvation of the electron would be expected to be the same in each solvent. The molar volume of NH8 (—33.7 ° C.) is 24.3 cc./mole and that of ND (—33.7 ° C.) is 25.0 cc./mole (12). These quantities differ by only 3% and are in the wrong direction to account for the observed difference in intensity. It is possible that the transition responsible for the absorption is less forbidden in liquid ND8 than in liquid NH8, but the most likely explanation is that some decomposition to amide had already occurred when the measurements in NH8 were made. 

A rough measure of the intensity of an electronic transition is provided by the maximum value of the molar extinction coefficient. A physically more meaningful quantity is the total area under the absorption band, given by the integral /cdi>, or the oscillator strength... 

The oxidation of a substrate by any Pd species in principle is a stoichiometric reaction, consuming first of all molar amounts of the Pd present, thus forming equivalent quantities of Pd°. If catalytic oxidative carbonylations are required with respect to the palladium compound, appropriate conditions for the reoxidation of Pd have to be found. This may be achieved by the presence of suitable co-catalysts, for example of certain transition metal salts, which are capable of changing their oxidation state. 

The two-dimensional gas model assumes no mutual interaction of the adsorbed molecules. It is believed that the adsorbent creates a constant (across the surface) adsorption potential. Thus, in the framework of statistical thermodynamics, the model describes adsorption as the transition of a gas with three translational degrees of freedom into an adsorbed state with one vibrational and two translational degrees. Assuming ideal behavior and using molar quantities, one obtains the standard entropy in the adsorbed phase as the sum of the translational and vibrational entropies from Eqs. 5.28 and 5.29 ... 

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