Standard state radius

The term 7 7 ln p/p° is clearly the chemical potential of a surface of radius r with respect to a flat surface of the same material as standard state. It follows that the difference in chemical potential between two surfaces, where... 

For estimation of thermodynamic properties of dissolved species, one can use the Entropy Correspondence Principle ( ), where the entropy of an ion at a given temperature is regarded as a function of the charge, the dielectric constant, mass, radius, and other variables. The function depends mainly upon the choice of the standard state, solvent, and temperature. The temperature dependency of entropy was derived based on the above principles and experimental data. By conducting the a square regression on Criss-Cobble s data ( ), we obtained the following eqtiation for calculating the entropies of species in aqueous solution. 

In these equations, the referenee states of H tq) are, by convention, equal to zero as are the functions AHf e[g ) and AG ( ( ,). The absolute entropies for the gaseous ions are calculated from statistical mechanics (Bratsch and Lagowski 1985a) and agree fairly well with the experimental values reported by Bertha and Choppin (1969), who interpreted the S-shaped dependence of standard state entropies on ionic radius in terms of a change in the overall hydration of the cation across the lanthanide series. Hinchey and Cobble (1970) proposed that this S-shaped relationship was an artifact of the method of data treatment and calculated a set of entropies from lanthanide... 

It was shown later that a mass transfer rate sufficiently high to measure the rate constant of potassium transfer [reaction (10a)] under steady-state conditions can be obtained using nanometer-sized pipettes (r < 250 nm) [8a]. Assuming uniform accessibility of the ITIES, the standard rate constant (k°) and transfer coefficient (a) were found by fitting the experimental data to Eq. (7) (Fig. 8). (Alternatively, the kinetic parameters of the interfacial reaction can be evaluated by the three-point method, i.e., the half-wave potential, iii/2, and two quartile potentials, and ii3/4 [8a,27].) A number of voltam-mograms obtained at 5-250 nm pipettes yielded similar values of kinetic parameters, = 1.3 0.6 cm/s, and a = 0.4 0.1. Importantly, no apparent correlation was found between the measured rate constant and the pipette size. The mass transfer coefficient for a 10 nm-radius pipette is > 10 cm/s (assuming D = 10 cm /s). Thus the upper limit for the determinable heterogeneous rate constant is at least 50 cm/s. 

To interpret the electronic structure of a material, it is often useful to understand what states are important in the vicinity of specific atoms. One standard way to do this is to use the local density of states (LDOS), defined as the number of electronic states at a specified energy weighted by the fraction of the total electron density for those states that appears in a specified volume around a nuclei. Typically, this volume is simply taken to be spherical so to calculate the LDOS we must specify the effective radii of each atom of interest. This definition cannot be made unambiguously. If a radius that is too small is used, information on electronic states that are genuinely associated with the nuclei will be missed. If the radius is too large, on the other hand, the LDOS will include contributions from other atoms. 

Symbol Ca atomic number 20 atomic weight 40.078 a Group IIA (Group 2) alkaline-earth metaUic element ionic radius 1.06 A (Ca2+) electron configuration [Ar]4s2 valence state +2 standard electrode potential, E° = -2.87V stable isotopes and their abundance Ca-40 (97.00%), Ca-44 (2.06%) Ca-42 (0.64%), Ca-48 (0.18%), Ca-43 (0.145%), and Ca-46 (0.003%) also the element has six unstable isotopes of which Ca-41 has the longest half-life, l.lxlO yr (decay mode electron capture), and Ca-38 has shortest half life 0.66 sec (P-decay). 

Symbol Nd atomic number 60 atomic weight 144.24 a rare earth lanthanide element a hght rare earth metal of cerium group an inner transition metal characterized by partially filled 4/ subshell electron configuration [Xe]4/35di6s2 most common valence state -i-3 other oxidation state +2 standard electrode potential, Nd + -i- 3e -2.323 V atomic radius 1.821 A (for CN 12) ionic radius, Nd + 0.995A atomic volume 20.60 cc/mol ionization potential 6.31 eV seven stable isotopes Nd-142 (27.13%), Nd-143 (12.20%), Nd-144 (23.87%), Nd-145 (8.29%), Nd-146 (17.18%), Nd-148 (5.72%), Nd-150 (5.60%) twenty-three radioisotopes are known in the mass range 127-141, 147, 149, 151-156. 

Symbol Ni atomic number 28 atomic weight 58.693 a transition metal element in the first triad of Group VIll(Group 10) after iron and cobalt electron configuration [Ar]3d 4s2 valence states 0, -i-l, +2, and -f-3 most common oxidation state +2 the standard electrode potential, NF+ -1- 2e Ni -0.237 V atomic radius 1.24A ionic radius (NF+) 0.70A five natural isotopes Ni-58 (68.08%), Ni-60 (26.22%), Ni-61 (1.14%), Ni-62 (3.63%), Ni-64 (0.93%) nineteen radioactive isotopes are known in the mass range 51-57, 59, 63, 65-74 the longest-lived radioisotope Ni-59 has a half-life 7.6x10 years. 

Symbol Th atomic number 90 atomic weight 232.04 an actinide series radioactive element electron configuration XRn]6d27s2 valence state +4 atomic radius 1.80 A ionic radius, Th4+ 1.05 A for coordination number 8 standard electrode potential, E° for Th4+ -1- 4e Th is -1.899V all isotopes are radioactive the only naturally-occurring isotope, Th-232, ti/2 1.4xl0i° year twenty-six isotopes are known in the mass range 212-237. 

Symbol Sn atomic number 50 atomic weight 118.69 a Group IV A (Group 14) metaUic element of carbon family electron configuration Kr] 4di°5s25p2 valence states +2, +4 atomic radius 1.41 A electronegativity 1.7 standard electrode potential, E° for Sn2+ -i- 2e Sn is -0.1375 V ten naturally-occurring sta-... 

The most stable state of nitrogen in acidic solution is the ammonium ion, NH4(aq), which is isoelectronic with CH4 and H30+. It is a tetrahedral ion with strong N-H bonds. The mean N-H bond enthalpy in NH4(aq) is 506 kJ mol 1 (that of the O-H bonds in H30 + is 539 kJ mol" ). The enthalpy of hydration of the ammonium ion is — 345 kJ mol V This value placed into the Born equation (3.32) gives an estimate of the radius of the ammonium ion of 135 pm, a value insignificantly different from its thermochemical radius of 136 pm. The value is comparable to that estimated for the smaller H30+ ion (99 pm) from its more negative enthalpy of hydration (— 420 kJ mol -see Section 2.6.1). The proton affinity of the ammonia molecule is of interest in a comparison of its properties with those of the water molecule. The proton affinity is defined as the standard enthalpy change for the reaction ... 

As it was mentioned above, up to now only the dynamic interaction of dissimilar particles was treated regularly in terms of the standard approach of the chemical kinetics. However, our generalized approach discussed above allow us for the first time to compare effects of dynamic interactions between similar and dissimilar particles. Let us assume that particles A and B attract each other according to the law U v(r) = — Ar-3, which is characterized by the elastic reaction radius re = (/3A)1/3. The attraction potential for BB pairs is the same at r > ro but as earlier it is cut-off, as r ro. Finally, pairs AA do not interact dynamically. Let us consider now again the symmetric and asymmetric cases. In the standard approach the relative diffusion coefficient D /D and the potential 1/bb (r) do not affect the reaction kinetics besides at long times the reaction rate tends to the steady-state value of K(oo) oc re. 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

---