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Legendre transforms constant derivation

The fractional saturation of tetramer YT and the fractional saturation of dimer YD are functions only of [02] at specified T, P, pH, etc., as shown by equations 7.1-18 and 7.3-6. However, since the tetramer form is partially dissociated into dimers, the fractional saturation of heme Y is a function of both [02] and [heme]. Ackers and Halvorson (1974) derived an expression for the function Y([02], [heme]). When Legendre transforms are used, a simpler form of this function is obtained, and it can be used to derive limiting forms at high and low [heme]. These limiting forms are of interest because they show that if data can be obtained in regions where Y is linear in some function of [heme], extrapolations can be made to obtain YT and YD. These fractional saturations can be analyzed separately to obtain the Adair constants for the tetramer and the dimmer (Alberty, 1997a). [Pg.129]

The equilibrium relations of the preceding section were derived on the assumption that the charge transferred Q can be held constant, but that is not really practical from an experimental point of view. It is better to consider the potential difference between the phases to be a natural variable. That is accomplished by use of the Legendre transform (Alberty, 1995c Alberty, Barthel, Cohen, Ewing, Goldberg, and Wilhelm, 2001)... [Pg.148]

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. [Pg.148]

We have postulated the existence of a function of state, the entropy, which achieves a maximum at constant U and V for a system at equilibrium. Using Legendre transforms we have derived the additional functions of state from the following fundamental equation ... [Pg.107]

Note that in Eq. (1,187), X and X2 must remain constant. In. Eq. (1.187), the order of the Jacobian has been reduced from (c + 2) to c. The above process can be reversed, going from order c to c + 2, which is also true from Eq. (1.187). The transformation properties of Jacobians is useful in the manipulation of Legendre transformation derivatives and can be utilized by the introduction of a new set of independent variables... [Pg.37]


See other pages where Legendre transforms constant derivation is mentioned: [Pg.419]    [Pg.67]    [Pg.469]    [Pg.470]    [Pg.691]    [Pg.41]    [Pg.39]    [Pg.304]   
See also in sourсe #XX -- [ Pg.74 , Pg.75 ]




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