Logarithms, natural

The difference in 5 values for two substances (P,Q) measured against a standard substance is approximately equal to 1000 times the natural logarithm of their fractionation factor (app). 

Here, t is the time taken for to fall to 1 /e of its initial value (where e is the base of natural logarithms) and is referred to as the lifetime of state n. If spontaneous emission is the only process by which M decays, comparison with Equation (2.9) shows that... 

H2O/100 kg of adsorbent. At equilibrium and at a given adsorbed water content, the dew point that can be obtained in the treated fluid is a function only of the adsorbent temperature. The slopes of the isosteres indicate that the capacity of molecular sieves is less temperature sensitive than that of siUca gel or activated alumina. In another type of isostere plot, the natural logarithm of the vapor pressure of water in equiUbrium with the desiccant is plotted against the reciprocal of absolute temperature. The slopes of these isosteres are proportional to the isosteric heats of adsorption of water on the desiccant (see... 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

Napierian (natural) logarithm base Eulers constant... 

Napierian (natural) logarithm of pi, base e Briggsian (common logarithm of pi, base 10... 

The common logarithm (base 10) is denoted log a or logi(/7, The natural logarithm (base e) is denoted In a (or in some texts loge ). 

Dijferential Operations The following differential operations are valid /, g, are differentiable functions of x, c is a constant e is the base of the natural logarithms. 

In Natural logarithm (logarithm to the base e) Dimensionless Dimensionless Dimensionless... 

Figure 6.8. Plot of the quasi-static reloaded yield stress of shock-loaded copper versus the natural logarithm of residual strain for a 10 GPa symmetric shock with 1 /is pulse duration.

By plotting the natural logarithm (In) of against the reciprocal of the absolute temperature (1/T) at constant stress, as shown in Fig. 17.6, we find that ... 

Equation 1-218 can be expressed in the form Y = AX . Using natural logarithm, Equation Y = AX gives... 

Taking the natural logarithm of both sides of Equation 3-90 gives lnti/2=ln(2"-i-l) -H(l-n)lnCAo-lnk-ln(n-l) (3-92)... 

Treating k2/(k2 - kj) as an exponent and removing the natural logarithm gives... 

E = Joint efficiency in cylindrical or spherical shells or ligaments between openings (see ASME Code Par.lJW-12 or UG-53) e = natural logarithm base, e = 2.718 e, = TNT equivalent (explosion) (see Table 7-26)... 

B = parameter in correlating equation In = natural logarithm log = logarithm to the base 10 N = actual theoretical stages required for a given separation... 

AP(j = dry bed pressure drop, in. water/ft AP = operating pressure drop, in. liquid/ft e = base of natural logarithms Xi,X2 = curve fit coefficients for C2, Table 9-32. 

In = Natural logarithm to base e MTD = Mean Temperature Difference, °F, see Figure 10-33 = Log mean temperature difference LTD = Atj = Least terminal temperature difference GTD = Atj = Greater terminal temperature difference... 

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