Logarithm differentiation

The average energy of the molecules follows by logarithmic differentiation ... 

The reader may now wish to verify that the activation energy calculated by logarithmic differentiation contains a contribution Sk T/l in addition to A , whereas the pre-exponential needs to be multiplied by the factor e in order to properly compare Eq. (139) with the Arrhenius equation. Although the prefactor turns out to have a rather strong temperature dependence, the deviation of a In k versus 1/T Arrhenius plot from a straight line will be small if the activation energy is not too small. 

Fig. 10.5. Logarithmic differential Th/Eu ratios plotted against stellar age. The crosses represent the average of two UMP r-process-rich stars CS 22892-052 and HD 115444 (Westin et al. 2000), and a third one BD +17° 3248 (Cowan et al. 2002) which are a kind of Rosetta stone for the r-process, assuming an age of 13 1 Gyr. Curves show predictions from various models discussed in the text.

Inserting the logarithmic differential of concentration into equation (9.4.14) gives... 

A logarithmic differentiation of this equation with respect to pressure gives us... 

As an example let us study the errors involved in a calculation of the constant fe in Eq. (V.2.5). By applying the methods of Sec. IV.6A we take the logarithmic differential of both sides of this equation. 

From this figure we may also see how variations in the reference composition and in the equilibrium constant affect the equilibrium extent. If we differentiate F(f) partially with respect to a particular component of the initial composition, say we have (by logarithmic differentiation as before)... 

The decrease of temperature with increasing altitude is called the lapse rate. Combining the hydrostatic equation and the expression for potential temperature, we can evaluate the dry adiabatic lapse rate (Td), i.e., the temperature decrease which would be associated with a vertical adiabatic displacement. Logarithmic differentiation of equation (3.9)... 

In seeking the differential coefficient of a complex function containing products and powers of polynomials, the work is often facilitated by taking the logarithm of each member separately before differentiation. The compound process is called logarithmic differentiation. 

After taking logarithms, differentiating with respect to T and using equation 3.41, the equation... 

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. 

NOTE For Nr, < 3 convective contributions which are not included may become important. Use with logarithmic couceutratiou difference (integrated form) or with arithmetic couceutratiou difference (differential form). 

From the Differential Equation Linear regression can be apphedwith the differential equation to obtain constants. Taking logarithms of Eq. (7-25),... 

Christensen differentiates Eq. (2-57) with respect to time and then takes logarithms, obtaining Eq. (2-58). 

Taking logarithms of Eq. (4.3.24) and differentiating with respect to lna yields... 

If we take the natural logarithm of both sides of Eq. (15) and differentiate with respect to one of the component velocities, say vx, we have... 

The variation of k with temperature may be determined by differentiating the logarithmic form of this equation. 

The determination of rate change of the logarithm of the neutron level, as in the source range, is accomplished by the differentiator. The differentiator measures reactor period or startup rate. Startup rate in the intermediate range is more stable because the neutron level signal is subject to less sudden large variations. For this reason, intermediate-range startup rate is often used as an input to the reactor protection system. 

LHSFs are determined at the center p of each shell. These LHSFs are then used to obtain the coupling matrix V i /nr(p p) given in Eq. (102). The coupled hyperradial equations in Eq. (101) are transformed into the coupled first-order nonlinear Bessel-Ricatti logarithmic matrix differential equation... 

---