Homogeneous functions first-order

This property is described by the statement that entropy is a homogeneous first-order function of the extensive parameters. The expression is readily interpreted to define molar entropy (s = S/N), internal energy (it) and... 

Equation (6.27) merely says that if the independent extensive arguments of U are multiplied by A [cf. (6.25b-d)], then U itself must be multiplied by the same factor [cf. (6.25a)]. [Mathematically, the property (6.27) identifies the internal energy function (6.26) as a homogeneous function of first order, and the consequence to be derived is merely a special case of what is called Euler s theorem for homogeneous functions in your college algebra textbook.]... 

There exists a homogeneous first order state variable, the entropy, which for isolated systems (those having constant U and V) achieves a maximum when the system is at stable equilibrium. Entropy and its derivatives are single-valued, continuous and differentiable functions of the other state variables. Entropy is a monotonically increasing function of the energy U. 

Another example for a homogeneous function is a homogeneous polynomial. For instance, (x + y) is a homogeneous polynomial of tth order. In addition, various functions in mechanics and thermodynamics are homogeneous functions of first order. 

Because of the thermodynamic similarity, i.e., the volume is a homogeneous function of first order... 

The last identity in Eq. (5.22) arises from the Euler relation of homogeneous functions of first order U = TS - pV + iJ,n and the definition of the enthalpy as... 

Generalized-function formulations of GPT for homogeneous systems are the source of sensitivity functions for different integral parameters Equation (189) for reactivity worths, and Eq. (162) for ratios of linear and bilinear functionals. The first-order perturbation theory expression for reactivity [Eq. (132)] can also be used for sensitivity studies. 

It should be noted that the von Mises (/i = s/7 ) and Drucker-Prager (/i = v + 0111) yield functions are first order homogeneous functions however... 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

Like the entropy expression the fundamental relation as a function of U is also a first-order homogeneous function, such that for constant A,... 

A kinetics study of the homogeneous reaction with [Rh(CO)2(amine)2]PF6 indicated that (1) the H2 rate was close to first order in Pc0 (2) the Arrhenius plots were segmented, such that E above 120 °C was 1.8 times higher than Ea below 120 °C. This was explained by a change in the rate limiting step (3) a decrease in H2 rate as a function of increasing Rh concentration, suggesting the presence of... 

Applying the usual steady-state treatment for consecutive first-order reactions kt at 16 torr pressure over the temperature range 597-701 °C is given by 1.8 x 1011 exp(—47,000/Kr) sec Within experimental error, reactions (1) and (2) were homogeneous processes. However, both k2 and k2 were functions of the total pressure in the system. This dependence is shown in Fig. 1. The methyl zinc decomposition is apparently in its second-order region. Therefore, assuming four effective oscillators and a mean temperature of 1050 °K, = Eohs.+i nRT... 

In the CE mechanism (Scheme 2.2), a first-order (or pseudo-first-order) homogeneous reaction precedes the electron transfer step. In the case where the initial electron transfer is fast enough not to interfere kinetically, the electrochemical response is a function of two parameters the first-order (or pseudo-first-order) equilibrium constant, K, and a dimensionless kinetic... 

Quantitative estimation of ventilation by indirect methods in mussels requires four assumptions (16) a) reduction of concentration results from uptake, b) constant ventilation (pumping) rate, c) uptake of a constant percentage of concentration (first order process), d) homogeneity of the test solution at all times. Our transport studies have utilized antipy-rine (22, 23) a water soluble, stable chemical of low acute toxicity to mussels. It is readily dissolved in ocean water or Instant Ocean and is neither adsorbed nor volatilized from the 300 ml test system. Mussels pump throughout the 4 hour test period and this action is apparently sufficient to insure homogeneity of the solution. Inspection of early uptake and elimination curves (antipyrine concentration as a function of time) prompted use of Coughlan s equation (16) for water transport. 

Consider a crystal which is in equilibrium having n chemical components (k = 1,2,..., ). We can define (at any given P and T) a Gibbs function, G, as a homogeneous function that is first order in the amount of components... 

It is always convenient to use intensive thermodynamic variables for the formulation of changes in energetic state functions such as the Gibbs energy G. Since G is a first order homogeneous function in the extensive variables V, S, and rtk, it follows that [H. Schmalzried, A.D. Pelton (1973)]... 

As the Gibbs energy is a first-order homogenous function of the extensive variables A7 and n. the application of Euler s theorem yields... 

Since a and 3 are represented by 4 x 4 matrices, the wave function / must also be a four-component function and the Dirac wave equation (3.9) is actually equivalent to four simultaneous first-order partial differential equations which are linear and homogeneous in the four components of P. According to the Pauli spin theory, introduced in the previous chapter, the spin of the electron requires the wave function to have only two components. We shall see in the next section that the wave equation (3.9) actually has two solutions corresponding to states of positive energy, and two corresponding to states of negative energy. The two solutions in each case correspond to the spin components. 

The substitution moment Isy differs from the equilibrium moment ley by first order terms of the expansion. Since g is a homogeneous function of degree one-half of the atomic masses [24], the second term on the right-hand side of Eq. 91a is, by Euler s theorem ... 

Example. Nitrous oxide, N20, decomposes into N2 and 02, the reactants and the products being all gaseous, (This is an example of a homogeneous gaseous reaction). If the reaction is first-order, develop expression for the rate constant as a function of time, initial pressure and the total pressure. 

Properties like mass m and volume Vare defined by the system as a whole. Such properties are additive, and are called extensive properties. Separation of the total change for a species into the external and internal parts may be generalized to any extensive property. All extensive properties are homogeneous functions of the first order in the mass of the system. For example, doubting the mass of a system at constant composition doubles the internal energy. 

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