Intensive quantities

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Integration of Eq. UI-87 holding constant the intensive quantities T, /i, and 7 gives... 

Units Free energy is an extensive quantity, but the standard free energy for 1 mol AGt gK (in units of J mol" ) defined above is an intensive quantity. 

Pressure is a potential, and as such is an intensive quantity. We will see later that it is a driving force that transfers energy in the form of work into or out of a system. 

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

In the display of the field in Section V, we apply the above separation into parts generated by the various components of the shielding tensor. The secular part of the shielding vector, eqs.(23,24), and the intensity quantities / (R us) and 7 (R ujg), obtained from eq.(21) as discussed above, lend themselves immediately to the response graph technique described in ref. [14], as illustrated in Figures (1-3,5), while the antisymmetry vector is illustrated in Figure 4. 

The mass density of a substance is an intensive quantity defined by ... 

The heat capacity Cv is an extensive quantity, so its value depends on how much of a material we want to warm up. As chemists, we usually want a value of Cv expressed per mole of material. A molar heat capacity is an intensive quantity. 

Comparison of Equations (1) and (2) shows that the chemical potentials are intensive quantities, that is, they do not depend on the amount of each species, because if all the nt are increased in the same proportion at constant T and p, the /x, must remain unchanged for G to increase in the same rate as the nt. This invariance property of the /x, is of the utmost importance in restricting the possible forms that the /x- may take. 

The current 7 is an extensive quantity, in that it depends on the size of the electrode. For this reason, the reaction rate is conveniently referred to the unit surface area (7/S=j, current density). Even so, the current density continues to be an extensive quantity if referred to the geometric (projected) surface area since electrodes are as a rule rough and the real surface does not coincide with the geometric surface [23]. Conversely, b is an intensive quantity, in that it depends only on the reaction mechanism and not on the size of the electtode. The term b is the most important kinetic parameter in electrochemistry also because of the easy and straightforward procedure for its experimental determination. Most electrode mechanisms can be resolved on the basis of Tafel lines only. 

The advantage of the chemical potential over the other thermodynamic quantities, U, H, and G, is that it is an intensive quantity—that is, is independent of the number of moles or quantity of species present. Internal energy, enthalpy, free energy, and entropy are all extensive variables. Their values depend on the extent of the system—that is, how much there is. We will see in the next section that intensive variables such as p., T, and P are useful in defining equilibrium. 

Similarly, the superficial velocity v or vq of the gas throughput as an intensity quantity is a reliable scale-up criterion only in mass transfer in gas/liquid systems in bubble columns. In mixing operations in bubble columns, requiring the whole liquid content be back mixed (e.g., in homogenization), this criterion completely loses its validity (10). 

The process parameters must be formulated as intensive quantities. In appliances where liquid throughput q and the power input P are separated from each other as two freely adjustable process parameters, the volume-related power input PjV and the period of its duration (r= V/q) must be considered ... 

Since the density of a pure solid is an intensive quantity (the concentration of a pure solid will also be a constant), it is not written. 

On the other hand, it may be expected that, after the transient period (1.3), an approximation method could be based on the idea that P is a smoothly varying function of X with a width of order Q. Accordingly we now choose as our variable the intensive quantity x = X/Q and write the... 

If an extensive quantity can be exchanged between two bodies, a condition necessary for equilibrium is that the conjugate potential, which is an intensive quantity, must have the same value throughout both bodies. 

Therefore, any result that follows from considerations of the form of Fick s second law applies to evolution of heat as well as concentration. However, the thermal and mass diffusion equations differ physically. The mass diffusion equation, dc/dt = V DVc, is a partial-differential equation for the density of an extensive quantity, and in the thermal case, dT/dt = V kVT is a partial-differential equation for an intensive quantity. The difference arises because for mass diffusion, the driving force is converted from a gradient in a potential V/u to a gradient in concentration Vc, which is easier to measure. For thermal diffusion, the time-dependent temperature arises because the enthalpy density is inferred from a temperature measurement. 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

We have limited the discussion here to three different quantities each with a different composition variable. Of course, there are other intensive quantities, such as the volume of a solution containing a fixed quantity of solvent, that could be used to express the same data. Moreover, each intensive... 

The values of extensive quantities (in contrast to intensive quantities) depend on the system size (the amount of solute depends on the volume of the solution taken its concentration does not). 

Both of these partial derivatives are divided by V to make them intensive quantities. The SI units of a are K-1 and those of k are Pa 1. A negative sign is used in the definition of k, because volumes always decrease as pressure increases, and we would prefer to tabulate positive quantities. 

Internal energy is an extensive property of a system. If we double the size of a system, keeping intensive variables such as temperature and pressure constant, we double the system s internal energy. If we divide the internal energy of a system by the number of moles in the system, we obtain the molar internal energy, Um = VIn, which is an intensive quantity. Other molar properties, such as the molar volume, are also indicated by the subscript m. 

This equation 5.20, called the Gibbs-Duhem equation, is unique among a variety of the thermodynamic equations of state in that the characteristic variables are all intensive quantities, each multiplied by its conjugate extensive quantity. 

Extensive quantities show an exact proportionality to n, only when intensive quantities, like T and P, remain constant. 

We have made mention earlier (Frame 5, section 5.4) albeit very briefly, of the definition of the chemical potential, //, of a substance i. For a two component system having components labelled as 1 and 2, this intensive quantity (Frame 1, section 1.3) is defined as the rate of change of Gibbs energy per mole of substance present ... 

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