Quantity molar

Typical notation distinguishes between intensive and extensive variables. Intensive quantities such as temperature and pressure do not scale with the system size extensive quantities such as internal energy and entropy do scale with system size. For example, if the size of a box of gas molecules is doubled and the number of molecules in the box doubles, then the internal energy and entropy double while the temperature and pressure are constant. Frequently, certain conventions are used to denote intensive versus extensive properties, such as using lowercase for intensive variables (e.g., p for pressure) and uppercase for extensive variables (e.g., S for entropy). Unfortunately, such nomenclature is not standardized and is frequently inconsistent. For example, temperature is almost always represented with an uppercase T, although it is an intensive quantity. 

Many extensive properties can be normalized by the size of the system to give specific properties, that is, properties per unit mass, or per mole, or per volume. For example, consider the extensive Gibbs free-energy change (AG xn) involved in a chemical reaction, which can be normalized to an intensive quantity (AGjjjj) via 

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If the process is carried out at constant volume, the heat evolved Qi will be equal to an energy change AE2 or, per mole of adsorbate, qi = Ae2 (small capital letters will be used to denote mean molar quantities). Alternatively, the process may be... 

In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

From cross-differentiation identities one can derive some additional Maxwell relations for partial molar quantities ... 

The high reactivity of the 5-position in 1.3-selenazoles toward electrophilic substitution was also observed on azocoupling. By reacting molar quantities of an aqueous solution of a diazonium salt with an ethanolic solution of a 2-arylamino selenazole. for example, the corresponding 2-arylamino-5 azoselenazoles are formed in a smooth reaction (100). They deposit from the deeply colored solution and form intenselv red-colored compounds after their recrystallization from a suitable solvent (Scheme 36l. 

Given that the p/Ca of imidazolium ion is 7 is a 1 M aqueous solution of imidazolium chloride acidic basic or neutraP What about a 1 M solu tion of imidazole A solution containing equal molar quantities of imidazole and imidazolium chloride ... 

Neutron magnetic moment Partial molar quantity A... 

The chemical potential is an example of a partial molar quantity /ij is the partial molar Gibbs free energy with respect to component i. Other partial molar quantities exist and share the following features ... 

Partial molar quantities have per mole units, and for Yj this is understood to mean per mole of component i. The value of this coefficient depends on the overall composition of the mixture. Thus Vj o the same for a water-alcohol mixture that is 10% water as for one that is 90% water. 

For a pure component the partial molar quantity is identical to the molar (superscript °) value of the pure substance. Thus for pure component i... 

Relationships which exist between ordinary thermodynamic variables also apply to the corresponding partial molar quantities. Two such relationships are... 

As noted above, all of the partial molar quantities are concentration dependent. It is convenient to define a thermodynamic concentration called the activity aj in terms of which the chemical potential is correctly given by the relationship... 

This is converted to a partial molar quantity by differentiation ... 

From the definition of the partial molar quantities [Eq. (8.8)] we write... 

The Gibbs-Duhem equation also follows from the definition of partial molar quantities nid/Hi + r 2d 2 0. With the Gibbs-Duhem equation, d G/dc2 becomes... 

Figure 3 is the humidity chart diagram in molar quantities where enthalpy deviations are not given. Enthalpy may also be calculated from the enthalpy of saturated air and of dry air using % saturation ... 

The NOBS system undergoes an additional reaction that forms a diacyl peroxide as a result of the nucleophilic attack of the peracid anion on the NOBS precursor as shown in equation 21. This undesirable side reaction can be minimized by the use of an excess molar quantity of hydrogen peroxide (91,96) or by the use of shorter dialkyl chain acid derivatives. However, the use of these acid derivatives also appears to result in less efficient bleaching. The dependence of the acid group on the side product formation is apparentiy the result of the proximity of the newly formed peracid to unreacted NOBS in the micellar environment (91). A variety of other peracid precursor stmctures can be found (97—118). 

The chemical potential is a partial molar quantity defined by the last term in equation 4 ... 

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. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

The well-known Gibbs-Duhem equation (2,3,18) is a special mathematical redundance test which is expressed in terms of the chemical potential (3,18). The general Duhem test procedure can be appHed to any set of partial molar quantities. It is also possible to perform an overall consistency test over a composition range with the integrated form of the Duhem equation (2). 

In some cases, reported data do not satisfy a consistency check, but these may be the only available data. In that case, it may be possible to smooth the data in order to obtain a set of partial molar quantities that is thermodynamically consistent. The procedure is simply to reconstmct the total molar property by a weighted mole fraction average of the n measured partial molar values and then recalculate normalised partial molar quantities. The new set should always be consistent. 

Considering any extensive property K, the partial molar quantity is defined by... 

These quantities (which are standard molar quantities) describe the process initial state transition state... 

The almost dry residue is cooled to 0°C and made strongly alkaline with a 50% potassium hydroxide solution. The amine is extracted into several portions of ether, dried over potassium hydroxide, the solvent removed, and the base fractioned. Reaction of the base with a half-molar quantity of sulfuric acid gives the sulfate. 

Since only catalytic quantities of selenium dioxide are required, the danger of handling large quantities of this material (Note 1) is avoided. Furthermore, the problems associated with the formation of selenium and organoselenides, which commonly arise in oxidations using molar quantities of selenium dioxide, are not encountered. 

For pure substances, n is usually held constant. We will usually be working with molar quantities so that n = 1. The number of moles n will become a variable when we work with solutions. Then, the number of moles will be used to express the effect of concentration (usually mole fraction, molality, or molarity) on the other thermodynamic properties. 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

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