Mole relationship with volume

The key relationship is provided by the ideal gas law, which yields the ability to calculate density. To see this, think about n/V as the number of moles per unit volume. This is equal to the (mass) density divided by the molar mass. Replacing n/V with that relationship gives us a gas law in terms of density ... 

This case involves constant temperature T and total pressure n. In this case, the density changes since tlie numher of moles change during the reaction, and the volume of a fluid element changes linearly with conversion or V = Vo(l -i- a a)- The relationship between C and is as follows ... 

Beck, et al. have used the permeation technique to study the effect of uniaxial tensile stresses in the elastic region on hydrogen permeation through pure iron, and have shown that it increases with increase in stress. The partial molar volume of hydrogen (cubic centimetres of hydrogen per mole of iron) in ferrous alloys can be evaluated from the variation of permeation with applied stress, and from the relationship... 

P the total pressure, aHj the mole fraction of hydrogen in the gas phase, and vHj the stoichiometric coefficient of hydrogen. It is assumed that the hydrogen concentration at the catalyst surface is in equilibrium with the hydrogen concentration in the liquid and is related to this through a Freundlich isotherm with the exponent a. The quantity Hj is related to co by stoichiometry, and Eg and Ag are related to - co because the reaction is accompanied by reduction of the gas-phase volume. The corresponding relationships are introduced into Eqs. (7)-(9), and these equations are solved by analog computation. 

The process to reach a quantitative solution to the problem requires working with moles. Thus, we need the relationship linking moles to molarity and volume n — M V We must use the equation In two ways ... 

Because moles are the currency of chemistry, all stoichiometric computations require amounts in moles. In the real world, we measure mass, volume, temperature, and pressure. With the ideal gas equation, our catalog of relationships for mole conversion is complete. Table lists three equations, each of which applies to a particular category of chemical substances. 

Because we know we are dealing with a buffer solution made from a specific conjugate acid-base pair, we can work directly with the buffer equation. We need to calculate the ratio of concentrations of conjugate base and acid that will produce a buffer solution of the desired pH. Then we use mole-mass-volume relationships to translate the ratio into actual quantities. 

It is thus seen that heat capacity at constant volume is the rate of change of internal energy with temperature, while heat capacity at constant pressure is the rate of change of enthalpy with temperature. Like internal energy, enthalpy and heat capacity are also extensive properties. The heat capacity values of substances are usually expressed per unit mass or mole. For instance, the specific heat which is the heat capacity per gram of the substance or the molar heat, which is the heat capacity per mole of the substance, are generally considered. The heat capacity of a substance increases with increase in temperature. This variation is usually represented by an empirical relationship such as... 

Addition of a cosolvent is an alternative mechanism to increase contaminant solubility in an aqueous solution. When a contaminant with low solubility enters an aqueous solution containing a cosolvent (e.g., acetone), the logarithm of its solubility is nearly a linear function of the mole fraction composition of the cosolvent (Hartley and Graham-Bryce 1980). The amount of contaminant that can dissolve in a mixture of two equal amounts of different solvents, within an aqueous phase, is much smaller than the amount that can dissolve solely by the more powerful solvent. In the case of a powerful organic solvent miscible with water, a more nearly linear slope for the log solubility versus solvent composition relationship is obtained if the composition is plotted as volume fraction rather than mole fraction. 

In contrast to template polycondensation or ring-opening polymerization, template radical polymerization kinetics has been a subject of many papers. Tan and Challa proposed to use the relationship between polymerization rate and concentration of monomer or template as a criterion for distinguishing between Type I and Type II template polymerization. The most popular method is to examine the initial rate or relative rate, Rr, as a function of base mole concentration of the template, [T], at a constant monomer concentration, [M]. For Type I, when strong interactions exist between the monomer and the template, Rr vs. [T] shows a maximum at [T] = [M]q. For type II, Rr increases with [T] to the critical concentration of the template c (the concentration in which template macromolecules start to overlap with each other), and then R is stable, c (concentration in mols per volume) depends on the molecular weight of the template. 

In this relationship, Vi is the initial (feed) volume of the gas. This is the case of Levenspiel s simplification where the volume of the reacting system varies linearly with conversion (Levenspiel, 1972). The last equation shows that even if we have a change in moles (sR / 0), if the conversion of the limiting reactant is veiy low, the volume of the reaction mixture could be taken as constant and eR is not involved in the solutions of the models (since eRjcA can be taken as approximately zero). 

Since both the osmotic pressure of a solution and the pressure-volume-temperature behavior of a gas are described by the same formal relationship of Equation (25), it seems plausible to approach nonideal solutions along the same lines that are used in dealing with nonideal gases. The behavior of real gases may be written as a power series in one of the following forms for n moles of gas ... 

The relationship dpy/dT is the rate of change of vapor pressure with temperature. Thus it represents the slope of the vapor-pressure line. Ly is the heat of vaporization of one mole of liquid, T is the absolute temperature, and VMg - Vml represents the change in volume of one mole as it goes from liquid to gas. 

Some very interesting ideas concerning the relationship between free-volume formation and the energy of one mole of hole formation were developed in detail by Kanig42. Kanig introduced some improvements to the definition of free-volume, On the basis of Frenkel s ideas43 he divided the free-volume into two parts, one of which is determined only by the thermal vibrations of atoms in the lattice of a real crystal while the other is connected with inherent free-volume, i.e. voids and holes. It is the latter that makes possible the exchange of particles, i.e. the very existence of the liquid state. He introduced some new definitions of fractions of free-volume ... 

On the other hand, irreversible thermodynamics has provided us with the insight that entropy generation is related to process flow rates like those of volume, V, mass in moles, h, chemical conversion, vl h, and heat, Q, and their so-called conjugated forces A(P/T), -A(p/T), A/T, and A(l/T). Although irreversible thermodynamics does not specify the relationship between these forces X and their conjugated flow rates /, it leaves no doubt about the... 

A chemical equation represents the relationship of the reactants and products through a numerical relationship expressed by the coefficients associated with the participants. The coefficients can be interpreted as telling us the number of molecules or moles of materials involved but they also represent the volumes of those participants that are gases, assuming a constant temperature and pressure (T and P). An example of these relationships is as follows ... 

In Chapter 1, the assumption that gases and gas mixtures behave ideally at low pressures (1 bar and below) was stated. (Deviation from this with large amounts of readily condensable vapours under compression near atmospheric pressure was dealt with in Chapter 3.) The ideal gas equation, expressing the relationship between the variables pressure, volume, temperature and amount (number of moles) of gas, together with the expression of pressure in terms of particle number density (n) and Dalton s law of partial pressures, allow many calculations useful to vacuum technology to be carried out (Examples... 

You have already encountered problems involving moles, molecules, and molar masses earlier in this book. There is still one other relationship that needs to be connected with the mole and that is molar volume. Once you make a connection between moles and volume, mass, and molecules you will be able to solve problems easily. One very helpful mnemonic device to use is the Mole-Go-Round. Some think of this method as a way of cheating the system, but because the SAT II exam does not require you to show work, the Mole-Go-Round is a perfectly acceptable method for achieving better results. 

The frequency with which A molecules in a solution will encounter B molecules is this frequency multiplied by ns, the mole fraction of B. For very dilute solutions ub = Nb/Nb, the ratio of the molecular densities of B to S molecules. But l/Ns, the volume per solvent molecule, can be written as t ab, with y determined by the packing factor for the lattice. By substituting these relationships in Eq, (XV.2.2) we can write for the encounter frequency of A and B in such a lattice s... 

By combining thermodynamically-based monomer partitioning relationships for saturation [170] and partial swelling [172] with mass balance equations, Noel et al. [174] proposed a model for saturation and a model for partial swelling that could predict the mole fraction of a specific monomer i in the polymer particles. They showed that the batch emulsion copolymerization behavior predicted by the models presented in this article agreed adequately with experimental results for MA-VAc and MA-Inden (Ind) systems. Karlsson et al. [176] studied the monomer swelling kinetics at 80 °C in Interval III of the seeded emulsion polymerization of isoprene with carboxylated PSt latex particles as the seeds. The authors measured the variation of the isoprene sorption rate into the seed polymer particles with the volume fraction of polymer in the latex particles, and discussed the sorption process of isoprene into the seed polymer particles in Interval III in detail from a thermodynamic point of view. 

This definition of entropy shows its exact relationship to probability. However, it is not useful in a practical sense for the typical types of samples used by chemists, because these samples contain so many components. For example, a mole of gas contains 6.022 X 1023 individual particles. In addition, according to one estimate, describing the positions and velocities of this mole of particles would require a stack of paper 10 light years tall—and this description would apply for only an instant. Clearly, we cannot deal directly with this definition of entropy for typical-sized samples. We must find a way to connect entropy to the macroscopic properties of matter. To do so, we will consider an ideal gas that expands isothermally from volume V] to volume 2V] (see Fig. 10.10). 

Then in the general case, for Ch moles of strong acid + Cha moles of weak acid HjrA in an initial volume Vq, titrated with V milliliters of standard base of concentration CqH/ the relationship is ... 

You need to be familiar with the Kelvin temperature scale and the relationship between mass, volume and density. You should also have covered the relationship between mass and moles in Unit 43, and the explanation of pressure in Unit 1.1. 

By using this fact in combination with the relationship density = massl volume, we can work out how much gas in moles or in grammes is present in any volume under different conditions. It is probably easiest to show this using an example. 

---