Mole and Mass Fractions

The amount of a substance in a fuel sample may be indicated by its mass or by the number of moles of that substance. A mole is defined as the mass of a substance equal to its molecular mass or molecular weight. Gram-mole or pound mole of C H N 0 S is 12 2 28 32 32 kg or Ib. The mass fraction of a component i, mfi is defined as the ratio of the mass of the component, m to the mass of the mixture, m, that is. 

An analogous definition for the mole fraction of a component, i, X, is the ratio of the number of moles of i, rtf, to the total number of moles in the mixture, n, that is. 

Thus the mass of a component i of a mixture is the product of the number of moles of i and the molecular weight, M,-, that is, the total mass is therefore the sum 

Dividing and multiplying the right-hand side by the total number of moles, n, and invoking Equation (6.4) defines the average molecular weight, that is. 

For a mixture at a given temperature and pressure, the ideal gas law shows that pVj = riidtT for any component, and pV = nWT for the mixture as a whole. The ratio of these two equations gives 

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What are the compositions (mole and mass fractions) and volumetric flow rates (m /kmol CH4 fed to burners) of (a) the effluent gas from the reformer burners and (b) the gas entering the stack What is the specific gravity, relative to ambient air (30X, 1 atm, 70% rh), of the stack gas as it enters the stack Why is this quantity of importance in designing the stack Why might there be a lower limit on the temperature to which the gas can be cooled prior to introducing it to the stack ... 

This then gives the following relationship between the mole and mass fractions... 

Our first calculation is to convert the mole fractions to mass fractions. The formulas for converting between mole and mass fractions are given in Table 1.1 and the results of the conversion are... 

The number of state variables is actually equal to 1+J in each case, according to equation (10) between mole and mass fractions. 

When heated, both barium carbonate (BaC03) and calcium carbonate (CaC03) release carbon dioxide (CO2), leaving barium oxide (BaO) and calcium oxide (CaO), respectively. In an experiment, a mixture of BaCOs and CaCOs with a combined mass of 63.67 g produces 19.67 g of CO2. Calculate the mole and mass fractions of barium carbonate and calcium carbonate in the original mixture, assuming complete decomposition of the compounds. 

Mole and mass fractions are appropriate to either the mixture or the solution point of view. The other quantities are appropriate to the solution point of view only. Conversions among these quantities can be carried out using the equations given in Table I-l following this Introduction. Other useful quantities will be defined in the prefaces to individual volumes or on specific data sheets. 

The dependent variable — also termed a state variable — is, for mass transfer operations, usually represented by the concentration of a system, or its total mass. Concentration can be expressed in a variety of ways, the most common of which is kg/m or mol/m or in terms of mole and mass fractions and ratios, whereas total mass is represented in terms of kg or mol. The reader is reminded that tire number of dependent variables equals the number of unknowns. For a system to be fully specified, the number of equations must tiierefore equal the nmnber of unlmowns, in other words, the number of dependent variables. The model is then said to be complete. 

We can face the problem of lack of dimension checking while working with other dimensionless quantities that have different dimensions. While we cannot add mass and moles, addition of mole and mass fractions will not result in an error message. 

This is not the case for mass transfer coefficients, where a variety of definitions are accepted. Four of the more common of these are shown in Table 8.2-2. This variety is largely an experimental artifact, arising because the concentration can be measured in so many different units, including partial pressure, mole and mass fractions, and molarity. 

It is recommended that concentration measurements for this type of modeling work are based on analytical standards of mole or mass fraction, to avoid the conversion error caused by density effects. The excess solid phase should always be characterized by a suitable analytical technique, before and after the equilibrium solubility measurements, to confirm that the polymorphic form is unchanged. It should be noted that the crystal shape (habit) does not always change significantly between different polymorphic forms, and visual assessments can be misleading. 

Since (M) is a concentration, it may be written in terms of the total density p and the mole or mass fraction e that is,... 

Using Eq. 3.84, we provide the relationship between mole fraction and mass fraction as... 

Evaluate the mass-fraction profiles and graph the major-species mass fractions (H2, O2, H2O, and N2). Discuss briefly the relationship between the mole-fraction and mass-fraction profiles. 

The overall reaction rate has a temperature dependence governed by the specific reaction rate k(T) and a concentration dependence that is expressed in terms of several concentration-based properties depending on the suitability for the particular reaction type mole or mass concentration, component vapor partial pressure, component activity, and mole or mass fraction. For example, if the dependence is expressed in terms of molar concentrations for components A(Ca) and B(Cb), the overall reaction rate can be written as... 

In this section, the concentration is represented by C. Mass balance accounting in terms of the number of moles and the fractional conversion is discussed in Sec. 7 and can be very useful. The rate of reaction is r the flow rate in moles is Na the volumetric flow rate is V reactor volume is Vr. Several equations are presented without specification of units. Use of any consistent unit set is appropriate. 

The parameters controlling the rate of entropy production in the tower are now obvious the vapor flow rate V (a function of the reflux ratio), the inlet and outlet mole (or mass) fractions, and the relationship between yA and y g (a function of the reflux ratio and the relative volatility). 

When JV = 2, the phase rule becomes F = 4 - it. Since there must be at least one phase (it = 1), the maximum number of phase-rule variables which must be specified to fix the intensive state of the system is three namely, P, T, and one mole (or mass) fraction. All equilibrium states of the system can therefore be... 

The number of unknowns and the number of equations relating these unknowns can become very large in a process-design problem. The number of unknowns and independent equations must be equal in order that a unique solution to a problem exists. Therefore, it is necessary to have a systematic method for enumerating them. The total number of independent extensive and intensive variables associated with each stream in a process is C + 2, where C is the number of independent chemical components in the stream. The quantity and the condition of the stream are completely determined by fixing the flow rate of each component in the stream (or, equivalently, the total flow rate and the mole or mass fractions of C — 1 components) and two additional variables, usually the temperature and pressure, although other choices are possible. This number includes situations where physical and chemical equilibrium exist.t... 

Finally, if a process involves the sublimation of a pure solid (such as ice or solid CO2) or the evaporation of a pure liquid (such as water) in a different medium such as air. the mole (or mass) fraction of the substance in the liquid or solid phase is simply taken to be 1.0, and the partial pressure and thus the mole fraction of the substance in the gas phase can readily be determined from the saturation data of the substance at the specified temperature. Also, the assumption of thermodynamic equilibrium at the interface is very rca.sonablc for pure solids, pure liquids, and solutions, except when chemical reactions are occurring at the interface. 

The isobaric phase diagrams of three-component systems are determined by four variables three concentration coordinates and the temperature. With regard to the condition that the sum of mole or mass fractions equals one, it is possible to show these phase diagrams in three-dimensional space, where the concentration coordinates are shown in the x-y plane and the temperature on the z axis. 

Other measures of composition such as mole fraction and mass fraction are less commonly used to express chemical reaction rates. Weight measurements are frequently used to prepare solutions or fill reactors. The resulting composition will have a known ratio of moles and masses of the various components, but the numerical value for concentration requires that the density be known. Good practice is to prepare solutions in mass units and then convert to standard concentration units based on the known or observed density of the solution under reaction conditions. To avoid ambiguity, modern analytical chemists frequently define both molarity and molality in weight units as moles per kilogram of solution or moles per kilogram of solvent. 

Let the moles and mass of component A per unit volume of mixture be cA and pA respectively. Then the mole fraction of A is cAfe or xA, and the mass fraction is pAfp or wA. 

NOTE ON CONCENTRATIONS. Strictly speaking, concentration means mass per unit volume. Mass may be in kilograms or pounds and volume in cubic meters or cubic feet. Kilogram moles or pound moles are often used for mass. In this book the context will make clear what quantity— molc or ordinary mass—is used. It is convenient to extend the use of the word concentration to include mole or mass fractions. The relation between concentration and mole or mass fraction of a component i is... 

In a countercurrent multistage section, the phases to be contacted enter a series of ideal or equilibrium stages from opposite ends. A contactor of this type is diagramatically represented by Fig. 8.1, which could be a series of stages in an absorption, a distillation, or an extraction column. Here L and V are the molal (or mass) flow rates of the heavier and lighter phases, and x,- and y,- the corresponding mole (or mass) fractions of component /, respectively. This chapter focuses on binary or pseudobinary systems so the subscript / is seldom required. Unless specifically stated, y and x will refer to mole (or mass) fractions of the lighter component in a binary mixture, or the species that is transferred between phases in three-component systems. 

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