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4.2: Species Mass Density

  • Page ID
    44476
  • The mass of species \(A\) per unit volume in a mixture of several components is known as the species mass density, and it is represented by \(\rho_{A}\). The species mass density can range from zero, when no species \(A\) is present in the mixture, to the density of pure species \(A\), when no other species \(A\)re present. In order to understand what is meant by the species mass density, we consider a mixing process in which three pure species \(A\)re combined to create a uniform mixture of species \(A\), \(B\), and \(C\). This mixing process is illustrated in Figure \(\PageIndex{2}\) where we have indicated that three pure species \(A\)re combined to create a uniform mixture having a measured volume of \(45 cm^{3}\). The total volume of the three pure species is \(50 cm^{3}\), thus there is a change of volume upon mixing as is usually the case with liquids. We denote this change of volume upon mixing by \(\Delta V_{mix}\), and for the process illustrated in Figure \(\PageIndex{2}\) we express this quantity as

    \[\Delta V_{mix} =V - \left(V_{A} +V_{B} +V_{C} \right) \label{22}\]

    The densities of the pure species have been denoted by a superscript zero, thus \(\rho_{A}^{o}\) represents the mass density of pure species \(A\). The species mass density of species \(A\) is defined by

    \[\left\{\begin{array}{c} \text{species mass density} \\ \text{ of species }A \end{array}\right\}=\left(\text{ mass of species }A\right) / \left(\begin{array}{c} \text{ volume in which species }A \\ \text{ is contained} \end{array}\right) \label{23}\]

    and this definition applies to mixtures in which species \(A\) is present as well as to the case of pure species \(A\).

    image
    Figure \(\PageIndex{1}\). Mixing process

    If we designate the mass of species \(A\) as \(m_{A}\) and the volume of the uniform mixture as \(V\), the species mass density can be expressed as

    \[\rho_{A} ={m_{A} / V} \label{24}\]

    For the mixing process illustrated in Figure \(\PageIndex{2}\), we are given that the mass of species \(A\) is

    \[m_{A} =\rho_{A}^{o} V_{A} =(0.85 \ g/cm^{3} )(15 \ cm^{3} )=12.75 \ g \label{25}\]

    and this allows us to determine the species mass density in the mixture according to

    \[\rho_{A} ={m_{A} / V} =\frac{12.75 \ g}{ 45 \ cm^{ 3} } =0.283 \ g / cm^{ 3} \label{26}\]

    This type of calculation can be carried our for species \(B\) and \(C\) in order to determine \(\rho_{B}\) and \(\rho_{C}\).

    The total mass density is simply the sum of all the species mass densities and is defined by

    \[\left\{\begin{array}{c} \text{ total mass} \\ \text{ density} \end{array}\right\}=\rho = \sum_{A = 1}^{A = N}\rho_{A} \label{27}\]

    The total mass density can be determined experimentally by measuring the mass, \(m\), and the volume, \(V\), of a mixture. For any a particular mixture, it is difficult to measure directly the species mass density; however, one can prepare a mixture in which the species mass densities can be determined as we have suggested in Figure \(\PageIndex{2}\). When working with molar forms, we often need the total molar concentration and this is defined by

    \[\left\{\begin{array}{c} \text{ total molar} \\ \text{ concentration} \end{array}\right\}=c=\sum_{A = 1}^{A = N}c_{A} \label{28}\]

    Mass Fraction and Mole Fraction

    For solid and liquid systems it is sometimes convenient to use the mass fraction as a measure of concentration. The mass fraction of species \(A\) can be expressed in words as

    \[\omega_{A} =\left\{\begin{array}{c} \text{ mass of species }A\text{ per} \\ \text{ unit mass of the mixture} \end{array}\right\} \label{29}\]

    and in precise mathematical form we have

    \[\omega_{A} =\frac{\rho_{A} }{\rho } ={\rho_{A} / \sum_{G = 1}^{G = N}\rho_{G} } \label{30}\]

    Note that the indicator, \(G\), is often referred to as a dummy indicator since any letter would suffice to denote the summation over all species in the mixture. In this particular case, we would not want to use A as the dummy indicator since this could lead to confusion. The mole fraction is analogous to the mass fraction and is defined by

    \[x_{A} =\frac{c_{A} }{c} ={c_{A} / \sum_{G = 1}^{G = N}c_{G} } \label{31}\]

    If one wishes to avoid the mixed-mode nomenclature in Equations \ref{30} and \ref{31}, one must express the mass fraction as

    \[\omega_{A} =\frac{\rho_{A} }{\rho } =\frac{\rho_{A} }{\rho_{A} + \rho_{B} + \rho_{C} + \rho_{D} + .... + \rho_{N} } \label{32}\]

    while the mole fraction takes the form

    \[y_{A} =\frac{c_{A} }{c} =\frac{c_{A} }{c_{A} + c_{B} + c_{C} + c_{D} + .... + c_{N} } \label{33}\]

    Very often \(x_{A}\) is used to represent mole fractions in liquid mixtures and \(y_{A}\) to represent mole fractions in vapor mixtures, thus Equation \ref{33} represents the mole fraction in a vapor mixture while Equation \ref{31} represents the mole fraction in a liquid mixture.

    Example \(\PageIndex{1}\): Conversion of mole fractions to mass fractions

    Sometimes we may be given the composition of a mixture in terms of the various mole fractions and require the mass fractions of the various constituents. To convert from \(x_{A}\) to \(\omega_{A}\) we proceed as follows:

    \[ x_A = c_A/c \label{1}\tag{1}\]

    \[c_A = x_A c \label{2}\tag{2}\]

    \[\rho_A = MW_A c_A = MW_A x_A c \label{3}\tag{3}\]

    \[\rho = \sum^{G=N}_{G=1} \rho_G = \left( \sum^{G=N}_{G=1} MW_G x_G \right) c \label{4}\tag{4}\]

    \[ \omega_{A} =\frac{\rho_{A} }{\rho } =\frac{MW_{A} x_{A} c}{\left(\sum_{G=1}^{G=N} MW_{G} x_{G} \right) c} = \frac{MW_{A} x_{A} }{\sum_{G=1}^{G=N}MW_{G} x_{G} } \label{5}\tag{5}\]

    Total mass balance

    Given the total density defined by Equation \ref{27}, we are ready to recover the total mass balance for multicomponent, reacting systems. For a fixed control volume, this is developed by summing Equation \((4.1.7)\) over all species to obtain

    \[\sum_{A = 1}^{A = N}\frac{d}{dt} \int_{\mathscr{V}}\rho_{A} dV + \sum_{A = 1}^{A = N}\int_{\mathscr{A}}\rho_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\sum_{A = 1}^{A = N}\int_{\mathscr{V}}r_{A} dV \label{34}\]

    The summation procedure can be interchanged with differentiation and integration so that this result takes the form

    \[\frac{d}{dt} \int_{\mathscr{V}}\sum_{A = 1}^{A = N}\rho_{A} dV + \int_{\mathscr{A}}\sum_{A = 1}^{A = N}\rho_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\int_{\mathscr{V}}\sum_{A = 1}^{A = N}r_{A} dV \label{35}\]

    On the basis of definition of the total mass density given by Equation \ref{27} and the axiom given by Equation \((4.1.11)\), this result simplifies to

    \[\frac{d}{dt} \int_{\mathscr{V}}\rho dV + \int_{\mathscr{A}}\sum_{A = 1}^{A = N}\rho_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =0 \label{36}\]

    At this point, we define the total mass flux according to

    \[\left\{\begin{array}{c} \text{ total} \\ \text{ mass flux} \end{array}\right\}=\rho \mathbf{v}=\sum_{A = 1}^{A = N}\rho_{A} \mathbf{v}_{A} \label{37}\]

    Since \(\rho\) is defined by Equation \ref{27}, this result actually represents a definition of the velocity \(\mathbf{v}\) that can be expressed as

    \[\mathbf{v}=\sum_{A = 1}^{A = N}\omega_{A} \mathbf{v}_{A} \label{38}\]

    This velocity is known as the mass average velocity and it plays a key role both in our studies of macroscopic mass balances and in subsequent studies of fluid mechanics, heat transfer, and mass transfer. Use of this definition for the mass average velocity allows us to express Equation \ref{36} as

    \[\frac{d}{dt} \int_{\mathscr{V}}\rho dV + \int_{\mathscr{A}}\rho \mathbf{v}\cdot \mathbf{n} dA=0 \label{39}\]

    This is identical in form to the mass balance for a fixed control volume that was presented in Chapter 3; however, this result has greater physical content than our previous result for single-component systems. In this case, the density is not the density of a single component but the sum of all the species densities as indicated by Equation \ref{27} and the velocity is not the velocity of a single component but the mass average velocity defined by Equation \ref{38}.