Upon studying this section, you should be familiar with the following:

    • Units of mass and a mole concentrations
      • Either a mass or mole concentration divided by a unit of volume
    • The mathematical relationships for molarity, mass fractions, mole fractions, ppm, and ppb
    • How to distinguish between mass and mole fraction notation
      • x for solids and liquids, y for gases
      • Each x or y can be for mole or mass fractions


Concentration tells us how much of one component (in a mixture) we have relative the mixture (or another component in the mixture). That is, for a mixture of sugar and water, a concentration can specify how much water we have relative to the solution (water and sugar), or how much water we have relative to the other component (sugar). These concentrations can be expressed using units of amount (mols, molecules, ...), mass (kg, lb, ...), and volume (m3, L, ...). The following are some ways to express concentration, where the numerator signifies the solute and the denominator signifies the solution.

Mass Concentration: g / cm3, lbm / ft3, kg / in3

Molar Concentration: kmol / m3, lb-mol / ft3, g-mol / L

The last molar concentration listed, g-mol/L is the Molarity of the solute in the mixture. Thus, the definition of molarity follows:

Molarity of component A, MA = g-molA / Ltotal

Mass and Mole Fractions

Mass and mole fractions will be used frequently in material and energy balance problems. The notation used throghout the majority of this course and in this online textbook differs from the notation introduced in chapter 3. Here, x designates a liquid or solid mass or mole fraction, and y designates mass or mole fraction for a gas. Here are the definitions for mass and mole fractions:

Equation 1: xA = mA / mtotal

Equation 2: xA = nA / ntotal

Equation 3: yA = mA / mtotal

Equation 4: yA = nA / ntotal

xA, yA - mass or mole fraction of component A.
mA - the mass of component A in the mixture
nA - the mole of component A in the mixture
mtotal - the mass of the total mixture
ntotal - the moles of the total mixture

"x" is used for liquids and solids, thus the first two equations are the mass and mole fractions for liquids and solids. Accordingly, the last two equations refer to gases. As you can see, "xA" can donote several different things. For example, xH2O can refer to either the mass or mole fraction of water or ice in a mixture. It will be up to the problem solver to keep track of how "xA" is defined in working a problem.


Related to the mass and mole fractions are the definitions for parts per million (ppm) and the parts per billion (ppb). Here, we simply multiply the mass or mole fraction by 106 for ppm, and by 109 for ppb.

Parts Per Million (ppm) = xA*106 or yA*106

Parts Per Billion (ppb) = xA*109 or yA*109

note #1: The mole and mass fractions must always be unitless, so the mole/mass of the component (numerator) and the total mole/mass (denominator) must both have the same units.

note #2: Mass ratios are usually used for liquids and solids, while mole rations are usually used for gases.

Example 1

A mixture of gases has the following composition by mass:

O2 16.0%
CO 4.0%
CO2 17.0%
N2 63.0%
To reach mass fractions from mass percent composition, we would move the decimal place over two places. For example, the mixture is 16% O2, or we could say, the mass fraction of O2 in the mixture is 0.16. Determine the molar composition of this mixture. (hint: you first need to find the total moles of the mixture)
Mole Fractions:

Example 2

Air is approximately 79 mole% N2 and 21 mole% O2. Calculate the average molecular weight of air by using the following formula:

    MWavg = yN2 * MWN2 + yO2 * MWO2

g/mole air

Example 3

A material balance problem from chapter 4 gives us the following information about streams going in and out of a condenser. Determine the mole fractions of each stream.

Check your calculated Mole Fractions:

Goto | Interrelations Home | Dr. Blowers Home | U of A Home

The University of Arizona. All copyrights reserved.


2007 Arizona Board of Regents for The University of Arizona