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

Explanation:

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 (m^{3}, L, ...). The following are some ways to
express concentration, where the numerator signifies the solute and the
denominator signifies the solution.

Mass Concentration: g / cm^{3}, lb_{m} /
ft^{3}, kg / in^{3}

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, M_{A} = g-mol_{A}
/ L_{total}

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: x_{A} = m_{A} /
m_{total}

Equation 2: x_{A} = n_{A} /
n_{total}

Equation 3: y_{A} = m_{A} /
m_{total}

Equation 4: y_{A} = n_{A} /
n_{total}

x_{A}, y_{A} - mass or mole fraction of component A.
m_{A} - the mass of component A in the mixture
n_{A} - the mole of component A in the mixture
m_{total} - the mass of the total mixture
n_{total} - 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, "x_{A}" can donote
several different things. For example, x_{H2O} 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 "x_{A}" is
defined in working a problem.

PPM and PPB

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 10^{6} for ppm, and by
10^{9} for ppb.

Parts Per Million (ppm) = x_{A}*10^{6} or y_{A}*10^{6}

Parts Per Billion (ppb) = x_{A}*10^{9} or y_{A}*10^{9}

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:

O_{2}

16.0%

CO

4.0%

CO_{2}

17.0%

N_{2}

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%
O_{2}, or we could say, the mass fraction of O_{2} 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:

O_{2}:

CO:

CO_{2}:

N_{2}:

Example 2

Air is approximately 79 mole% N_{2} and 21 mole% O_{2}. Calculate the average molecular weight of air by using the following formula:

MW_{avg} = y_{N2} * MW_{N2} + y_{O2} * MW_{O2}

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.