What The Frac!
Just The Facts

DOFPro Team

Mass

Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied. An object’s mass also determines the strength of its gravitational attraction to other bodies. (https://en.wikipedia.org/wiki/Mass)

The SI unit of mass, the kilogram, is defined from Planck’s constant

\[h=6.626070150 \times 10^{-34}\ \mathrm{kg \cdot m^2/s}\]

or

\[ 1\ \mathrm{kg}=\frac{h}{6.626070150 \times 10^{-34}\ \mathrm{m^2/s}} \]

Mass (cont.)

For the curious, the second is defined as 9 192 631 770 times the period of the unperturbed ground-state hyperfine transition of the cesium 133 atom, \(\Delta \nu_\mathrm{Cs}\). And the meter is defined as 1/299 792 458th of the distance light travels in a vacuum in one second, \(c\), or

\[1\ \mathrm{kg} = \frac{(299\ 792\ 458)^2}{(6.626\ 070\ 150 \times 10^{-34})(9\ 192\ 631\ 770)} \frac{h\ \Delta \nu_\mathrm{Cs}}{c^2}\]

The American engineering unit of mass, the pound mass, or \(\mathrm{lb_m}\) is defined as

\[1\ \mathrm{lb_m} \equiv 0.453\ 592\ 37\ \mathrm{kg}\]

Mole

  • See What Ia A Mole? for a more thorough explanation

  • The most common moles are

    • \(\text{g-mol}\) or \(\text{mol}\)
    • \(\text{kg-mol}\) or \(\text{kmol}\)
    • \(\text{lb-mol}\)
    • \(\text{ton-mol}\)
    • \(\text{tonne-mol}\)

You should NEVER convert your mass in \(\text{kg}\) or \(\mathrm{lb_m}\) to \(\mathrm{g}\), and then to \(\text{g-mol}\). You should ALWAYS use the kind of mole your mass is in. The conversion factor is simply the molecular weight.

Mole (cont.)

The molecular weight of methane is:

\[M_\mathrm{CH_4} = 16.0425\ \frac{\text{g}}{\text{g-mol}} = 16.0425\ \frac{\text{kg}}{\text{kg-mol}} = 16.0425\ \frac{\mathrm{lb_m}}{\text{lb-mol}}\]

To calculate the pounds of oxygen (\(M = 31.9988\)) required to completely burn three pounds of methane:

\[\mathrm{CH_4} + 2 \mathrm{O_2} \rightarrow \mathrm{CO_2} + 2 \mathrm{H_2O} \implies \frac{2\ \text{lb-mol }\mathrm{O_2}}{1\ \text{lb-mol }\mathrm{CH_4}}\]

\[\frac{3\ \mathrm{lb_m\ CH_4}}{16.0425\ \frac{\mathrm{lb_m\ CH_4}}{\text{lb-mol CH}_4}} \frac{2\ \text{lb-mol }\mathrm{O_2}}{1\ \text{lb-mol }\mathrm{CH_4}} \frac{31.9988\ \mathrm{lb_m\ O_2}}{1\ \text{lb-mol O}_2} = 11.9678\ \mathrm{lb_m\ O_2}\]

Calculating it any other way is just a waste of time and brain effort.

Mass Fractions and Mole Fractions

\(x\), \(y\), and \(z\) are all used for both mass fractions and mole fractions.

\(x_\mathrm{A}\) can be the mass fraction of species \(\mathrm{A}\) or the mole fraction of species \(\mathrm{A}\).

Likewise for \(y_\mathrm{A}\) and \(z_\mathrm{A}\).

Convention usually uses:

  • \(x\) for fractions in a solid or liquid mixture,
  • \(y\) for fractions in a gas mixture, and
  • \(z\) for the overall fraction in a gas-liquid, liquid-solid, or gas-solid mixture.

Conversions

To convert from mass fraction, \(x\), to mole fraction, \(y\):

\[y_\mathrm{A} = \frac{\frac{x_\mathrm{A}}{M_\mathrm{A}}}{\sum \frac{x_i}{M_i}}\]


To convert from mole fraction, \(y\), to mass fraction, \(x\):

\[x_\mathrm{A} = \frac{y_\mathrm{A} M_\mathrm{A}}{\sum y_i M_i}\]

Mole-Averaged and Mass-Averaged Molecular Weight


The mole-averaged (or number-averaged) molecular weight is

\[\bar{M} = \sum y_i M_i = \left(\sum \frac{x_i}{M_i}\right)^{-1}\]

The mass-averaged molecular weight is

\[\bar{M}_m = \sum x_i M_i = \frac{\sum y_i M_i^2}{\sum y_i M_i}\]

The mole-averaged molecular weight is most often used for mixtures of gases.


The mass-averaged molecular weight is most often used for polymers.

The Takeaways

  1. Materials in chemical processes are measured in mass, moles, or both.
  2. Always use the mole unit, e.g. \(\text{lb-mol}\), corresponding to your mass unit, e.g., \(\mathrm{lb_m}\).
  3. Learn how to convert from mass fractions to mole fractions and back.
  4. Learn how to calculate mole- or number-averaged molecular weights, and mass-averaged molecular weights.






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The DOFPro Team