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The driving force, which produces equilibrium distributions, is considered to be proportional at any time to the difference between the actual con-centration and equilibrium concentration of the component being separated.. Thus, concentrations in contact equilibrium separation processes are linked with the general driving force concept.

Consider a case in which initially all of the molecules of some component A of a gas mixture are confined by a partition in one region of a system. The partition is then removed. Random movement among the gas molecules will, in time, distribute component A through the mixture. The greater the concentration of A in the partitioned region, the more rapidly will diffusion occur across the boundary once the partition is removed.

The relative proportions of the components in a mixture or a solution are expressed in terms of the concentrations. Any convenient units may be used for concentration, such as g g-1, g kg-1, mg g-1, percentages, parts per million, and so on.

Because the gas laws are based on numbers of molecules, it is often convenient to express concentrations in terms of the relative numbers of molecules of the components. The unit in this case is called the molecular fraction, shortened to mole fraction, which has been introduced in Chapter 2. The mole fraction of a component in a mixture is the proportion of the number of molecules of the component present to the total number of the molecules of all the components.

In a mixture which contains wA kg of component A of molecular weight MA and wB kg of component B of molecular weight MB, the mole fraction:

xA                       =                       number of moles of A                   
                                  number of moles of A + number of moles of B

                          =           wA /MA                                                                            (9.1)
                                  wA /MA + wB /MB

xB                      =            wB /MB                                                                           (9.2)
                                   wA /MA + wB/MB

Notice that (xA + xB) = 1, and so, xB = (1 - xA)

The definition of the mole fraction can be extended to any number of components in a multicomponent mixture. The mole fraction of any one component again expresses the relative number of molecules of that component, to the total number of molecules of all the components in the mixture. Exactly the same method is followed if the weights of the components are expressed in grams. The mole fraction is a ratio, and so has no dimensions.

EXAMPLE 9.1. Mole fractions of ethanol in water
A solution of ethanol in water contains 30% of ethanol by weight. Calculate the mole fractions of ethanol and water in the solution.

Molecular weight of ethanol, C2H5OH, is 46 and the molecular weight of water, H2O, is 18.
Since, in 100 kg of the mixture there are 30 kg of ethanol and 70 kg of water,

mole fraction of ethanol  =  (30/46) / [(30/46 + (70/18)]
                                   = 0.144
mole fraction of water    =   (70/18) / [(30/46 + (70/18)]
                                   = 0.856
                                   = (1 - 0.144)

Concentrations of the components in gas mixtures can be expressed as weight fractions, mole fractions, and so on. When expressed as mole fractions, they can be related to the partial pressure of the components. The partial pressure of a component is that pressure which the component would exert if it alone occupied the whole volume of the mixture. Partial pressures of the components are additive, and their sum is equal to the total pressure of the mixture. The partial pressures and the mole fractions are proportional, so that the total pressure is made up from the sum of all the partial pressures, which are in the ratios of the mole fractions of the components.

If a gas mixture exists under a total pressure P and the mixture comprises a mole fraction xA of component A, a mole fraction xB of component B, a mole fraction xC of component C and so on, then

P = PxA + PxB + PxC + …..
   = pA + pB + pC + …..                                                                                           (9.3)

where pA, pB, pC, are the partial pressures of components A, B, C ...

In the case of gas mixtures, it is also possible to relate weight and volume proportions, as Avogadro's Law states that under equal conditions of temperature and pressure, equal volumes of gases contain equal numbers of molecules. This can be put in another way by saying that in a gas mixture, volume fractions will be proportional to mole fractions.

EXAMPLE 9.2. Mole fractions in air
Air is reported to contain 79 parts of nitrogen to 21 parts of oxygen, by volume. Calculate the mole fraction of oxygen and of nitrogen in the mixture and also the weight fractions and the mean molecular weight.

Since mole fractions are proportional to volume fractions,

              mole fraction of nitrogen  =  79 / [79 + 21]
                                                  = 0.79
              mole fraction of oxygen  =  21 / [79 + 21]
                                                  = 0.21

The molecular weight of nitrogen, N2, is 28 and of oxygen, O2, is 32.

The weight fraction of nitrogen is given by:
                weight of nitrogen                =           79 x 28           
weight of nitrogen + weight of oxygen         79 x 28 + 21 x 32

                                                  = 0.77

Similarly the weight fraction of oxygen   =  (21 x 32) / [(79 x 28) + (21 x 32)]
                                                          = 0.23.

As the sum of the two weight fractions must add to 1, the weight fraction of the oxygen could have been found by the subtraction of      (1 - 0.77) = 0.23.

To find the mean molecular weight, we must find the weight of one mole of the gas:

0.79 moles of N2 weighing 0.79 x 28 kg = 22.1 kg
plus 0.21 moles of O2 weighing 0.21 x 32kg = 6.7 kg
make up 1 mole of air weighing 28.8 kg and so
Mean molecular weight of air is 28.8, say 29.

Contact-Equilibrium Processes - THEORY > GAS-LIQUID EQUILIBRIA

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Unit Operations in Food Processing. Copyright © 1983, R. L. Earle. :: Published by NZIFST (Inc.)
NZIFST - The New Zealand Institute of Food Science & Technology