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4.6 Volumetric Fractions and Mole Fractions

One major benefit of the behavior of gases is that the volume of one ideal gas in a mixture of ideal gases is equivalent to its mole fraction. For all practical purposes, the volume fractions and the mole fractions of the components of an ideal gas mixture are interchangeable.

We can prove the preceding statement by using the Ideal Gas Law. First, the definition of the mole fraction is given by Equation 4.17.

Equation 4.17

04equ17.jpg

For Equation 4.17, if yi is the mole fraction of component i of a mixture k components of gas, ni is the number of moles of i, and nj is the number of moles of component j summed from component 1 to component k. We are using yi for the mole fraction of a gas so as to be consistent with the nomenclature used later in this text where we will use xi for the mole fraction of component i in a liquid.

Next, we substitute the Ideal Gas Law to calculate the number of moles n for any component, which gives us Equation 4.18.

Equation 4.18

04equ18.jpg

Note that all of the terms include pbyrt.jpg, which we can now divide out to obtain Equation 4.19.

Equation 4.19

04equ19.jpg

Realizing that the total volume V is the sum of the partial volumes common2.jpg, we can substitute V and rewrite Equation 4.19 as Equation 4.20.

Equation 4.20

04equ20.jpg

Equating the volume fraction of gas to its mole fraction is analogous to Dalton’s Law of Partial Pressures, which equates the partial pressure of a mixture’s gas component to the mole fraction of that mixture times the total pressure. We use the total pressure of the gas in Equation 4.18 and not the partial pressure because we are using the volume fraction based on the total volume and total pressure of our system. If we used the total volume of the system instead of the volume fraction, then we would use the partial pressure of the gas and Equation 4.18 would look something like Equation 4.21.

Equation 4.21

04equ21.jpg

However, we normally don’t think in terms of pressure fractions; instead, we think in terms of volume fractions, which is why we correlate mole fractions with volumetric fractions. In the end, it really makes no difference whether we view the fractions as either volume fractions or pressure fractions.

If the volume and temperature are considered to be constant, then we can rewrite Equation 4.21 to give the partial pressure of component I, which is found by Equation 4.22.

Equation 4.22

04equ22.jpg

When considering gas mixtures, the most important is our atmosphere, which consists of approximately 78% nitrogen, 21% oxygen, and 1% other gases. Those percentages are both the volumetric percentages and the mole percentages. To make a gas equivalent to our atmosphere, we would need to take about 4 volumes of nitrogen, 1 volume of oxygen, and add just a touch of “other.”

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