What is the most convenient basis to use? (For example, suppose that the composition of a given material is known in mole percent Then selecting 100 kg moles of the material as basis

would make sense. On the other hand, if the composition of the material in terms of mass is known, then 100 kg of the material would be an appropriate basis.)

These questions and their answers will suggest suitable bases. Sometimes when several bases seem appropriate, you may find it is best to use a unit basis of 1 or 100 of something, for example,

kilograms, hours, moles, or cubic feet. For liquids and solids in which a mass (weight) analysis

applies, a convenient basis is often 1 or 100 lb or kg; similarly, because gas compositions are usually provided in terms of moles, 1 or 100 moles is often a good choice for a gas.

Always state the basis you have chosen for your calculations by writing it prominently on your calculation sheets or in the computer program used to solve the problem.

Example 2.14. Choosing a Basis

The dehydration of the lower-molecular-weight alkanes can be carried out using a ceric oxide (CeO) catalyst. What are the mass fraction and mole fraction of Ce and O in the catalyst?

Solution

Start the solution by selecting a basis. Because no specific amount of material is specified, the question “What do I have to start with?” does not help determine a basis. Neither does the question about the desired answer. Thus, selecting a convenient basis becomes the best choice. What do you know about CeO? You know from the formula that 1 mole of Ce is combined with 1 mole of O. Consequently, a basis of 1 kg mol (or 1 g mol, or 1 lb mol, etc.) would make sense. You can get the atomic weights for Ce and O from Appendix B, and then you are prepared to calculate the

respective masses of Ce and O in CeO. The calculations for the mole and mass fractions for Ce and O in CeO are presented in the following table:

Basis: 1 kg mole of CeO

Example 2.15. Choosing a Basis

Most processes for producing high-energy-content gas or gasoline from coal include some type of gasification step to make hydrogen or synthesis gas. Pressure gasification is preferred because of its greater yield of methane and higher rate of gasification.

Given that a 50.0 kg test run of gas averages 10.0% H₂, 40.0% CH₄, 30.0% CO, and 20.0% CO₂, what is the average molecular weight of the gas?

Solution

Let’s choose a basis. The answer to question 1 is to select a basis of 50.0 kg of gas (“What do I have to start with?”), but is this choice a good basis? A little reflection will show that such a basis is of no use. You cannot multiply the given mole percent of this gas (remember that the

composition of gases is given in mole percent unless otherwise stated) times kilograms and expect the result to mean anything. Try it, being sure to include the respective units. Thus, the next step is to choose a “convenient basis,” which is, say, 100 kg mol of gas, and proceed as follows:

Basis: 100 kg mol or lb mol of gas

Set up a table such as the following to make a compact presentation of the calculations. You do not have to, but making individual computations for each component is inefficient and more prone to errors.

Check the solution by noting that an average molecular weight of 23.8 is reasonable because the molecular weights of the components range only from 2 to 44 and the answer is intermediate to these values.

To sum up, be sure to state the basis of your calculations so that you will keep clearly in mind their real nature, and so that anyone checking your problem solution will be able to understand on what basis your calculations were performed.

You no doubt have heard the story of Ali Baba and the 40 thieves. Have you heard about Ali Baba and the 39 camels? Ali Baba gave his four sons 39 camels to be divided among them so that the oldest son got one-half of the camels, the second son a quarter, the third an eighth, and the youngest a tenth. The four brothers were at a loss as to how they should divide the inheritance without killing camels until a stranger came riding along on his camel. He added his own camel to Ali Baba’s 39 and then divided the 40 among the sons. The oldest son received 20; the second, 10; the third, 5; and the youngest, 4. One camel was left. The stranger mounted it—for it was his own—and rode off. Amazed, the four brothers watched him ride away. The oldest brother was the first to start calculating. Had his father not willed half of the camels to him? Twenty camels are obviously more than half of 39. One of the four sons must have received less than his due. But figure as they would, each found that he had more than his share. Cover the next few lines of text. What is the answer to the paradox?

After thinking over this problem, you will realize that the sum of ½, ¼, ⅛, and is not 1 but is 0.975. By adjusting (normalizing) the camel fractions (!) so that they total 1, the division of camels is

What we have done is to change the calculations from a basis total of 0.975 to a new basis of 1.000. More frequently than you probably would like, you will have to change from your original selection of a basis in solving a problem to one or more different bases in order to put together the information needed to solve the entire problem. Consider the following example.

Example 2.16. Changing Bases

Considering a gas containing O₂ (20%), N₂ (78%), and SO₂ (2%), find the composition of the gas on an SO₂-free basis, meaning gas without the SO₂ in it.

Solution

First choose a basis of 1 mol of gas (or 100 mol). Why? The composition for the gas is in mole percent. Next you should calculate the moles of each component, remove the SO₂, and adjust the basis for the calculations so that the gas becomes composed of only O₂ and N₂ with a percent composition totaling 100%:

Basis: 1.0 mol of gas

Self-Assessment Test

Questions