The Mole

Mole is abbreviated mol. Do not use m or M for mole these symbols are used for other quantities related to moles, and so you will be confused if you use either of them. Note A mole is referred to by some authors as a gram molecular mass because 1 mol of molecules has a mass in grams equal to its molecular mass. In this terminology, a gram atomic mass is 1 mol of atoms, and a gram formula mass is 1 mol of formula units. 

The formula mass of a substance is equal to its number of grams per mole. Avogadro s number is the number of atomic mass units in 1 g. It is defined in that manner so that the atomic mass of an element (in amu) is 

Avogadro s number appears in both the numerator and the denominator of this expression the values reduce to a factor of 1 (they cancel), and the numeric value in grams per mole is equal to the numeric value of the atomic mass in amu per atom. 

A similar argument leads to the conclusion that the formula mass of any element or compound is equal to the number of grams per mole of the element or compound. 

EXAMPLE 7.3. How many feet taU is a stack of a dozen shoe boxes, each 4 in. taU  

The atom is an incredibly tiny object. Its mass is far too small to measure on an ordinary balance. In Chapter 5 (Section 5.9), we learned to compare atoms using a table of atomic mass units. These units are valuable when we compare the masses of individual atoms (mentally), but they have no practical use in the laboratory. The mass in grams for an average carbon atom (atomic mass 12.01 amu) would be 2.00 X 10 g, which is much too tiny for the best laboratory balance. 

So how can we confidently measure masses for these very tiny atoms We increase the number of atoms in a sample until we have an amount large enough to measure on a laboratory balance. The problem then is how to count our sample of atoms. 

Fie can now weigh 64.6 kg of apples and 65.1 kg of oranges and pack them without actually counting them. Manufacturers and suppliers often count by weighing. Other examples of counting by weighing include nuts, bolts, and candy. 

Chemists also count atoms by weighing. We know the average masses of atoms, so we can count atoms by defining a unit to represent a larger number of atoms. Chemists have chosen the mole (mol) as the unit for counting atoms. The mole is a unit for counting just as a dozen or a ream or a gross is used to count  

1 dozen = 12 objects 1 ream = 500 objects 1 gross = 144 objects 1 mole = 6.022 x 10 objects 

On the last two pages you read about relative atomic masses and formula masses. These are not just boring numbers—they are very important things for a chemist to know  

If you work out the RAM or formula mass of a substance, and then weigh out that number of grams of the substance, you can say how many atoms or molecules it contains. 

This is very useful, since single atoms and molecules are far too small to be counted. 

For example, the RAM of carbon is 12. The photograph on the right shows 12 grams of carbon. 

This huge number of atoms is called a mole of atoms. 

At the grocery store, you buy eggs by the dozen or soda by the case. In an office-supply store, pencils are ordered by the gross and paper by the ream. Conunon terms such as dozen, case, gross, and ream are used to count the number of items present For example, when you buy a dozen eggs, you know you will get 12 eggs in the carton. 

In chemistry, particles such as atoms, molecules, and ions are counted by the mole (abbreviated mol in calculations), a unit called Avogadro s number that contains 6.022 X 10 items. Avogadro s number is a very big number because atoms are so small that it takes an extremely large number of atoms to provide a sufficient amount to weigh and use in chemical reactions. Avogadro s number is named for Amedeo Avogadro (1776-1856), an Italian physicist. 

Collections of items include dozen, gross, and mole. 

One mole of any element always contains Avogadro s number of atoms. For example, 1 mol of carbon contains 6.022 X 10 carbon atoms 1 mol of aluminum contains 

022 X 10 aluminum atoms 1 mol of sulfur contains 6.022 X 10 sulfur atoms. 

The mole represents a large number of extremely small particles. 

A molecular formula of a compound is a whole-number multiple of its empirical formula. 

Mint has never officially produced a coin called the penny the official name is the United States one-centcoin. 

The present-day penny is copper-plated zinc, and has a composition of 97.5% Zn and 2.5% Cu. 

The Denver and Philadelphia Mints produce 65 million to 80 million coins a day. 

As in the case of carbon, the mass for each element listed in the table inside the front cover of the text is an average value based on the isotopic composition of the naturally occnrring element. For instance, the mass listed for hydrogen (1.008) is the average mass for natural hydrogen, which is a mixture of and (deuterium). No atom of hydrogen actually has the mass 1.008. 

In addition to being useful for determining accurate mass values for individual atoms, the mass spectrometer is used to determine the isotopic composition of a natural element. For example, when a sample of natural neon is injected into amass spectrometer, the mass spectrum shown in Fig. 3.2 is obtained. The areas of the peaks or the heights of the bars indicate the relative abundances of loNe, loNe, and atoms. 

When a sample of natural copper is vaporized and injected into a mass spectrometer, the results shown in Fig. 3.3 are obtained. Use these data to compute the average mass of natural copper. (The mass values for Cu and Cu are 62.93 u and 64.93 u, respectively.) 

As shown by the graph, of every 100 atoms of natural copper, 69.09 are and 30.91 

This mass value is used in doing calculations involving the reactions of copper and is the value given in the table inside the front cover of this book. 

Identify and calculate the number of representative particles in each of the following quantities. 

Calculate the number of moles of the substance that contains the following number of representative particles. 

The SI definition of the mole is the amount of a substance that contains as many entities as there are in exactly 0.012 kg (12 g) of carbon-12. 

One-mole samples of copper, sulfur, mercury, and carbon. 

The mass of 1 mole of an element is equal to its atomic mass in grams. 

The magnitude of the number 6.022 X 1023 is very difficult to imagine. To give you some idea, 1 mole of seconds represents a span of time 4 million times as long as the earth has already existed, and 1 mole of marbles is enough to cover the entire earth to a depth of 50 miles However, since atoms are so tiny, a mole of atoms or molecules is a perfectly manageable quantity to use in a reaction (see Fig. 3.4). 

How do we use the mole in chemical calculations Recall that Avogadro s number is defined as the number of atoms in exactly 12 grams of 12C. Thus 12 grams of 12C contains 6.022 X 1023 atoms. Also, a 12.01-gram sample of natural carbon contains 6.022 X 1023 atoms (a mixture of 12C, 13C, and 14C atoms, with an average mass of 12.01). Since the ratio of the masses of the samples (12 g/12.01 g) is the same as the ratio of the masses of the individual components (12 amu/12.01 amu), the two samples contain the same number of components. 

Avogadro s number is 6.022 x 10 1 One mole of anything is 6.022 x 10 units of that substance. 

To understand the mole concept and Avogadro s number. To learn to convert among moles, mass, and number of atoms in a given sample. 

In the previous section we used atomic mass units for mass, but these are extremely small units. In the laboratory a much larger unit, the gram, is the convenient unit for mass. In this section we will learn to count atoms in samples with masses given in grams. 

Let s assume we have a sample of aluminum that has a mass of 26.98 g. What mass of copper contains exactly the same number of atoms as this sample of aluminum  

To answer this question, we need to know the average atomic masses for aluminum (26.98 amu) and copper (63.55 amu). Which atom has the greater atomic mass, aluminum or copper The answer is copper. If we have 26.98 g of aluminum, do we need more or less than 26.98 g of copper to have the same number of copper atoms as aluminum atoms We need more than 26.98 g of copper because each copper atom has a greater mass than each aluminum atom. Therefore, a given number of copper atoms will weigh more than an equal number of aluminum atoms. How much copper do we need Because the average masses of aluminum and copper atoms are 26.98 amu and 63.55 amu, respectively, 26.98 g of aluminum and 63.55 g of copper 

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In chemistry, we calculate and measure the amounts of substances to use in lab. Actually, measuring the amount of a substance is something you do every day. When you cook, you measure out the proper amounts of ingredients so you don t have too much of one or too little of another. At the gas station, you pump a certain amount of fuel into your gas tank. If you paint the walls of a room, you measure the area and purchase the amount of paint that will cover the walls. In the lab, the chemical formula of a substance tells us the number and kinds of atoms it has, which we then use to determine the mass of the substance to use in an experiment. 

Chemical reactions occur everywhere. The fuel in our cars burns with oxygen to make the car move and run the air conditioner. When we cook our food or bleach our hair, chemical reactions take place. In our bodies, chemical reactions convert food into molecules that build muscles and move them. In the leaves of trees and plants, carbon dioxide and water are converted into carbohydrates. 

Some chemical reactions are simple, whereas others are quite complex. However, they can all be written with chemical equations that chemists use to describe chemical reactions. In every chemical reaction, the atoms in the reacting substances, called reactants, are rearranged to give new substances called products. 

Writing Conversion Factors from Equalities (2.5) Using Conversion Factors (2.6) 

Use Avogadro s number to determine the number of particles in a given number of moles. 

All these samples of pure elements contain the same number (a mole) of atoms 6.022 x 10 atoms. 

We need to digress to define the mole, which is just a number, in the same way that a gross is 144 or a ream is 500 sheets of paper. First, we need to review atomic weight. The atomic weights in the periodic table (see Appendix 1) represent the average mass of an atom of the element in atonfic mass units. Some atomic weights are given below in Table 5  

These numbers tell us the relative weights of the atoms. 

How much heavier is an average atom of titanium than an average atom of hydrogen  

What mass of titanium will contain the same number of atoms as l.Og of hydrogen  

Atoms of titanium are 47.5 times heavier than those of hydrogen. Therefore, 48 g (we are rounding off to two significant figures) of titanium will contain the same number of atoms as l.Og of hydrogen. 

5 What is the total number of atoms contained in 2.00 moles of helium  

This question covers NSCS Bl. This question tests the material that was covered in the textbook on page 311. 

6 A compound has the formula MgS04 7H20. Its chemical name is — 

7 Indium (In) is a relatively rare element that never occurs as a free metal. It is usually found in a compound that contams 70.48% In and 29.52% S. What is the empirical formula for this compound  

Each copper atom has a greater mass than each aluminum atom. We will need more mass of copper than aluminum for the same number of atoms. 

These masses contain the same number of atoms. 

The mass of copper found is greater than the mass of aluminum. This makes sense because a copper atom has a greater mass than an aluminum atom. 

People in different professions often use special counting units. You and I eat eggs one at a time, but farmers sell them by the dozen. We spend dollar bills one at a time, but Congress distributes them by the billion. Chemists have their own counting unit, Avogadro s number (Section 2.3). 

A mole represents not only a speciflc number of particles but also a definite mass of a substance as represented by its formula (O, O2, H2O, NaCl,. . . ). The molar mass, MM, in grams per mole, is numerically equal to the sum of the masses (in amu) of the atoms in the formula. 

Notice that the formula of a substance must be known to find its molar mass. It would be ambiguous, to say the least, to refer to the molar mass of hydrogen. One mole of hydrogen atoms, represented by the symbol H, weighs 1.008 g the molar mass of H is 1.008 g/mol. One mole of hydrogen molecules, represented by the formula H2, weighs 2.016 g the molar mass of H2 is 2.016 g/mol. 

As you will see later in this chapter, it is often necessary to convert from moles of a substance to mass in grams or vice versa. Such conversions are readily made by using the general relation 

The mass is in grams, MM is the molar mass (g/mol), and n is the amount in moles. In effect, the molar mass, MM, is a conversion factor that allows you to calculate moles from mass or vice versa. 

One-mole amounts of sugar (C12H22O11), baking soda (NaHC03), and copper nails (Cu). 

The size of the mole was chosen to make this relation true. 

Formula Sum of Atomic Masses Molar Mass, MM CENGAGENOW  

New materials, new products, new consumer goods of all kinds come on the market regularly. But before manufacturing begins on most new products, calculations involving the mole must be done. 

Visit the Chemistry Web site at chemistrymc.com to find links to the mole. 

Florists often sell flowers, such as roses, carnations, and tulips, by the dozen. A dozen is a counting unit for 12 items. 

If a mole is 6.02 x 10 items, how far will a mole of paper clips, placed end to end lengthwise, reach into space  

How many light-years (ly) would the paper clips extend into space  

The concept of amount of substance is central to chemical measurement. The amount of substance of a system is proportional to the number of elementary entities of that substance present in the system. The elementary entities must be described they may be atoms, molecules, ions, or specified groups of such particles. The entity itself is a natural unit for measuring the amount of substance for example, we can describe the amount of substance in a sample of iron by saying that there are 2.0 x 10 Fe atoms in the sample. The amount of substance in a crystal of NaCl can be described by saying that there are 8.0 X 10 ° ion pairs, Na Cl , in the crystal. 

Since any tangible sample of matter contains such an enormous number of atoms or molecules, a unit larger than the entity itself is needed to measure the amount of substance. The SI unit for amount of substance is the mole. The mole is defined as the amount of substance in exactly 0.012 kg of carbon-12. One mole of any substance contains the same number of elementary entities as there are carbon atoms in exactly 0.012 kg of carbon-12. This number is the Avogadro constant, = 6.022045 x 10 mol . 

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In vapor-liquid equilibria, if one phase composition is given, there are basically four types of problems, characterized by those variables which are specified and those which are to be calculated. Let T stand for temperature, P for total pressure, for the mole fraction of component i in the liquid phase, and y for the mole fraction of component i in the vapor phase. For a mixture containing m components, the four types can be organized in this way ... 

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

According to Equation (14), the fugacity of component i becomes equal to the mole fraction multiplied by the standard-... 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Finally, Table 2 shows enthalpy calculations for the system nitrogen-water at 100 atm. in the range 313.5-584.7°K. [See also Figure (4-13).] The mole fraction of nitrogen in the liquid phase is small throughout, but that in the vapor phase varies from essentially unity at the low-temperature end to zero at the high-temperature end. In the liquid phase, the enthalpy is determined primarily by the temperature, but in the vapor phase it is determined by both temperature and composition. 

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

The total enthalpy correction due to chemical reactions is the sum of all the enthalpies of dimerization for each i-j pair multiplied by the mole fraction of dimer i-j. Since this gives the enthalpy correction for one mole of true species, we multiply this quantity by the ratio of the true number of moles to the stoichiometric number of moles. This gives... 

If a reaction is reversible, there is a maximum conversion that can be achieved, the equilibrium conversion, which is less than 1.0. Fixing the mole ratio of reactants, temperature, and pressure fixes the equilibrium conversion. ... 

This is an exothermic, reversible, homogeneous reaction taking place in a single liquid phase. The liquid butadiene feed contains 0.5 percent normal butane as an impurity. The sulfur dioxide is essentially pure. The mole ratio of sulfur dioxide to butadiene must be kept above 1 to prevent unwanted polymerization reactions. A value of 1.2 is assumed. The temperature in the process must be kept above 65°C to prevent crystallization of the butadiene sulfone but below lOO C to prevent its decomposition. The product must contain less than 0.5 wt% butadiene and less thM 0.3 wt% sulfur dioxide. 

Raoult s law When a solute is dissolved in a solvent, the vapour pressure of the latter is lowered proportionally to the mole fraction of solute present. Since the lowering of vapour pressure causes an elevation of the boiling point and a depression of the freezing point, Raoult s law also applies and leads to the conclusion that the elevation of boiling point or depression of freezing point is proportional to the weight of the solute and inversely proportional to its molecular weight. Raoult s law is strictly only applicable to ideal solutions since it assumes that there is no chemical interaction between the solute and solvent molecules. 

The value of coefficient depends on the composition. As the mole fraction of component A approaches 0, approaches ZJ g the diffusion coefficient of component A in the solvent B at infinite dilution. The coefficient Z g can be estimated by the Wilke and Chang (1955) method ... 

The data in Table III-2 have been determined for the surface tension of isooctane-benzene solutions at 30°C. Calculate Ff, F, F, and F for various concentrations and plot these quantities versus the mole fraction of the solution. Assume ideal solutions. 

The following data (for 25°C) were obtained at the pzc for the Hg-aqueous NaF interface. Estimate and plot it as a function of the mole fraction of salt in solution. In the table,/ is mean activity coefficient such that a = f m , where m is mean molality. 

The surface excess per square centimeter F is just n/E, where n is the moles adsorbed per gram and E is the specific surface area. By means of the Gibbs equation (111-80), one can write the relationship... 

The moles of a solute species adsorbed per gram of adsorbent nl can be expressed in terms of the mole fraction of the solute on the surface N and the moles of adsorption sites per gram as... 

A surfactant solution is a mixture of DTAC (dodecyltrimethylammonium chloride) and CPC (cetyl pyridinium chloride) the respective CMCs of the pure surfactants are 2 X lO M and 9 x IO M (Ref. 140). Make a plot of the CMC for mixtures of these surfactants versus the mole fraction of DTAC. 

These concluding chapters deal with various aspects of a very important type of situation, namely, that in which some adsorbate species is distributed between a solid phase and a gaseous one. From the phenomenological point of view, one observes, on mechanically separating the solid and gas phases, that there is a certain distribution of the adsorbate between them. This may be expressed, for example, as ria, the moles adsorbed per gram of solid versus the pressure P. The distribution, in general, is temperature dependent, so the complete empirical description would be in terms of an adsorption function ria = f(P, T). 

Alternatively, 6 can be replaced by n/n , where denotes the moles per gram adsorbed at the monolayer point. Thus... 

Equation (A2.1.23) can be mtegrated by the following trick One keeps T, p, and all the chemical potentials p. constant and increases the number of moles n. of each species by an amount n. d where d is the same fractional increment for each. Obviously one is increasing the size of the system by a factor (1 + dQ, increasing all the extensive properties U, S, V, nl) by this factor and leaving the relative compositions (as measured by the mole fractions) and all other intensive properties unchanged. Therefore, d.S =. S d, V=V d, dn. = n. d, etc, and... 

The molar Helmholtz free energy of mixing (appropriate at constant volume) for such a synnnetrical system of molecules of equal size, usually called a simple mixture , is written as a fiinction of the mole fraction v of the component B... 

The treatment of such order-disorder phenomena was initiated by Gorsky (1928) and generalized by Bragg and Williams (1934) [5], For simplicity we restrict the discussion to the synnnetrical situation where there are equal amounts of each component (x = 1/2). The lattice is divided into two superlattices a and p, like those in the figure, and a degree of order s is defined such that the mole fraction of component B on superlattice p is (1 +. s)/4 while that on superlattice a is (1 -. s)/4. Conservation conditions then yield the mole fraction of A on the two superlattices... 

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