Solutions to General Chemistry (Linus Pauling)/Printable version

Solutions to General Chemistry (Linus Pauling)

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The Nature and Properties of Matter

1-1

What is the difference between matter and radiant energy?

The essential distinction between these two forms of mass-energy is that matter moves at a velocity of less than the speed of light, and that radiant energy moves at the speed of light.

1-2

What is the Einstein relation between mass and energy? Indicate the IS units of the terms in this relation. This is of course the cliched $E=mc^{2}$. In terms of IS units, the equation reads:

$\mathrm {J} =\mathrm {kg} \cdot \left({\frac {\mathrm {m} }{\mathrm {s} }}\right)^{2}$

Note that the term $\mathrm {J}$ (for joules) on the left hand resolves to:

$\mathrm {N} \cdot \mathrm {m} =\mathrm {kg} \cdot {\frac {\mathrm {m} }{\mathrm {s} ^{2}}}\cdot \mathrm {m} =\mathrm {kg} \cdot \left({\frac {\mathrm {m} }{\mathrm {s} }}\right)^{2}$

As such the units are equal on either side, as required.

1-3

Approximately how much energy, in IS units, is needed to raise 1 liter (1 kg) of liquid water from 273.15°K to 373.15°K? (See the discussion of the calorie, Section 1-3.)

The answer will have to be written in joules. However, the most convenient unit for the purposes of calculation is the thermochemical calorie, which is equal to 4.184 J. The thermochemical calorie, in turn, is slightly smaller than the 15°C calorie, the unit of energy required to raise the temperature of a gram of water from 14.5 to 15.5°C at standard temperature. Since only an approximate answer is needed, though, simply bear in mind that the Kelvin and the Celsius scale have the same magnitude. Then the required number of (thermochemical) calories is 100,000 thermochemical calories, or 418,400 joules (418.4 kJ).

1-4

Verify the following... To convert celsius to farenheight we must multiply by 1.8 and addition 32, to convert farenheight to celsius we must substract 32 and then divide by 1.8.

1-5

Mercury freezes at -40°C. What is its freezing point on the Fahrenheit scale?

-40°F (the only temperature which is the same on both scales).

1-6

For each of the following systems systems ([1]) state how many phases are present in the system; ([2]) state for each phase whether it is a pure substance or a mixture; ([3]) give the constituents of the system; ([4]) give a set of components for system:

1. A flask containing a saturated aqueous solution of salt and several crystals of salt.
   1. Two.
   2. The aqueous phase is a mixture, as it contains water and salt. The solid phase, however, can be considered a pure substance.
   3. The constituents are the aqueous and solid phases of salt.
   4. The components are salt and water.
2. An evacuated, sealed quartz tube of 100-ml volume containing 10 g of pure zinc heated until about one half the zinc is melted.
   1. Disregarding the quartz tube itself, there are two phases present in the system.
   2. Both phases exist in the form of pure substances.
   3. The constituents are the liquid and solid phases of zinc.
   4. There is one component, zinc.
3. As ([2]) , but containing 10 g of a copper-gold alloy instead of 10 g of zinc.
   1. There are two phases in the system (using the same assumption regarding the container as before).
   2. Both phases are mixtures, specifically, alloys.
   3. The constituents are the liquid and solid phases of the copper-gold alloy.
   4. There are two components, copper and gold.

1-7

What is meant by "intrinsic property" of a substance? Are odor, shape, density, color, weight, taste, luster, area, magnetic susceptibility, and heat capacity intrinsic properties? Which of these are properties that can be quantitatively measured?

An intrinsic property of a substance is one which is not significantly affected by the size of any given amount of the substance, or its state of subdivision. In other words, a mountain of pulverized salt (sodium chloride) shares in common with a baseball-sized salt crystal certain invariant, intrinsic properties, such as density and cleavage. The color of a substance is an important physical property. It is interesting to note that the apparent color of a substance depends upon its state of subdivision: the color becomes lighter as large particles are ground up into smaller ones, because the distance through which the light penetrates before it is reflected back from the interfaces (surfaces) becomes less as the particles become smaller.

Odor, density, color, taste, magnetic susceptibility and heat capacity are intrinsic properties, though odor and color may be somewhat magnified when the material is finely subdivided. Shape, weight and area are not intrinsic, since they depend on the amount of material.

Density, color, weight, area, magnetic susceptibility and heat capacity can be quantitatively measured.

The Atomic and Molecular Structure of Matter

1. What are the differences between a hypothesis, a theory, a law, and the fact? Classify the following statements as hypotheses, theories, laws, or facts:
   1. The interior of the moon consists of granite and similar rocks.

      This can be considered a fact. The Apollo missions brought back hundreds of kilograms of diverse lunar rocks, including some with a composition not unlike granite.

   2. Hydrogen, nitrogen, oxygen, and neon are all gases under ordinary conditions.

      This statement is a conjunction of directly observed experimental facts.

   3. The force f acting on a body with mass $m$ causes it to be accelerated by the amount fm^-1.

      This statement is an inductive generalization over all forces acting on bodies with mass, which have been borne out by many observations from the seventeenth century onwards. It does not explain why this happens.

   4. The properties of gases can be explained by considering the motion of molecules comprising them.

      The keyword in this statement is 'explain'. Hypotheses and theories aim to explain observations, with the distinction being that theories are supported by a large body of empirical evidence. Specifically, the term 'theory' is ostensibly being used in the second sense given in the text, "a systematic body of knowledge, compounded of facts, laws, theories in the limited sense [of a hypothesis borne out], deductive arguments, and so on." This "systematic body" would encompass areas of physics and chemistry, such as thermodynamics.

   5. All crystals contain atoms or molecules arranged in a regular way.

      On the surface, this statement seems to be a definition, true de facto. However, Pauling appears to imply early on in the chapter that he is referring to the atomic theory of crystals, which explains why crystals have readily apparent regular, highly ordered structures. (Remember that the distinction between a law and a hypothesis or theory is that a law does not explain the facts it summarizes, while a hypothesis or theory does.)

2. Discuss some of the evidence for the atomic nature of matter.

   open-ended

3. The metal indium forms tetragonal crystals. The unit of structure is a rectangular parallelepiped, with edges a = 3.24 Å, b = 3.24 Å, and c = 4.94 Å... TODO

Problem 3 (as written) appears to have no solution. The positions of the atoms are given in the problem as 0,0,0, and 1/2, 1/2, 1/2 which implies a body-centered structure. Indium has a face-centered (not body-centered) tetragonal structure that can be solved. Each atom then has 4 closest neighbors and 8 more a little farther away. Do the math.

The Electron the Nuclei of Atoms and the Photon

1. An ordinary electric light bulb is operated under conditions such that one ampere of current (1 C s^-1) is passing through the filament. How many electrons pass through the filament each second? (Remember that the charge of the electron is -0.160 × 10^-18 C.)

   6.25 × 10¹⁸ electrons are passing through the filament each second. The calculation should be fairly self-evident.

2. According to the law of gravitation, the gravitational force of attraction between two particles with masses m₁ and m₂ a distance r apart is Gm₁m₂ / r², where G, the constant of gravitation has been found by experiment to have the value 0.6673 × 10^-10 N m² kg ^-2; that is, the force of attraction between two particles each with mass 1 kg and the distance 1 m apart is 0.6673 × 10^-10 N
   1. Calculate the force of electrostatic attraction between an electron and a proton 10 Å apart.

      The appropriate rule to apply here is (the scalar form of) Coulomb's Law. To the reader's ire, Coulomb's law is given using Stoney units, while the charges of electrons and protons are typically given in coulombs. The constant required to calculate the electrostatic force is given later on in the chapter and is 8.9876 × 10⁹. That being said, the electrostatic force between the two particles is 8.9876 × 10⁹ × -1.60217646 × 10^-19 C × 1.60217646 × 10^-19 C / (10^-9)², or about -2.30708943 × 10^-10 N. Note the sign; it's important. A negative sign implies an attractive force, while a positive sign implies a repulsive force. Because the electron and proton are of opposite polarity, they attract.

   2. Calculate the force of gravitational attraction between an electron and a proton 10 Å apart. What is the relationship of the electrostatic attraction to the gravitational attraction at this distance?

      Here the formula is given directly and is straightforward once the masses are known. The mass of an electron is 9.10938188 × 10^-31 kg; the mass of a proton, 1.67262158 × 10^-27 kg. The gravitational attraction then is 1.0167349 × 10^-49 N, according to the law of gravitation.

   3. What is the dependence on distance of the ratio of electrostatic attraction and gravitational attraction of an electron and proton?

      This question seems vague, but the author appears to be referring to only the ratio of electrostatic and gravitational force given the same 10 Å distance. It is 2.269116 × 10³⁹.

3. Calculate the velocity with which the electrons would move in the apparatus used by J.J. Thomson, operated at an accelerating voltage of 6000 V. Assume that each electron has kinetic energy equal to eV, where e is the charge of the electron in coulombs and V is the accelerating potential in volts.

   The kinetic energy of the electron is given by:

   $eV={\frac {1}{2}}mv^{2}$

   Inserting known values:

   $1.60217646\cdot 10^{-19}\cdot 6000={\frac {1}{2}}\cdot 9.10938188\cdot 10^{-31}\cdot v^{2}$

   Accordingly, the velocity is about 45,941,095 meters per second. Is it necessary to use relativity here?

Elements and Compounds Atomic and Molecular Masses

4-5

Define isotope, isobar, nuclide, mass number, N, A, Z, the dalton.

• Nuclide: either a nucleus with a certain value of Z and A (see below) or an atom containing such a nucleus.
• Isotope: a nuclide which shares in common with a different nuclide the same value of Z, but differs in N (see below)
• Isotone: a nuclide which shares in common with a different nuclide the same value of N, but differs in Z (see below)
• Isobar: a nuclide which shares in common with a different nuclide the same combined number of protons and neutrons, for example, boron-12 (an exotic radioactive isotope of boron) and carbon-12 (the most common stable form of carbon).
• Mass number: same as A?
• N: the number of neutrons in a nuclide
• Z: the number of protons in a nuclide, also known as atomic number
• A: the sum of neutrons and protons in a nuclide, $N+Z$
• Dalton: exactly one-twelfth the mass of a neutral carbon atom, or approximately, 1.66033 × 10^-27 kg. Used as the standard for atomic mass measurement.

4-6

What are the atomic number and approximate atomic weight of the element each of whose nuclei contains 81 protons and 122 neutrons? Give the complete symbol for this nuclide, including chemical symbol, atomic number, mass number.

The atomic number would be 81; the atomic weight, approximately equal to the sum of protons and neutrons in the nuclide, so what again will work out 203. The complete symbol for this nuclide is then $\displaystyle _{81}^{203}\mathrm {Tl}$.

4-8

An atom of ⁹⁰Sr emits a beta ray. What are the atomic number and mass number of the resulting nucleus? What element is it? This nucleus also emits a beta ray. What nucleus does it produce?

Because strontium-90 undergoes beta decay, specifically, beta minus decay, one of its neutrons becomes a proton and a free electron. As such it becomes yttrium-90, with an atomic number of 39, and a mass number of 90. In turn, yttrium-90 undergoes a further beta minus decay, resulting in zirconium-90, with an atomic number of 40, and a mass number of 90.

4-9

Argon, potassium, and calcium all have nuclides with mass number 40. How many protons and how many neutrons constitute each of the three nuclei?

The number of protons in each nuclide is given by the atomic number. Then the neutrons are whatever number is left over. Argon has 18 protons and 22 neutrons; potassium has 19 and 21; and calcium, 20 and 20.

4-11

What would be the implications of defining Avogadro's number as 1.00000 × 10²⁴? What units would have to be changed?

The redefinition of Avogrado's number, arbitrary as it is, would upset a great many chemical formulae. Because atomic mass would no longer be convenient for use with the dalton, which would have to be altered to restore the relationship between moles and atomic mass. is it possible to elaborate here?

4-13

The molecule of the anesthetic agent nitrous oxide, N₂O, contains two atoms of nitrogen and one atom of oxygen. Using Avogrado's number and the atomic weights of nitrogen and oxygen, calculate the weight in kilograms of one atom of oxygen and that of two atoms of nitrogen. Also calculate the weight of the nitrous oxide molecule. What is the percentage composition by weight of nitrous oxide in terms of nitrogen and oxygen?

The atomic weight (or, better said, atomic mass) of oxygen is 15.9994. In other words, a mole of oxygen is 15.9994 g. Dividing this number by Avogadro's constant, the number of molecules of a substance in a mole, or simply N, the mass of an oxygen atom is 2.65676255 × 10^-23 grams. (Remember units.) So, the mass of an oxygen atom is 2.65676255 × 10^-26 in kilograms. Through virtually identical calculations, the mass in kilograms of a nitrogen atom is 2.32586697 × 10^-26, and of two, it is 4.65173394 × 10^-26. The mass of a nitrous oxide molecule is then 7.30849649 × 10^-26 kg. The percentage (by mass) of oxygen in a quantity of nitrous oxide is then the quotient of the mass of an oxygen atom and the mass of nitrous oxide molecule, approximately 36.35%. Likewise, the percentage by mass of nitrogen in nitrous oxide is approximately 63.65%.

4-14

Balance the following equations of chemical reactions. The molecular formulas are correct.

• Fe₂O₃ + C → Fe + CO₂
• Ag + S₈ → Ag₂S
• C₁₂H₂₂O₁₁ + O₂ → CO₂ + H₂O
• H₃PO₄ + NaOH → Na₃PO₄ + H₂O
• Li + H₂O → LiOH + H₂
• HCl + Ba(OH)₂ → BaCl₂ + H₂O
• CuO + NH₃ → N₂ + Cu + H₂O
• N₂ + H₂ → NH₃
• H₃BO₃ → B₂O₃ + H₂O
• Fe + F₂ → FeF₃

The author informs the reader that the only things that need to be changed are the amounts of each substance. The most general way to do this is through the solution of a system of linear equations. Like most algorithms, the method is best shown by example.

First, assign an unknown variable to each substance in the reaction. For instance:

wFe₂O₃ + xC → yFe + zCO₂

Then, for each element mentioned in the reaction, write a linear equation equating the relative amount of each element on each side and relating the variables given above:

${\begin{aligned}2w&=&y\\3w&=&2z\\x&=&z\end{aligned}}$

The equations are written for the balances of iron, oxygen, and carbon, respectively. Taking iron, for example, the appropriate equation is $2w=y$, because ferric oxide (Fe₂O₃) on the left contains two iron atoms, whereas its product on the right is pure iron. So, for a given amount w of reactant ferric oxide molecules, there are 2w iron atoms on the left side, which must be equal to the number of iron atoms on the right side, y.

Finally, these equations must be combined into a system of linear equations which is then solved to give the relative amounts of each substance in the reaction

${\begin{aligned}2w&&&-&y&&&=&0\\3w&&&&&-&2z&=&0\\&&x&&&-&z&=&0\end{aligned}}$

The solution leaves z free, parameterizing the other variables like so:

${\begin{aligned}x&=&z\\w&=&{\frac {2}{3}}z\\y&=&{\frac {4}{3}}z\end{aligned}}$

Because the reaction coefficients are necessarily integers, let z equal 3. x then is three; w, 2; and y, 4. The reaction is 2Fe₂O₃ + 3C → 4Fe + 3CO₂.

The remaining reactions, which can be determined in the same manner, are:

• 16Ag + S₈ → 8Ag₂S
• C₁₂H₂₂O₁₁ + 12O₂ → 12CO₂ + 11H₂O
• H₃PO₄ + 3NaOH → Na₃PO₄ + 3H₂O
• 2Li + 2H₂O → 2LiOH + H₂
• 2HCl + Ba(OH)₂ → BaCl₂ + 2H₂O
• 3CuO + 2NH₃ → N₂ + 3Cu + 3H₂O
• N₂ + 3H₂ → 2NH₃
• 2H₃BO₃ → B₂O₃ + 3H₂O
• 2Fe + 3F₂ → 2FeF₃

4-15

How much coal (assumed to be pure carbon) is needed to reduce one ton of Fe₂O₃ to iron? How much iron is produced?

First, notice that the reaction is already given: 2Fe₂O₃ + 3C → 4Fe + 3CO₂. Because the amount of ferric oxide given in the equation is in tons, it is necessary to convert it to grams. Then there are about 907,184.74 grams of ferric oxide in the reaction. In itself, however, the mass is not useful for the purposes of this problem. What is needed is the number of molecules of ferric oxide in the reaction, a number manageable in the form of moles. The atomic mass of ferric oxide is 2 × 55.845 + 3 × 15.9994 = 159.6882. There are then about 5,680.98 moles of ferric oxide in the reaction. The reaction specifies that for every two moles of ferric oxide, there must also be three moles of carbon. So there 8,521.47 moles of carbon are needed, or 8,521.47 × 12.0107 = 102,348.82 g. Note that, although the molar amount of carbon needed for the reaction is greater by half than the amount of ferric oxide, a chemist ordering the coal in Imperial units would ask for a little more than 225 pounds, due to the lower molar mass of coal.

4-16

Exactly what is meant by the statement that the atomic weight of samarium is 150.33?

In the broadest sense, this means that that the average mass of naturally occurring samarium atoms is 150.33 daltons (see above), as the mass of an individual samarium atom varies according to its isotope. Specifically, this number is the sum of the masses of the 62 protons, 88 electrons, and (to a much lesser extent) the 62 electrons in the average samarium atom.