Part 225

The invoice value of 48 bottles is 48 × 90 cents = $43.20 Tare and leakage are 7% of $43.20 = $ 3.024 -------- Value on which duty is paid $40.176

Ad valorem duty is 25% of $40.176 = $10.044 Specific duty is 48 × 20 cents = 9.60 -------- Total duty $19.644

The total cost is

Invoice value $43.20 Ad valorem duty 10.04 Specific duty 9.60 ------ $62.84

The total cost per bottle is 1/48 of $62.84, or $1.31-.

SQUARE ROOT AND CUBE ROOT

POWERS AND ROOTS.--When a product consists of the same factor repeated any number of times it is called a _power_ of that factor.

7 × 7 is the _second power_, or the _square_ of 7.

7 × 7 × 7 is the _third power_, or the _cube_ of 7.

A power of a number is generally expressed by writing the number only once, and placing after it, above the line, a small figure to show how many factors are to be taken. The small figure is called an _index_.

Thus, 7² = 49; 7³ = 343; 7⁴ = 2401.

A number is called the _square root_ of its square.

Since 7² = 49, the square root of 49 is 7.

The “square root of 49” is written √49.

Again, a number is called the _cube root_ of its cube. 7³ = 343. Therefore, the cube root of 343 is 7.

The “cube root of 343” is written ∛343.

A _perfect square_ is a number whose square root is a whole number. A _perfect cube_ is a number whose cube root is a whole number.

SQUARE ROOT.--If a number can be put into prime factors, its square root can be written down by inspection.

EXAMPLE: Find the square root of 27225.

Since 27225 = 3² × 5² × 11².

∴ √27225 = 3 × 5 × 11 = 165 _Ans._

RULE FOR DIGITS.--We know that √1 = 1, and √100 = 10. Therefore, the square root of any number which lies between 1 and 100 lies between 1 and 10; _i.e._, if a number contains one or two digits, its square root consists of one digit.

Similarly, since √100 = 10 and √10000 = 100, the square root of a number between 100 and 10000 lies between 10 and 100. That is, if a number contains three or four digits, its square root consists of two digits.

Proceeding in this way, we obtain a general result--viz., the square of a number has either twice as many digits as the number, or one less than twice as many.

Hence, to ascertain the number of digits in the square root of a perfect square, mark off the digits in pairs, beginning from the right. Each pair marked off gives a digit in the square root; and, if there is an odd digit remaining, that digit also gives a digit in the square root.

EXAMPLES: There are three digits in the square root of 546121, and four in the square root of 5774409.

For, marking off the digits from the right, we get in the first case 54,61,21, giving three digits in the square root, and in the second case 5,77,44,09, the odd digit giving the fourth in the square root.

The method of finding the square root of a given number depends on the _form_ of the square of the sum of two numbers.

EXPLANATION: The square root of 144 is 12. Let us see how we found it.

12 = 1 ten + 2 units.

12² is the same as (10 + 2)².

Let us square (10 + 2), that is, multiply 10 + 2 by 10 + 2.

10 + 2 10 + 2 --------------- 10² + (10 × 2) + (10 × 2) + 2² -------------------- 10² + 2(10 × 2) + 2²

Then, 12² = 10² + 2(10 × 2) + 2²

RULE.--_The square of any number made up of tens and units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units._

ANOTHER EXPLANATION: Find the square root of 45369.

SOLUTION:

4·53·69)213 4 --- ---+-------- 41|53 |41 +---- 423|1269 |1269 +---- |

(1) Point off the number into periods of two figures each, as before.

(2) The square root of the first period is 2. 2 × 2 = 4. Write the 2 in the root and subtract the 4 from 4. Bring down the next period, 53.

(3) 2 × 2 = 4. (Remember the 4 is to be used as a trial divisor, being 2 × the _tens_.)

4 is contained in 5 about 1 time.

Place 1 in the root, also on the right of the 4 in the divisor. Multiply 41 by 1. Subtract and bring down the next period.

(4) 2 × 21 = 42. 42 is the _trial divisor_. 126 ÷ 42 = _about_ 3 times. Place the 3 in the root also at the right of the 42 in the divisor. Multiply out.

Square root = 213.

CUBE ROOT.--The _cube root_ of a number is one of the three equal factors of that number.

Thus, 5 is the cube root of 125, because 5 × 5 × 5 = 125.

The _radical sign_ with a figure 3 over it (∛) means that the cube root of the number following it is to be taken.

∛125 reads, “The cube root of 125.”

If we can find the prime factors of any perfect cube, we can write down its cube root by inspection.

EXAMPLE: Find the cube root of 74088.

8|74088 +----- 9| 9261 +----- ∴ 74088 = 8 × 9 × 3 × 7 × 7 × 7 3| 1029 = 2³ × 3³ × 7³ +----- 7| 343 ∴ ∛74088 = 2 × 3 × 7 +----- = 42 _Ans._ 7| 49 +----- | 7 +-----

RULE FOR DIGITS.--Since 1³ = 1 and 10³ = 1000, therefore the cube of a number which lies between 1 and 10 lies between 1 and 1000, _i. e._, the cube of a number of one digit contains either one, two or three digits.

Again, since 10³ = 1000 and 100³ = 1000000, the cube of a number of two digits contains either four, five, or six digits.

Proceeding in this way, we see that the cube of a number contains three times, or one less or two less than three times, as many digits as the number.

Hence, to find the number of digits in the cube root of a given number, we mark off the digits in sets of three, beginning at the decimal point, and marking both to the right and to the left.

Thus, 289383 will be pointed off into two periods--289·383--and we readily see there will be only 2 figures in the root.

The simplest method of finding the cube root of numbers whose prime factors are not known is analogous to the method of finding square root, being based upon the form of the cube of the sum of two numbers.

EXPLANATION: The cube root of 1728 is 12. Let us see how we found it.

12 = 1 ten + 2 units 12³ = (10 + 2)³ (10 + 2)³ means 10 + 2 × 10 + 2 × 10 + 2 10 + 2 10 + 2 -------------- 10² + (10 × 2) + (10 × 2) +2² -------------------- 10² + 2(10 × 2) + 2² 10 + 2 ----------------------------- 10³ + 2(10² × 2) + (10 × 2²) + (10² × 2) + 2(10 × 2²) + 2³ ---------------------------------- 10³ + 3(10² × 2) + 3(10 × 2²) + 2³

That is, the cube of any number made up of tens and units equals--

_The cube of the tens + three times the product of the square of the tens by the units + three times the product of the tens by the square of the units + the cube of the units_,

or

tens³ + 3(tens² × units) + 3(tens × units²) + units³.

For graphic illustration the geometrical representation of the cube of units and tens in the drawings is helpful.

After the process is understood, this short method of writing the work may be used by the pupil:

EXAMPLE: Find the cube root of .0163956, carrying the root to 3 decimal places.

WORK:

.016·395·600).254+ 8 ------ ------+----- 1200|8395 300| 25| ----+ 1525|7625 +------- | 770600 187500| 3000| 16| ------+ 190516| 762064 +------- | 8536

CHEMISTRY

ITS USE AND IMPORTANCE -- WHAT IT IS -- HOW IT DIFFERS FROM PHYSICS -- ITS DIVISIONS -- DISTINCTION BETWEEN THEORETICAL AND PRACTICAL CHEMISTRY -- OUTLINE OF THEORETICAL CHEMISTRY -- LAWS OF CHEMISTRY -- ATOMIC THEORY -- CHEMICAL NOTATION -- MOLECULAR WEIGHTS -- REACTIONS -- CHEMICAL ARITHMETIC -- BASES -- QUANTIVALENCE -- TESTS -- TABLE OF CHEMICAL ELEMENTS -- -CHEMISTRY OF FAMILIAR THINGS -- COMMON NAMES OF CHEMICALS -- RADIO-ACTIVITY AND RADIO-ACTIVE SUBSTANCES -- RADIUM AND ITS USES -- THE SPINTHARISCOPE

IMPORTANCE OF CHEMISTRY

A certain amount of knowledge of chemistry is eminently useful in almost every walk of life. An intelligent knowledge of the chemistry involved in the processes of the kitchen, the dairy, the dye-house, the farm, or the manufactory, places the possessor engaged in any of these processes on a different level from the rule-of-thumb worker, who is as ignorant of the reason for adopting a particular method as he is of the properties of the materials he employs.

_Technical_ chemistry deals especially with the application of the principles and processes of chemistry to the arts and manufactures, and it is to those who are engaged in manufactures of almost every kind that a knowledge of chemistry is a particular advantage.

It is not a question of expediency alone, but one of absolute necessity that a technical education, including chemistry as one of its principal subjects, should form not the least important part of the equipment for his work of any artisan who is to excel in his employment in intelligence and skill.

_What is chemistry?_

Chemistry is that branch of science which treats of the _intimate composition of matter_, and the changes produced in it when subjected to particular conditions--such as _temperature_, _pressure_, _mass_, _light_, _catalysis_, etc.

_How does chemistry differ from physics?_

The two branches, physics and chemistry, overlap a great deal, it being very difficult to draw the line of demarcation between them, particularly in the higher stages of the _physical_ and _chemical_ changes of matter.

For example, a steel needle rubbed on a magnet in a definite way undergoes _physical_ change by means of which it acquires the power of the magnet. On the other hand, a match rubbed on a match-box undergoes a _chemical_ change by means of which flame is produced. Thus it is possible to make a distinction between the sciences of physics and chemistry. A chemical change involves some alteration in the essential nature of the substance. The match having been ignited has undergone a permanent change, whereby it is no longer combustible. The physical change quoted above involves no alteration in the substance itself, and the acquired property is further only temporary and can be continually lost and reacquired.

The difficulty occurs in this fact, however, that every chemical change is accompanied by physical change, and the physical change may often be the only sign that chemical change has taken place.

_What are the chief divisions of chemistry?_

ORGANIC AND INORGANIC CHEMISTRY.--There are two great divisions in the science of chemistry, organic and inorganic. The branch which is best known is that of inorganic chemistry, which covers the chemistry of all the purely mineral substances. Organic chemistry has to do primarily with that of substances obtained from animal or vegetable sources. Now, however, it has resolved itself into the study of the compounds of carbon, always bearing in mind the fact that many carbon compounds have no organic origin, and therefore really fall outside the scope of organic chemistry.

The fundamentals of both branches are the same, and the real reason for the division is the number of the carbon compounds and their highly complex character. It is in this realm that the graphic formula is of most service, and in its organic branch chemistry most nearly approaches biology.

The branch of inorganic chemistry which treats of the composition, etc., of naturally occurring minerals, receives the title of _mineralogical chemistry_.

PHYSICAL CHEMISTRY explains processes, formulates laws for these processes, and is divided within itself again into electro-chemistry and thermo-chemistry, etc. One branch of physical chemistry in which great strides have been made, is the study of the general properties of gases. But it is really as much in the realm of physics as it is in the realm of chemistry.

The study of the chemical nature of substances entering into the constitution of the animal organism, and the chemical changes taking place during the life processes of animals, forms the domain of _physiological chemistry_.

The investigation of the influence of soils, and manures, etc., of different compositions, upon vegetable life, and the chemical principles underlying the art of agriculture, are included in the province of _agricultural chemistry_.

_Pharmaceutical chemistry_ deals with the nature and mode of preparation of the various drugs, ointments, etc., employed for medicinal purposes.

The science in its relations to the arts, manufactures, and industrial processes is embraced in the wide titles of _technical_ and _applied chemistry_.

_What is the difference between theoretical and practical chemistry?_

There are in every science two great divisions. These are known as the “theory” and the “practice” (or, as they are sometimes called, the science and the art). The theory of any science is that part of it which forms the answer in any case to the question “Why?” The practice in the same way answers to the question “How?”

If we find, for example, that by putting a fire under a vessel of water, the water gradually begins to boil, as we say, “boils away,” we have learned something that relates to _practice_. We have learned how to change water into vapor. It is not necessary that we should know why the result is brought about, so long as we are satisfied with the result alone.

But as soon as we begin to wish to bring about any result in the best possible way, we must inquire why a certain course of action causes the result; and in the case of the water, we ask why heat should make water boil and then disappear. The answer to the question “How?” is usually a simple one. It can be found out by experiment. Once having found out, we may usually repeat the work as often as we choose.

But the question “Why?” lies deeper, and sometimes cannot be answered at all. The answer to it is in all cases merely a guess--an attempt to explain more or less fully and satisfactorily. If we find that our explanation or theory makes it possible to foretell what will happen in new cases, then we may safely trust it and believe in it.

_Give a clear, succinct outline of the essentials of theoretical chemistry._

The whole matter of molecules and atoms is one of _theory_. None of our senses can enable us to know directly either molecules or atoms. We can only imagine that they exist, and then give reasons why their existence makes clear to us the action of elements or of compounds one upon the other.

But in a course of descriptive chemistry, a good knowledge of theoretical chemistry is necessary in order to fully understand all that will be taken up.

THEORETICAL CHEMISTRY

(1) DEFINITIONS.--An element is a substance that _cannot_ be decomposed.

A compound is a substance that _can_ be decomposed into other different substances; and if the decomposition goes far enough, these substances will be elements.

A mixture is made up of two or more components (elements and compounds or both), _physically_ put together. It differs from a compound whose compounds are _chemically_ united.

(2) LAWS.--_Law of Definite Proportions_: All specimens of a compound contain the same elements in the same proportions.

_Law of Multiple Proportions_: When two compounds consist of the same elements, the proportion of one is a simple multiple of the proportion of the other.

_Law of Combining Proportions_: Each element enters into all its compounds by a fixed proportional weight.

The fundamental laws of chemistry are proved by experiment.

(3) THE ATOMIC THEORY.--The atomic theory teaches that matter is composed of minute particles which themselves cannot be divided, but which unite to form molecules which can be divided.

A _molecule_, then, is the smallest amount of a substance that can exist in a free state.

The diameters of molecules have been ascertained by Jeans to be--

Hydrogen 20.3 Nitrogen 29.1 Oxygen 27.3

These figures express number of billionths of a meter.

An _atom_ is an indivisible particle of an element, and goes to make up the molecule.

(4) CHEMICAL NOTATION.--The symbols used to represent the different elements (_e.g._ H for hydrogen, O for oxygen, etc.), when used in chemical compounds, refer to the number of atoms which go to make up the molecule of that particular compound. For example, the expression H₂SO₄ means that in one molecule of that acid there are 2 atoms of hydrogen, 1 of sulphur, and 4 of oxygen.

(5) MOLECULAR WEIGHTS.--To determine the molecular weight of a compound it is necessary to know _Avogadro’s Law_: Equal volumes of all gases under the same conditions contain the same number of molecules; and Molecular Weight = Vapor Density × 2.

(6) REACTIONS.--A reaction or chemical equation is a method of representing a chemical change.

In chemistry we have three kinds of reactions, namely:

(1) _Analytical_ reaction, which is the breaking up of compound bodies into simple, _e.g._, H₂CO₃ can be broken up into its components, H₂O and CO₂, _e.g._, H₂CO₃ = H₂O + CO₂.

(2) _Synthetical_ reaction is the building up of a compound body by the union of two or more simple bodies, _e.g._, H₂ + O = H₂O and H + Cl = HCl.

(3) _Metathetical_ reaction consists in the interchange of two radicals in two substances, _e.g._,

2HCl + Zn = ZnCl₂ + H₂. Here the H of the acid is replaced by the Zn.

KCl + AgNO₃ = AgCl + KNO₃. Here the Ag and the K change places.

(7) THE CHEMICAL ARITHMETIC by which from the molecular weights of two substances, and the weight of one substance we are enabled to get the weight of the required substance is called Stoichiometry.

EXAMPLE: Required the amount of zinc necessary to generate 10 grams of hydrogen.

Atomic weights of H, Cl, and Zn are respectively 1, 35.5, and 65.3.

The reaction is as follows:--

Zn + 2HCl = ZnCl₂ + H₂, and shows that 2 atoms of H are used for every 1 of Zn.

(Mol. Wt. Zn.) (Mol. Wt. H₂) (Wt. Zn.) (Wt. H₂.) 65.3 : 2 = x : 10

65.3 × 10 --------- = x = 326.5 grams of Zn. 2

(8) BERTHOLLET’S LAW.--Berthollet established the following law, which is of great importance. When two substances can form a substance insoluble or volatile, under the condition of the reaction, that substance will be formed till one of the factors is exhausted.

(9) RADICALS.--A radical is an atom or group of atoms which changes places in a reaction. A compound radical is made up of different sorts of radicals, as NH₄.

A basic radical is a metal, or a compound radical which behaves like a metal, _e.g._, Zn and NH₄.

(10) HYDRATES.--A hydrate is a substance formed from water by replacing half of its hydrogen by a radical, _e.g._, H₂O + 2Na = 2NaOH + H₂, where the sodium has taken the place of one atom of hydrogen.

(11) BASE.--If a hydrate is formed by a basic radical, the hydrate is called a base, _e.g._, ZnO₂H₂.

(12) ALKALI.--An alkali is a soluble base, _e.g._, NaOH, KOH, NH₄OH, LiOH.

(13) ACID.--An acid is a substance containing hydrogen which may be replaced by a basic radical, _e.g._, 2HCl + Zn = ZnCl₂ + H₂.

(14) SALTS.--A salt is a substance formed from an acid replacing its hydrogen by a basic radical, _e.g._ 2HCl + Zn = ZnCl₂ + H₂.

An acid salt is a compound derived from an acid which has not all of its hydrogen replaced, _e.g._, 2NaCl + H₂SO₄ = NaHSO₄ + HCl + NaCl.

(15) CHEMICAL NOMENCLATURE.--_Termination_ “--UM” is now applied to all _Metals_, though the older-known metals retain the former names, _e.g._--Aluminium, Tellurium, etc.

_Termination_ “--IDE” denotes a _Binary Compound_, that is, a substance composed of only two elements, _e.g._, Sodium Chloride (NaCl).

_Termination_ “--OUS” is applied to the first of two elements when it exists in a greater proportion than in another combination with the same element, _e.g._, one atom of phosphorus and three atoms of chlorine form _Phosphorous Chloride_.

_Termination_ “--IC,” when the first exists in a lesser proportion, _e.g._, one atom of phosphorus with five atoms of chlorine form _Phosphoric Chloride_.

_Prefixes_ “MONO--,” “BI--,” “TRI--,” etc., indicate the proportion of the latter of two elements, and are sometimes used instead of the above termination. Thus phosphorous chloride may also be called _Phosphorous Tri-Chloride_; so one atom of carbon with one atom of oxygen is _Carbon Monoxide_.

_Prefix_ “HYPO--” (under) and “PER--” (over), specify compounds formed by the same two elements containing less (or more) of an element than is in the usual compound.

_Nomenclature of Salts._--From the common acids we get the following salts:--

HCl forms chlorides. HNO₃ forms nitrates. H₂SO₄ forms sulphates. H₂S forms sulphides. H₂CO₃ forms carbonates. H₂O forms no salts. H₂SiO₄ forms silicates. H₃PO₄ forms phosphates.

A rough rule for the nomenclature of acids may be made from the above. Acids with the prefix _hydro_ and the suffix _ic_ form salts in _ide_; with suffix _ate_, salts in _ate_; with suffix _ous_, salts in _ite_.

(16) BASICITY.--The basicity of a substance is measured by the amount of hydrogen which it contains that can be replaced by a basic radical, _e.g._, H₂SO₄ is di-basic, _i.e._, the two atoms of hydrogen can be replaced by a basic radical. H₂SO₄ + CaCl₂ = CaSO₄ + 2HCl.

(17) QUANTIVALENCE.--The quantivalence of an element is measured by the number of atoms of hydrogen it combines with or replaces. _E.g._, Na is univalent, for when added to HCl it replaces one atom of hydrogen; Ca is bivalent, for, as seen in the above reaction, it replaces two atoms of hydrogen.

(18) TEST FOR A CHLORIDE.--To test for HCl or any chloride, add to the solution to be tested a little AgNO₃, and if a chloride is present, a milky-white precipitate will be formed. The reaction is as follows: HCl + AgNO₃ = AgCl (white precipitate) + HNO₃. A metal almost invariably changes places with hydrogen.

_Caution._--In diluting H₂SO₄ add the acid to the water; for the evolution of heat from the process will cause the water to boil, and reversing this process will cause the liquid to boil over and possibly result disastrously.

(19) IMPURITY IN H₂SO₄.--Commercial sulphuric acid contains PbSO₄ as an impurity. This gives it the colored appearance, plumbic sulphate being soluble in strong sulphuric acid.

(20) H₂S.--Sulphuretted hydrogen is somewhat soluble in water, slightly poisonous, and is a reducing agent.

(21) CARBONIC ACID.--H₂CO₃ does not exist as an acid. We infer its existence from the presence of its salts. Na₂CO₃ + 2HCl = 2NaCl + H₂CO₃, but the H₂CO₃ is so unstable that it breaks up at once into H₂O and CO₂.

(22) TEST FOR A CARBONATE.--To test for a carbonate, treat the substance with an acid; CO₂ is formed; pour the gas into a solution of lime-water, and a white insoluble precipitate is formed, CaCO₃.

=TABLE OF ALL THE KNOWN CHEMICAL ELEMENTS=

The _Chemical Elements_ are the simplest known constituents of all compound substances. Chemists regard them as elements or elementary substances only when they have been proved to be _not_ compound. The elements are somewhat arbitrarily divided into metals and non-metals, the former constituting by far the larger class. Several elements occupy positions on the border line. Below is a list of the elements at present known with certainty, and of their atomic weights as fixed by the various kinds of evidence obtained by very numerous, and in many cases varied, experiments. The values are all referred to oxygen as standard with atomic weight 16, and are those adopted, for 1910, by the International Commission on Atomic Weights. The standard O = 16 has been pretty generally adopted by chemists as, upon the whole, more satisfactory than H = 1.