The number of formula units. Formulas and units
When typing text in the Word editor, it is recommended to write formulas using the built-in formula editor, keeping the default settings in it. It is allowed to type formulas in larger type than the text, if it is necessary for the convenience of reading small indices. It is recommended to define a separate line for formulas with your own style (naming it, for example, Equation), in which you should set the necessary indents, intervals, alignment and style of the next line.
Formulas in the work are numbered with Arabic numerals. The formula number consists of the section number and the ordinal number of the formula in the section, separated by a dot. The number is indicated on the right side of the sheet at the formula level in parentheses. For example, (2.1) is the first formula of the second section. The formulas themselves should be written in the center of the page. The letter designations of the quantities included in the formula must be deciphered (if this has not been done in the text of the work earlier). For example: full number M deaths from malignant tumors as a result of exposure in the population will be equal to
where n(e) is the distribution density of individuals in the population by age, R(e) is the lifetime risk of death from malignant neoplasms for an individual of age e at the time of a single exposure or the onset of chronic exposure.
The designations are deciphered in the sequence corresponding to the order in which they appear in the formula. It is allowed to decipher each of the designations to write in a separate line.
You should strictly follow the rules for punctuation after writing formulas.
Equations and formulas must be separated from the text by free lines. If the equation does not fit on one line, then it must be moved after the equal sign (=) or after the addition (+), subtraction (-), multiplication (x), and division (:) signs. Floating point numbers should be written in the form, for example: 2×10 -12 s, denoting the multiplication sign with the symbol (×) from the Symbol font. You should not denote the multiplication operation with the symbol (*).
Units of measurement of physical quantities must be given only in the International System of Units (SI) in the accepted abbreviations.
Construction work
The names of the structural parts of the work "Abstract", "Contents", "Denotations and abbreviations", "Normative references", "Introduction", "Main part", "Conclusion", "List of references" serve as headings of the structural elements of the work.
The main part of the work should be divided into chapters "Literature Review", "Research Material and Methods", "Research Results and Discussion", sections, subsections and paragraphs. Items, if necessary, can be divided into sub-items. When dividing the text of the work into paragraphs and subparagraphs, it is necessary that each paragraph contains complete information. Chapters, sections, subsections should have titles. Section headings are placed symmetrically to the text. Subsection headings begin 15-17 mm from the left margin. Word hyphenation in headings is not allowed. Do not put a dot at the end of the title. If the title consists of two sentences, then they are separated by a dot. The distance between the title, subtitle and text should be 15-17 mm (12 pt with the same font size). Headings should not be underlined. Each section (chapter) of the work must begin on a new sheet (page).
Chapters, sections, subsections, paragraphs and subparagraphs should be numbered in Arabic numerals. Sections should be numbered sequentially within the entire text of the chapter, with the exception of appendices.
Do not put a dot after the number of the section, subsection, paragraph and subparagraph in the text. If the heading consists of two or more sentences, they are separated by a dot(s).
Section headings are printed in lowercase letters (except for the first capital) with a paragraph indent in bold type with a size of 1-2 points more than in the main text.
Subheadings are printed with a paragraph indent in lowercase letters (except for the first capital) in bold type with the font size of the main text.
The distance between the title (except for the title of the paragraph) and the text should be 2-3 line spacing. If there is no text between two headings, then the distance between them is set to 1.5-2 line spacing.
Illustrations
Illustrations (diagrams, graphs, diagrams, photographs) are located, as a rule, on separate pages, which are included in the general numbering. When computer-based illustrations are allowed to place them in the general text.
Illustrations should be placed in the work directly after the text in which they are mentioned for the first time, or on the next page. All illustrations must be referenced in the work.
The number of illustrations is determined by the content of the work and should be sufficient to make the material presented clear and specific. Drawings must be printed using a computer or done in black ink or ink. It is forbidden to make drawings in a different color, as well as in pencil. Color printing of drawings and photographs is allowed.
Illustrations should be arranged so that they can be easily viewed without turning the work or turning it clockwise. Illustrations are placed in the text after the first reference to them.
Illustrations (diagrams and graphs) that cannot be placed on an A4 sheet are placed on an A3 sheet and then folded to A4 size.
All illustrations should be referenced in the text of the work. All illustrations are designated by the word "drawing" and numbered sequentially with Arabic numerals through numbering, with the exception of the illustrations given in the appendix. The word "figure" in the captions to the figure and in references to it is not abbreviated.
It is allowed to number illustrations within the section. In this case, the illustration number must consist of the section number and the sequence number of the illustration in the section. For example, Figure 1.2 is the second figure in the first section.
Illustrations, as a rule, have explanatory data (figure text) located in the center of the page. Explanatory data is placed under the illustration, and from the next line - the word "Figure", the number and name of the illustration, separating the number from the name with a dash. Do not put a dot at the end of the numbering and titles of illustrations. Wrapping of words in the title of the figure is not allowed. The word "Figure", its number and the name of the illustration are printed in bold, and the word "Figure", its number, as well as explanatory data to it, are reduced by 1-2 points in font size.
An example of illustration design is given in Appendix D.
tables
Digital material, as a rule, should be presented in the form of tables.
The digital material of the dissertation is presented in the form of tables. Each table must have a short title, which consists of the word "Table", its number and title, separated from the number by a dash. The heading should be placed above the table on the left, without paragraph indentation.
Headings of graphs and lines should be written with a capital letter in the singular, and subheadings of graphs should be written with a lowercase letter if they make up one sentence with the heading, and with capital letters if they have an independent meaning.
The table should be placed after its first mention in the text. Tables are numbered in the same way as illustrations. For example, table 1.2. is the second table of the first section. In the name of the table, the word "Table" is written in full. When referring to a table in the text, the word "table" is not abbreviated. If necessary, tables can be placed on separate sheets, which are included in the overall page numbering.
When designing tables, you must follow the following rules:
it is allowed to use in the table a font 1-2 points smaller than in the text of the dissertation;
the column "Sequence number" should not be included in the table. If it is necessary to number the indicators included in the table, the serial numbers are indicated in the sidebar of the table immediately before their name;
a table with a large number of rows can be transferred to the next sheet. When transferring part of the table to another sheet, its heading is indicated once above the first part, the word "Continuation" is written to the left above the other parts. If there are several tables in the dissertation, then after the word "Continuation" indicate the number of the table, for example: "Continuation of table 1.2";
a table with a large number of columns can be divided into parts and placed one part under the other within one page, repeating a sidebar in each part of the table. The heading of the table is placed only above the first part of the table, and above the rest they write "Continuation of the table" or "End of the table" indicating its number;
a table with a small number of columns can be divided into parts and placed one part next to the other on the same page, separating them from each other with a double line and repeating the head of the table in each part. With a large size of the head, it is allowed not to repeat it in the second and subsequent parts, replacing it with the corresponding column numbers. In this case, the columns are numbered with Arabic numerals;
if the text repeated in different lines of the column of the table consists of one word, then after the first writing it is allowed to replace it with quotation marks; if from two or more words, then it is replaced by the words "The same" at the first repetition, and then - quotes. It is not allowed to put quotation marks instead of repeated numbers, marks, signs, mathematical, physical and chemical symbols. If digital or other data is not given in any line of the table, then a dash is put in it;
column and line headings should be written with a capital letter in the singular, and graph subheadings should be written with a lowercase letter if they form one sentence with the heading, and with a capital letter if they have an independent meaning. It is allowed to number the columns with Arabic numerals, if it is necessary to give links to them in the text of the dissertation;
column headings, as a rule, are written parallel to the rows of the table. If necessary, it is allowed to place the headings of the columns parallel to the columns of the table.
An example of the design of the table is given in Appendix D.
Similar information.
This guide has been compiled from various sources. But its creation was prompted by a small book "Mass Radio Library" published in 1964, as a translation of the book by O. Kroneger in the GDR in 1961. Despite its antiquity, it is my reference book (along with several other reference books). I think time has no power over such books, because the foundations of physics, electrical and radio engineering (electronics) are unshakable and eternal.
Units of measurement of mechanical and thermal quantities.
Units of measurement of electromagnetic quantities
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Relations between units of magnetic quantities
in CGSM and SI systems
| In electrical and reference literature published before the introduction of the SI system, the magnitude of the magnetic field strength H often expressed in oersteds (uh) magnetic induction value AT - in gauss (gs), magnetic flux Ф and flux linkage ψ - in maxwells (µs). |
| 1e \u003d 1/4 π × 10 3 a / m; 1a / m \u003d 4π × 10 -3 e; 1gf=10 -4 t; 1tl=104 gs; 1mks=10 -8 wb; 1vb=10 8 ms |
| It should be noted that the equalities are written for the case of a rationalized practical MKSA system, which was included in the SI system as an integral part. From a theoretical point of view, it would be better to about in all six relationships, replace the equal sign (=) with the match sign (^). For example |
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1e \u003d 1 / 4π × 10 3 a / m |
| which means: a field strength of 1 Oe corresponds to a strength of 1/4π × 10 3 a/m = 79.6 a/m |
| The point is that the units gs and ms belong to the CGMS system. In this system, the unit of current strength is not the main one, as in the SI system, but a derivative. Therefore, the dimensions of the quantities characterizing the same concept in the CGSM and SI systems turn out to be different, which can lead to misunderstandings and paradoxes, if we forget about this circumstance. When performing engineering calculations, when there is no basis for misunderstandings of this kind |
Off-system units
Some mathematical and physical concepts
applied to radio engineering
| Like the concept - the speed of movement, in mechanics, in radio engineering there are similar concepts, such as the rate of change of current and voltage. They can be either averaged over the course of the process, or instantaneous. |
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i \u003d (I 1 -I 0) / (t 2 -t 1) \u003d ΔI / Δt |
| With Δt -> 0, we get the instantaneous values of the current change rate. It most accurately characterizes the nature of the change in the quantity and can be written as: |
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i=lim ΔI/Δt =dI/dt |
| And you should pay attention - the average values and instantaneous values \u200b\u200bcan differ by dozens of times. This is especially evident when a changing current flows through circuits with a sufficiently large inductance. |
decibell |
| To assess the ratio of two quantities of the same dimension in radio engineering, a special unit is used - the decibel. |
| K u \u003d U 2 / U 1 Voltage gain; K u [dB] = 20 log U 2 / U 1 Voltage gain in decibels. Ki [dB] = 20 log I 2 / I 1 Current gain in decibels. Kp[dB] = 10 log P 2 / P 1 Power gain in decibels. |
| The logarithmic scale also allows, on a graph of normal sizes, to depict functions that have a dynamic range of parameter changes in several orders of magnitude. |
| To determine the signal strength in the reception area, another logarithmic unit of DBM is used - dicibells per meter. |
| P [dbm] = 10 log U 2 / R +30 = 10 log P + 30. [dbm]; |
| The effective load voltage at a known P[dBm] can be determined by the formula: |
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Dimensional coefficients of basic physical quantities
| In accordance with state standards, the following multiple and submultiple units - prefixes are allowed: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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It is possible to establish the type of chemical formula according to structural data (i.e., according to the model of the structure or according to its projection - drawing) in another way, counting the number of atoms of each type (chemical element) per unit cell . For example, in the CaF 2 fluorite structure, all eight F - ions are located inside the unit cell, i.e., they belong only to this cell. The location of the Ca 2+ ions is different: some of them are localized at eight vertices of the cubic cell of the mineral structure, the other part - at the centers of all six of its faces. Since each of the eight "top" Ca 2+ ions simultaneously belongs to eight neighboring elementary cells - cubes, then only a part of each of them belongs to the original cell. Thus, the contribution of the “top” Ca atoms to the initial cell will be equal to 1 Ca (1/8 x 8 = 1 Ca). Each of the six Ca atoms located at the centers of the cubic cell faces simultaneously belongs to two adjacent cells. Hence, the contribution of six Ca atoms centering the faces of the cube will be equal to 1/2 x 6 = 3 Ca. As a result, there will be 1 + 3 = 4 Ca atoms per unit cell. The calculation shows that there are four Ca atoms and eight F atoms per cell. This confirms the type of chemical formula (AX 2) of the mineral - CaF 2, where there are two times fewer Ca atoms than F atoms. It is easy to come to similar results if shift the origin of the elementary cell so that all atoms are within the same cell. Determining the number of atoms in the Bravais cell allows, in addition to the type of chemical formula, to obtain another useful constant - the number of formula units, denoted by the letter Z For simple substances consisting of atoms of one element ( Cu, Fe, Se, etc.), the number of formula units corresponds to the number of atoms in the unit cell. For simple molecular substances (I 2, S 8, etc.) and molecular compounds (CO 2, realgar As 4 S 4), the number Z is equal to the number of molecules in the cell. In the vast majority of inorganic and intermetallic compounds (NaCl, CaF 2 , CuAu, etc.) there are no molecules, and in this case, instead of the term "number of molecules", the term "number of formula units" is used. In our example, for fluorite 4, since four Ca atoms and eight F atoms per one Bravais cell will make up four formula units "CaF 2". The number of formula units can be determined experimentally in the process of X-ray examination of a substance. If the structure does not contain such microdefects as vacancies in the position of atoms or replacement of some particles by others, as well as macrodefects (fractures, inclusions, interblock voids), then Z should be an integer within the experimental error. By experimentally determining Z and rounding it to an integer, one can calculate the density of an ideal single crystal, which is called the X-ray density
Abstract keywords: chemical formula, index, coefficient, qualitative and quantitative composition, formula unit.
- this is a conditional record of the composition of a substance by means of chemical signs and indices.
The number that is in the formula at the bottom right of the sign of the element is called index. The index indicates the number of atoms of an element that make up a given substance.
If it is required to designate not one, but several molecules (or individual atoms), then before the chemical formula (or sign) they put the corresponding number, which is called coefficient. For example, three water molecules are denoted 3H 2 O, five iron atoms - 5Fe. Index 1 in chemical formulas and coefficient 1 do not write in front of chemical symbols and formulas.
The formulas shown in the figure read as follows: tri-cuprum-chloro-two, five-aluminum-two-o-three, tri-ferrum-chloro-tri . Recording 5H 2 O(five-ash-two-o) should be understood as follows: five water molecules are formed by ten hydrogen atoms and five oxygen atoms.
The chemical formula shows the atoms of which elements a substance consists of (that is, qualitative composition of matter); and what is the ratio of the atoms of these elements (i.e. quantitative composition of the substance).
formula unit
Chemical formulas of substances having a non-molecular structure, for example FeS, do not describe the composition of the molecule; but only show the ratio of the elements that form a given substance.
So, the crystal lattice of table salt - sodium chloride consists not of molecules, but of. For every positively charged sodium ion, there is one negatively charged chloride ion. It turns out that the ratio of indices in the record NaCl matches the relation; in which the chemical elements combine with each other to form a substance. In relation to substances having a non-molecular structure, it is more correct to call such a record not a formula, but formula unit.
Knowing the model of the crystal structure, i.e., the spatial arrangement of atoms relative to the symmetry elements in the unit cell - their coordinates, and, consequently, the characteristics of the regular systems of points that the atoms occupy, a number of crystal chemical conclusions can be drawn using fairly simple methods for describing structures. Since the 14 derived Bravais lattices cannot reflect the entire variety of crystal structures known to date, characteristics are needed that allow one to unambiguously describe the individual features of each crystal structure. Such characteristics, which give an idea of the geometric nature of the structure, include: coordination numbers (CN), coordination polyhedra (CM), or polyhedra (CP), and the number of formula units (Z). First of all, using the model, one can solve the question of the type of chemical formula of the compound under consideration, i.e., establish the quantitative ratio of atoms in the structure. This is not difficult to do on the basis of an analysis of the mutual environment - mutual coordination - of atoms of different (or identical) elements.
The term "atom coordination" was introduced in chemistry at the end of the 19th century. in the process of forming its new field - the chemistry of coordination (complex) compounds. And already in 1893, A. Werner introduced the concept of coordination number (CN) as the number of atoms (ligands - ions directly associated with central atoms (cations)) directly associated with the central one. Chemists at one time were faced with the fact that the number of bonds formed by an atom may differ from its formal valence and even exceed it. For example, in the ionic compound NaCl, each ion is surrounded by six ions of opposite charge (KN Na / Cl = 6, KN Cl / Na = 6), although the formal valence of Na and C1 atoms is 1. Thus, according to the modern concept, KN is the number neighboring atoms (ions) closest to a given atom (ion) in the crystal structure, regardless of whether they are atoms of the same type as the central one or another. In this case, interatomic distances are the main criterion used in calculating the cn.
For example, in the cubic structures of the a-Fe modification (Fig. 7.2.a) and CsCl (Fig. 7.2. c), the coordination numbers of all atoms are 8: in the a-Fe structure, Fe atoms are located at the sites of a body-centered cube, hence KN Fe = 8 ; in the CsCl structure, Cl - ions are located at the vertices of the unit cell, and the Cs + ion is located in the center of the volume, the coordination number of which is also equal to 8 (CN Cs / Cl = 8), just as each Cl ion is surrounded by eight Cs + ions in cube (CN Cl / Cs = 8). This confirms the Cs:C1 = 1:1 ratio in the structure of this compound.
In the α-Fe structure, the coordination number of the Fe atom in the first coordination sphere is 8, and, taking into account the second sphere, it is 14 (8 + 6). Coordination polyhedra - cube and rhombic dodecahedron respectively .
Coordination numbers and coordination polyhedra are the most important characteristics of a particular crystal structure, distinguishing it from other structures. On this basis, a classification can be carried out, referring a specific crystal structure to a specific structural type.
It is also possible to establish the type of chemical formula according to structural data (i.e., according to the model of the structure or according to its projection - drawing) in another way, by counting the number of atoms of each type (chemical element) per unit cell. This confirms the type of chemical formula NaCl.
In the structure of NaCl (Fig. 7.4), typical for ionic crystals of the AB type (where A are atoms (ions) of one type, B of another), 27 atoms of both types take part in the construction of the unit cell, of which 14 atoms are A (balls of large size) and 13 B atoms (smaller balls), but only one is completely included in the cell. an atom at its center. An atom located in the center of a face of an elementary cell belongs simultaneously to two cells - the given one and the one adjacent to it. Therefore, only half of this atom belongs to this cell. In each of the vertices of the cell, 8 cells converge simultaneously, therefore only 1/8 of the atom located at the vertex belongs to this cell. From each atom located on the edge of the cell, only 1/4 belongs to it.
Let us calculate the total number of atoms per unit cell NaCl:
So, for the cell shown in Fig. 7.4, there are not 27 atoms, but only 8 atoms: 4 sodium atoms and 4 chlorine atoms.
Determining the number of atoms in a Bravais cell allows, in addition to the type of chemical formula, to obtain another useful constant - the number of formula units, denoted by the letter Z. For simple substances consisting of atoms of one element (Cu, Fe, Se, etc.), the number of formula units corresponds to the number atoms in a unit cell. For simple molecular substances (I 2, S 8, etc.) and molecular compounds (CO 2), the number Z is equal to the number of molecules in the cell. In the vast majority of inorganic and intermetallic compounds (NaCl, CaF 2, CuAu, etc.) there are no molecules, and in this case, instead of the term "number of molecules", the term "number of formula units" is used.
The number of formula units can be determined experimentally in the process of X-ray examination of a substance.
