Vector can be divided into two types

1. Polar Vectors

These are those vectors which have a starting point or a point of application as a displacement, force etc.

2. Axial Vectors

These are those vectors which represent rotational effect and act along the axis of rotation in accordance with right hand screw rule as angular velocity, torque, angular momentum etc.

Scalars

Those physical quantities which require only magnitude but no direction for their complete representation, are called scalars.

Distance, speed, work, mass, density, etc are the examples of scalars. Scalars can be added, subtracted, multiplied or divided by simple algebraic laws.

Tensors

Tensors are those physical quantities which have different values in different directions at the same point.

Moment of inertia, radius of gyration, modulus of elasticity, pressure, stress, conductivity, resistivity, refractive index, wave velocity and density, etc are the examples of tensors. Magnitude of tensor is not unique.

Different Types of Vectors

(i) Equal Vectors Two vectors of equal magnitude, in same direction are called equal vectors.

(ii) Negative Vectors Two vectors of equal magnitude but in opposite directions are called negative vectors.

(iii) Zero Vector or Null Vector A vector whose magnitude is zero is known as a zero or null vector. Its direction is not defined. It is denoted by 0.

Velocity of a stationary object, acceleration of an object moving with uniform velocity and resultant of two equal and opposite vectors are the examples of null vector.

(iv) Unit Vector A vector having unit magnitude is called a unit vector.

A unit vector in the direction of vector A is given by

 = A / A

A unit vector is unitless and dimensionless vector and represents direction only.

(v) Orthogonal Unit Vectors The unit vectors along the direction of orthogonal axis, i.e., X – axis, Y – axis and Z – axis are called orthogonal unit vectors. They are represented by

(vi) Co-initial Vectors Vectors having a common initial point, are called co-initial vectors.

(vii) Collinear Vectors Vectors having equal or unequal magnitudes but acting along the same or Ab parallel lines are called collinear vectors.

(viii) Coplanar Vectors Vectors acting in the same plane are called coplanar vectors.

(ix) Localised Vector A vector whose initial point is fixed, is called a localised vector.

(x) Non-localised or Free Vector A vector whose initial point is not fixed is called a non-localised or a free vector.

(xi) Position Vector A vector representing the straight line distance and the direction of any point or object with respect to the origin, is called position vector.

Addition of Vectors

1. Triangle Law of Vectors

If two vectors acting at a point are represented in magnitude and direction by the two sides of a triangle taken in one order, then their resultant is represented by the third side of the triangle taken in the opposite order.

If two vectors A and B acting at a point are inclined at an angle θ, then their resultant

R = √A² + B² + 2AB cos θ

If the resultant vector R subtends an angle β with vector A, then

tan β = B sin θ / A + B cos θ

2. Parallelogram Law of Vectors

If two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram draw from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram draw from the same point.

Resultant of vectors A and B is given by

√A² + B² + 2AB cos θ

If the resultant vector R subtends an angle β with vector A, then

tan β = B sin θ / A + B cos θ

Polygon Law of Vectors

It states that if number of vectors acting on a particle at a time are represented in magnitude and – direction by the various sides of an open polygon taken in same order, their resultant vector E is represented in magnitude and direction by the closing side of polygon taken in opposite order. In fact, polygon law of vectors is the outcome of triangle law of vectors.

R = A + B + C + D + E

OE = OA + AB + BC + CD + DE

Properties of Vector Addition

(i) Vector addition is commutative, i.e., A + B = B + A

(ii) Vector addition is associative, i.e.,

A +(B + C)= B + (C + A)= C + (A + B)

(iii) Vector addition is distributive, i.e., m (A + B) = m A + m B

Rotation of a Vector

(i) If a vector is rotated through an angle 0, which is not an integral multiple of 2 π, the vector changes.

(ii) If the frame of reference is rotated or translated, the given vector does not change. The components of the vector may, however, change.

Resolution of a Vector into Rectangular Components

If any vector A subtends an angle θ with x-axis, then its

Horizontal component A_x = A cos θ

Vertical component A_y = A sin θ

Magnitude of vector A = √A_x² + A_y²

tan θ = A_y / A_x

Direction Cosines of a Vector

If any vector A subtend angles α, β and γ with x – axis, y – axis and z – axis respectively and its components along these axes are A_x, A_y and A_z, then

cos α= A_x / A, cos β = A_y / A, cos γ = A_z / A

and cos² α + cos² β + cos² γ = 1

Subtraction of Vectors

Subtraction of a vector B from a vector A is defined as the addition of vector -B (negative of vector B) to vector A

Thus, A – B = A + (-B)

Multiplication of a Vector

1. By a Real Number

When a vector A is multiplied by a real number n, then its magnitude becomes n times but direction and unit remains unchanged.

2. By a Scalar

When a vector A is multiplied by a scalar S, then its magnitude becomes S times, and unit is the product of units of A and S but direction remains same as that of vector A.

Scalar or Dot Product of Two Vectors

The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them. It is denoted by . (dot).

A * B = AB cos θ

The scalar or dot product of two vectors is a scalar.

Properties of Scalar Product

(i) Scalar product is commutative, i.e., A * B= B * A

(ii) Scalar product is distributive, i.e., A * (B + C) = A * B + A * C

(iii) Scalar product of two perpendicular vectors is zero.

A * B = AB cos 90° = O

(iv) Scalar product of two parallel vectors is equal to the product of their magnitudes, i.e., A * B = AB cos 0° = AB

(v) Scalar product of a vector with itself is equal to the square of its magnitude, i.e.,

A * A = AA cos 0° = A²

(vi) Scalar product of orthogonal unit vectors

and

(vii) Scalar product in cartesian coordinates

= A_xB_x + A_yB_y + A_zB_z

Vector or Cross Product of Two Vectors

The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by * (cross).

A * B = AB sin θ n

The direction of unit vector n can be obtained from right hand thumb rule.

If fingers of right hand are curled from A to B through smaller angle between them, then thumb will represent the direction of vector (A * B).

The vector or cross product of two vectors is also a vector.

Properties of Vector Product

(i) Vector product is not commutative, i.e.,

A * B ≠ B * A  [∴ (A * B) = — (B * A)]

(ii) Vector product is distributive, i.e.,

A * (B + C) = A * B + A * C

(iii) Vector product of two parallel vectors is zero, i.e.,

A * B = AB sin O° = 0

(iv) Vector product of any vector with itself is zero.

A * A = AA sin O° = 0

(v) Vector product of orthogonal unit vectors

(vi) Vector product in cartesian coordinates

Direction of Vector Cross Product

When C = A * B, the direction of C is at right angles to the plane containing the vectors A and B. The direction is determined by the right hand screw rule and right hand thumb rule.

(i) Right Hand Screw Rule Rotate a right handed screw from first vector (A) towards second vector (B). The direction in which the right handed screw moves gives the direction of vector (C).

(ii) Right Hand Thumb Rule Curl the fingers of your right hand from A to B. Then, the direction of the erect thumb will point in the direction of A * B.

• A definite amount of a physical quantity is taken as its standard unit.
• The standard unit should be easily reproducible, internationally accepted.

Fundamental Units

Those physical quantities which are independent to each other are called fundamental quantities and their units are called fundamental units.

Supplementary Fundamental Units

Radian and steradian are two supplementary fundamental units. It measures plane angle and solid angle respectively.

Derived Units

Those physical quantities which are derived from fundamental quantities are called derived quantities and their units are called derived units.
e.g., velocity, acceleration, force, work etc.

Definitions of Fundamental Units

The seven fundamental units of SI have been defined as under.

1. 1 kilogram A cylindrical prototype mass made of platinum and iridium alloys of height 39 mm and diameter 39 mm. It is mass of 5.0188 x 10²⁵ atoms of carbon-12.
2. 1 metre 1 metre is the distance that contains 1650763.73 wavelength of orange-red light of Kr-86.
3. 1 second 1 second is the time in which cesium atom vibrates 9192631770 times in an atomic clock.
4. 1 kelvin 1 kelvin is the (1/273.16) part of the thermodynamics temperature of the triple point of water.
5. 1 candela 1 candela is (1/60) luminous intensity of an ideal source by an area of cm’ when source is at melting point of platinum (1760°C).
6. 1 ampere 1 ampere is the electric current which it maintained in two straight parallel conductor of infinite length and of negligible cross-section area placed one metre apart in vacuum will produce between them a force 2 x 10^-7 N per metre length.
7. 1 mole 1 mole is the amount of substance of a system which contains a many elementary entities (may be atoms, molecules, ions, electrons or group of particles, as this and atoms in 0.012 kg of carbon isotope ₆C¹².

Systems of Units

A system of units is the complete set of units, both fundamental and derived, for all kinds of physical quantities. The common system of units which is used in mechanics are given below:

1. CGS System In this system, the unit of length is centimetre, the unit of mass is gram and the unit of time is second.
2. FPS System In this system, the unit of length is foot, the unit of mass is pound and the unit of time is second.
3. MKS System In this system, the unit of length is metre, the unit of mass is kilogram and the unit of time is second.
4. SI System This system contain seven fundamental units and two supplementary fundamental units.

Relationship between Some Mechanical SI Unit and Commonly Used Units

Some Practical Units

1. 1 fermi =10^-15 m
2. 1 X-ray unit = 10^-13 m
3. 1 astronomical unit = 1.49 x 10¹¹ m (average distance between sun and earth)
4. 1 light year = 9.46 x 10¹⁵ m
5. 1 parsec = 3.08 x 10¹⁶ m = 3.26 light year

Some Approximate Masses

Dimensions

Dimensions of any physical quantity are those powers which are raised on fundamental units to express its unit. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity, is called the dimensional formula.

Dimensional Formula of Some Physical Quantities

Homogeneity Principle

If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle.

Mathematically [LHS] = [RHS]

Applications of Dimensions

1. To check the accuracy of physical equations.
2. To change a physical quantity from one system of units to another system of units.
3. To obtain a relation between different physical quantities.

Significant Figures

In the measured value of a physical quantity, the number of digits about the correctness of which we are sure plus the next doubtful digit, are called the significant figures.

Rules for Finding Significant Figures

1. All non-zeros digits are significant figures, e.g., 4362 m has 4 significant figures.
2. All zeros occuring between non-zero digits are significant figures, e.g., 1005 has 4 significant figures.
3. All zeros to the right of the last non-zero digit are not significant, e.g., 6250 has only 3 significant figures.
4. In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00325 has only 3 significant figures.
5. All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.4750 has 5 significant figures.

Significant Figures in Algebric Operations

(i) In Addition or Subtraction In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. e.g.,

If l₁= 4.326 m and l₂ = 1.50 m

Then, l₁ + l₂ = (4.326 + 1.50) m = 5.826 m

As l₂ has measured upto two decimal places, therefore

l₁ + l₂ = 5.83 m

(ii) In Multiplication or Division In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values. e.g., If length 1= 12.5 m and breadth b = 4.125 m.

Then, area A = l x b = 12.5 x 4.125 = 51.5625 m²

As l has only 3 significant figures, therefore

A= 51.6 m²

Rules of Rounding Off Significant Figures

1. If the digit to be dropped is less than 5, then the preceding digit is left unchanged. e.g., 1.54 is rounded off to 1.5.
2. If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g., 2.49 is rounded off to 2.5.
3. If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g., 3.55 is rounded off to 3.6.
4. If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g., 3.750 is rounded off to 3.8 and 4.650 is rounded off to 4.6.

Error

The lack in accuracy in the measurement due to the limit of accuracy of the instrument or due to any other cause is called an error.

1. Absolute Error

The difference between the true value and the measured value of a quantity is called absolute error.

If a₁ , a₂, a₃ ,…, a_n are the measured values of any quantity a in an experiment performed n times, then the arithmetic mean of these values is called the true value (a_m) of the quantity.

The absolute error in measured values is given by

Δa₁ = a_m – a₁
Δa₂ = a_m – a₁

………….

Δa_m = Δa_m – Δa_n

2. Mean Absolute Error

The arithmetic mean of the magnitude of absolute errors in all the measurement is called mean absolute error.

3. Relative Error The ratio of mean absolute error to the true value is called relative

4. Percentage Error The relative error expressed in percentage is called percentage error.

Propagation of Error

(i) Error in Addition or Subtraction Let x = a + b or x = a – b

If the measured values of two quantities a and b are (a ± Δa and (b ± Δb), then maximum absolute error in their addition or subtraction.

Δx = ±(Δa + Δb)

(ii) Error in Multiplication or Division Let x = a x b or x = (a/b).
If the measured values of a and b are (a ± Δa) and (b ± Δb), then maximum relative error

• Importance of Chemistry
Chemistry has a direct impact on our life and has wide range of applications in different fields. These are given below:
(A) In Agriculture and Food:
(i) It has provided chemical fertilizers such as urea, calcium phosphate, sodium nitrate, ammonium phosphate etc.
(ii) It has helped to protect the crops from insects and harmful bacteria, by the use ‘ of certain effective insecticides, fungicides and pesticides.
(iii) The use of preservatives has helped to preserve food products like jam, butter, squashes etc. for longer periods.
(B) In Health and Sanitation:
(i) It has provided mankind with a large number of life-saving drugs. Today, dysentery and pneumonia are curable due to discovery of sulpha drugs and penicillin life-saving drugs. Cisplatin and taxol have been found to be very effective for cancer therapy and AZT (Azidothymidine) is used for AIDS victims.
(ii) Disinfectants such as phenol are used to kill the micro-organisms present in drains, toilet, floors etc.
(iii) A low concentration of chlorine i.e., 0.2 to 0.4 parts per million (ppm) is used ’ for sterilization of water to make it fit for drinking purposes.
(C) Saving the Environment:
The rapid industrialisation all over the world has resulted in lot of pollution.
Poisonous gases and chemicals are being constantly released in the atmosphere. They are polluting environment at an alarming rate. Scientists are working day and night to develop substitutes which may cause lower pollution. For example, CNG (Compressed Natural Gas), a substitute of petrol, is very effective in checking pollution caused by automobiles.
(D) Application in Industry:
Chemistry has played an important role in developing many industrially ^ manufactured fertilizers, alkalis, acids, salts, dyes, polymers, drugs, soaps,
detergents, metal alloys and other inorganic and organic chemicals including new materials contribute in a big way to the national economy.
• Matter
Anything which has mass and occupies space is called matter.
For example, book, pencil, water, air are composed of matter as we know that they have
mass and they occupy space.
• Classification of Matter
There are two ways of classifying the matter:
(A) Physical classification
(B) Chemical classification
(A) Physical Classification:
Matter can exist in three physical states:
1. Solids 2. Liquids 3. Gases
1. Solids: The particles are held very close to each other in an orderly fashion and there is not much freedom of movement.
Characteristics of solids: Solids have definite volume and definite shape.
2. Liquids: In liquids, the particles are close to each other but can move around. Characteristics of liquids: Liquids have definite volume but not definite shape.
3. Gases: In gases, the particles are far apart as compared to those present in solid or liquid states. Their movement is easy and fast.
Characteristics of Gases: Gases have neither definite volume nor definite shape. They completely occupy the container in which they are placed.
(B) Chemical Classification:
Based upon the composition, matter can be divided into two main types:
1. Pure Substances 2. Mixtures.
1. Pure substances: A pure substance may be defined as a single substance (or matter) which cannot be separated by simple physical methods.
Pure substances can be further classified as (i) Elements (ii) Compounds
(i) Elements: An element consists of only one type of particles. These particles may be atoms or molecules.
For example, sodium, copper, silver, hydrogen, oxygen etc. are some examples of elements. They all contain atoms of one type. However, atoms of different elements are different in nature. Some elements such as sodium . or copper contain single atoms held together as their constituent particles whereas in some others two or more atoms combine to give molecules of the element. Thus, hydrogen, nitrogen and oxygen gases consist of molecules in which two atoms combine to give the respective molecules of the element.
(ii) Compounds: It may be defined as a pure substance containing two or more elements combined together in a fixed proportion by weight and can be decomposed into these elements by suitable chemical methods. Moreover, the properties of a compound are altogether different from the constituting elements.
The compounds have been classified into two types. These are:
(i) Inorganic Compounds: These are compounds which are obtained from non-living sources such as rocks and minerals. A few
examples are: Common salt, marble, gypsum, washing soda etc.
(ii) Organic Compounds are the compounds which are present in plants and animals. All the organic compounds have been found to contain carbon as their essential constituent. For example, carbohydrates, proteins, oils, fats etc.
2. Mixtures: The combination of two or more elements or compounds which are not chemically combined together and may also be present in any proportion, is called mixture. A few examples of mixtures are: milk, sea water, petrol, lime water, paint glass, cement, wood etc.
Types of mixtures: Mixtures are of two types:
(i) Homogeneous mixtures: A mixture is said to be homogeneous if it has a uniform composition throughout and there are no visible boundaries of separation between the constituents.
For example: A mixture of sugar solution in water has the same sugar water composition throughout and all portions have the same sweetness.
(ii) Heterogeneous mixtures: A mixture is said to be heterogeneous if it does not have uniform composition throughout and has visible boundaries of separation between the various constituents. The different constituents of a heterogeneous mixture can be seen even with naked eye.
For example: When iron filings and sulphur powder are mixed together, the mixture formed is heterogeneous. It has greyish-yellow appearance and the two constituents, iron and sulphur, can be easily identified with naked eye.
• Differences between Compounds and Mixtures
Compounds
1. In a compound, two or more elements are combined chemically.
2. In a compound, the elements are present in the fixed ratio by mass. This ratio cannot change.
3. CompoUnds are always homogeneous i.e., they havethe same composition throughout.
4 In a compound, constituents cannot be separated by physical methods
5. In a compound, the constituents lose their identities i.e., i compound does not show the characteristics of the constituting elements.
Mixtures
1. In a mixture, or more elements or compounds are simply mixed and not combined chemically.
2. In a mixture the constituents are not present in fixed ratio. It can vary
3. Mixtures may be either homogeneous or heterogeneous in nature.
4. Constituents of mixtures can be separated by physical methods.
5, In a mixture, the constituents do not lose their identities i.e., a mixture shows the characteristics of all the constituents .
We have discussed the physical and chemical classification of matter. A flow sheet representation of the same is given below.

• Properties of Matter and Their Measurements
Physical Properties: Those properties which can be measured or observed without changing the identity or the composition of the substance.
Some examples of physical properties are colour, odour, melting point, boiling point etc. Chemical Properties: It requires a chemical change to occur. The examples of chemical properties are characteristic reactions of different substances. These include acidity, basicity, combustibility etc.
• Units of Measurement
Fundamental Units: The quantities mass, length and time are called fundamental quantities and their units are known as fundamental units.
There are seven basic units of measurement for the quantities: length, mass, time, temperature, amount of substance, electric current and luminous intensity.
Si-System: This system of measurement is the most common system employed throughout the world.
It has given units of all the seven basic quantities listed above.

• Definitions of Basic SI Units
1. Metre: It is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.
2. Kilogram: It is the unit of mass. It is equal to the mass of the international prototype
of the kilogram. ,
3. Second: It is the duration of 9192631, 770 periods of radiation which correspond to the transition between the two hyper fine levels of the ground state of caesium- 133 atom.
4. Kelvin: It is the unit of thermodynamic temperature and is equal to 1/273.16 of the thermodynamic temperature of the triple point of water.
5. Ampere: The ampere is that constant current which if maintained in two straight parallel conductors of infinite length, of negligible circular cross section and placed, 1 metre apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 N per metre of length.
6. Candela: It may be defined as the luminous intensity in a given direction, from a source which emits monochromatic radiation of frequency 540 x 1012 Hz and that has a radiant intensity in that direction of 1/ 683 watt per steradian.
7. Mole: It is the amount of substance which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon -12. Its symbol is ‘mol’.
• Mass and Weight
Mass: Mass of a substance is the amount of matter present in it.
The mass of a substance is constant.
The mass of a substance can be determined accurately in the laboratory by using an analytical
balance. SI unit of mass is kilogram.

Weight: It is the force exerted by gravity on an object. Weight of substance may vary from one place to another due to change in gravity.
Volume: Volume means the space occupied by matter. It has the units of (length)3. In SI units, volume is expressed in metre3 (m3). However, a popular unit of measuring volume, particularly in liquids is litre (L) but it is not in SI units or an S.I. unit.
Mathematically,
1L = 1000 mL = 1000 cm3 = 1dm3.
Volume of liquids can be measured by different devices like burette, pipette, cylinder, measuring flask etc. All of them have been calibrated.

Temperature: There are three scales in which temperature can be measured. These are known as Celsius scale (°C), Fahrenheit scale (°F) and Kelvin scale (K).

-> Thermometres with Celsius scale are calibrated from 0°C to 100°C.
-> Thermometres with Fahrenheit scale are calibrated from 32°F to 212°F.
-> Kelvin’scale of temperature is S.I. scale and is very common these days.Temperature on this scale is shown by the sign K.
The temperature on two scales are related to each other by the relationship

Density: Density of a substance is its amount of mass per unit volume. So, SI unit of density can be obtained as follows:

This unit is quite large and a chemist often expresses density in g cm³ where mass is expressed in gram and volume is expressed in cm³.
• Uncertainty in Measurements
All scientific measurements involve certain degree of error or uncertainty. The errors which arise depend upon two factors.
(i) Skill and accuracy of the worker (ii) Limitations of measuring instruments.
• Scientific Notation
It is an exponential notation in which any number can be represented in the form N x 10ⁿ where n is an exponent having positive or negative values and N can vary between 1 to 10. Thus, 232.508 can be written as 2.32508 x 10² in scientific notation.
Now let us see how calculations are carried out with numbers expressed in scientific notation.
(i) Calculation involving multiplication and division

(ii) Calculation involving addition and subtraction: For these two operations, the first numbers are written in such a way that they have the same exponent. After that, the coefficients are added or subtracted as the case may be. For example,

• Significant Figures
Significant figures are meaningful digits which are known with certainty. There are certain rules for determining the number of significant figures. These are stated below:
1. All non-zero digits are significant. For example, in 285 cm, there are three significant figures and in 0.25 mL, there are two significant figures.
2. Zeros preceding to first non-zero digit are not significant. Such zeros indicates the position of decimal point.
For example, 0.03 has one significant figure and 0.0052 has two significant figures.
3. Zeros between two non-zero digits are significant. Thus, 2.005 has four significant figures.
4. Zeros at the end or right of a number are significant provided they are on the right side of the decimal point. For example, 0.200 g has three significant figures.
5. Counting numbers of objects. For example, 2 balls or 20 eggs have infinite significant figures as these are exact numbers and can be represented by writing infinite number of zeros after placing a decimal.
i.e., 2 = 2.000000
or 20 = 20.000000
• Addition and Subtraction of Significant Figures
In addition or subtraction of the numbers having different precisions, the final result should be reported to the same number of decimal places as in the term having the least number of decimal places.
For example, let us carry out the addition of three numbers 3.52, 2.3 and 6.24, having different precisions or different number of decimal places.

The final result has two decimal places but the answer has to be reported only up to one decimal place, i.e., the answer would be 12.0.
Subtraction of numbers can be done in the same way as the addition.

The final result has four decimal places. But it has to be reported only up to two decimal places, i.e., the answer would be 11.36.
• Multiplication and Division of Significant Figures
In the multiplication or division, the final result should be reported upto the same number of significant figures as present in the least precise number.
Multiplication of Numbers: 2.2120 x 0.011 = 0.024332
According to the rule the final result = 0.024
Division of Numbers: 4.2211÷3.76 = 1.12263
The correct answer = 1.12
• Dimensional Analysis
Often while calculating, there is a need to convert units from one system to other. The method used to accomplish this is called factor label method or unit factor method or dimensional analysis.
• Laws of Chemical Combinations
The combination of elements to form compounds is governed by the following five basic laws.
(i) Law of Conservation of Mass
(ii) Law of Definite Proportions
(iii) Law of Multiple Proportions
(iv) Law of Gaseous Volume (Gay Lussac’s Law)
(v) Avogadro’s Law
(i) Law of Conservation of Mass
The law was established by a French chemist, A. Lavoisier. The law states:
In all physical and chemical changes, the total mass of the reactants is equal to that of the products.
In other words, matter can neither be created nor destroyed.
The following experiments illustrate the truth of this law.
(a) When matter undergoes a physical change.

It is found that there is no change in weight though a physical change has taken place.
(b) When matter undergoes a chemical change.
For example, decomposition of mercuric oxide.

During the above decomposition reaction, matter is neither gained nor lost.
(ii) Law of Definite Proportions
According to this law:
A pure chemical compound always consists of the same elements combined together in a fixed proportion by weight.
For example, Carbon dioxide may be formed in a number of ways i.e.,

(iii) Law of Multiple Proportions
If two elements combine to form two or more compounds, the weight of one of the elements which combines with a fixed weight of the other in these compounds, bears simple whole number ratio by weight.
For example,

(iv) Gay Lussac’s Law of Gaseous Volumes
The law states that, under similar conditions of temperature and pressure, whenever gases combine, they do so in volumes which bear simple whole number ratio with each other and also with the gaseous products. The law may be illustrated by the following examples.
(a) Combination between hydrogen and chlorine:

(b) Combination between nitrogen and hydrogen: The two gases lead to the formation of ammonia gas under suitable conditions. The chemical equation is

(v) Avogadro’s Law: Avogadro proposed that, equal volumes of gases at the same temperature and pressure should contain equal number of molecules.
For example,
If we consider the reaction of hydrogen and oxygen to produce water, we see that two volumes of hydrogen combine with one volume of oxygen to give two volumes of water without leaving any unreacted oxygen.

• Dalton’s Atomic Theory
In 1808, Dalton published ‘A New System of Chemical Philosophy’ in which he proposed the following:
1. Matter consists of indivisible atoms.
2. All the atoms of a given element have identical properties including identical mass. Atoms of different elements differ in mass.
3. Compounds are formed when atoms of different elements combine in a fixed ratio.
4. Chemical reactions involve reorganisation of atoms. These are neither created nor destroyed in a chemical reaction.
• Atomic Mass
The atomic mass of an element is the number of times an atom of that element is heavier than an atom of carbon taken as 12. It may be noted that the atomic masses as obtained above are the relative atomic masses and not the actual masses of the atoms.
One atomic mass unit (amu) is equal to l/12th of the mass of an atom of carbon-12 isotope. It is also known as unified mass.
Average Atomic Mass
Most of the elements exist as isotopes which are different atoms of the same element with different mass numbers and the same atomic number. Therefore, the atomic mass of an element must be its average atomic mass and it may be defined as the average relative mass of an atom of an element as compared to the mass of carbon atoms (C-12) taken as 12w.
Molecular Mass
Molecular mass is the sum of atomic masses of the elements present in a molecule. It is obtained by multiplying the atomic mass of each element by number of its atoms and adding them together.
For example,
Molecular mass of methane (CH4)
= 12.011 u + 4 (1.008 u)
= 16.043 u
Formula Mass
Ionic compounds such as NaCl, KNO₃, Na₂C0₃ etc. do not consist of molecules i.e., single entities but exist “as ions closely packed together in a three dimensional space as shown in -Fig. 1.5.

In such cases, the formula is used to calculate the formula mass instead of molecular mass. Thus, formula mass of NaCl = Atomic mass of sodium + atomic mass of chlorine
= 23.0 u + 35.5 u = 58.5 u.
• Mole Concept
It is found that one gram atom of any element contains the same number of atoms and one gram molecule of any substance contains the same number of molecules. This number has been experimentally determined and found to be equal to 6.022137 x 10²³ The value is generally called Avogadro’s number or Avogadro’s constant.
It is usually represented by NA:
Avogadro’s Number, NA = 6.022 × 10²³
• Percentage Composition
One can check the purity of a given sample by analysing this data. Let us understand by taking the example of water (H₂0). Since water contains hydrogen and oxygen, the percentage composition of both these elements can be calculated as follows:

• Empirical Formula
The formula of the compound which gives the simplest whole number ratio of the atoms of yarious elements present in one molecule of the compound.
For example, the formula of hydrogen peroxide is H₂0₂. In order to express its empirical formula, we have to take out a common factor 2. The simplest whole number ratio of the atoms is 1:1 and the empirical formula is HO. Similarly, the formula of glucose is C₆H₁₂0₆. In order to get the simplest whole number of the atoms,
Common factor = 6
The ratio is = 1 : 2 : 1 The empirical formula of glucose = CH₂0
• Molecular Formula
The formula of a compound which gives the actual ratio of the atoms of various elements present in one molecule of the compound.
For example, molecular formula of hydrogen peroxide = H₂0₂and Glucose = C₆H₁₂0₆
Molecular formula = n x Empirical formula
Where n is the common factor and also called multiplying factor. The value of n may be 1, 2, 3, 4, 5, 6 etc.
In case n is 1, Molecular formula of a compound = Empirical formula of the compound.
• Stoichiometry and Stoichiometric Calculations
The word ‘stoichiometry’ is derived from two Greek words—Stoicheion (meaning element) and metron (meaning measure). Stoichiometry, thus deals with the calculation of masses (sometimes volume also) of the reactants and the products involved in a chemical reaction. Let us consider the combustion of methane. A balanced equation for this reaction is as given below:

Limiting Reactant/Reagent
Sometimes, in alchemical equation, the reactants present are not the amount as required according to the balanced equation. The amount of products formed then depends upon the reactant which has reacted completely. This reactant which reacts completely in the reaction is called the limiting reactant or limiting reagent. The reactant which is not consumed completely in the reaction is called excess reactant.
Reactions in Solutions
When the reactions are carried out in solutions, the amount of substance present in its given volume can be expressed in any of the following ways:
1. Mass percent or weight percent (w/w%)
2. Mole fraction
3. Molarity
4. Molality
1. Mass percent: It is obtained by using the following relation:

2. Mole fraction: It is the ratio of number of moles of a particular component to the total number of moles of the solution. For a solution containing n2 moles of the solute dissolved in n1 moles of the solvent,

3. Molarity: It is defined as the number of moles of solute in 1 litre of the solution.

4. Molality: It is defined as the number of moles of solute present in 1 kg of solvent. It is denoted by m.

• All substances contain matter which can exist in three states — solid, liquid or gas.
• Matter can also be classified into elements, compounds and mixtures.
• Element: An element contains particles of only one type which may be atoms or molecules.
• Compounds are formed when atoms of two or more elements combine in a fixed ratio to each other.
• Mixtures: Many of the substances present around us are mixtures.
• Scientific notation: The measurement of quantities in chemistry are spread over a wide rhnge of 10^-31to 10²³. Hence, a convenient system of expressing the number in scientific notation is used.
• Scientific figures: The uncertainty is taken care of by specifying the number of significant figures in which the observations are reported.
• Dimensional analysis: It helps to express the measured quantities in different systems of units.
• Laws of Chemical Combinations are:
(i) Law of Conservation of Mass
(ii) Law of Definite Proportions
(iii) Law of Multiple Proportions
(iv) Gay Lussac’s Law of Gaseous Volumes
(v) Avogadro’s Law.
• Atomic mass: The atomic mass of an element is expressed relative to 12C isotope of carbon which has an exact value of 12u.
• Average atomic mass: Obtained by taking into account the natural aboundance of different isotopes of that element.
• Molecular mass: The molecular mass of a molecule is obtained by taking sum of atomic masses of different atoms present in a molecule.
• Avogadro number: The number of atoms, molecules or any other particles present in a given system are expressed in terms of Avogadro constant.
= 6.022 x 10²³
• Balanced chemical equation: A balanced equation has the same number of atoms of each element on both sides of the equation.
• Stoichiometry: The quantitative study of the reactants required or the products formed is called stoichiometry. Using stoichiometric calculations, the amounts of one or more reactants required to produce a particular amount of product can be determined and vice-versa.

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Module I Motion, Force and Energy - HW-02-01

Chapter Name - MOTION IN A STRAIGHT LINE - 2
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Important Questions for NIOS - ODE & PE

Q1. Is it possible for a moving body to have non-zero average speed but zero average velocity during any given interval of time? If so, explain.

Q2. A lady drove to the market at a speed of 8 km h–1. Finding market closed, she came back home at a speed of 10 km h–1. If the market is 2km away from her home, calculate the average velocity and average speed.

Q3. Can a moving body have zero relative velocity with respect to another body? Give an example.

Q4. A person strolls inside a train with a velocity of 1.0 m s–1 in the direction of motion of the train. If the train is moving with a velocity of 3.0 m s–1, calculate his
(a) velocity as seen by passengers in the compartment, and (b) velocity with respect to a person sitting on the platform.

Q5. A body starting from rest covers a distance of 40 m in 4s with constant acceleration along a straight line. Compute its final velocity and the time required to cover half of the total distance.

Q6. A car moves along a straight road with constant aceleration of 5 ms–2. Initially at 5m, its velocity was 3 ms–1 Compute its position and velocity at t = 2 s.

Q7. With what velocity should a body be thrown vertically upward so that it reaches a height of 25 m? For how long will it be in the air?

Q8. A ball is thrown upward in the air. Is its acceleration greater while it is being thrown or after it is thrown?

Q9. Distinguish between average speed and average velocity.

Q10. A car C moving with a speed of 65 km h–1 on a straight road is ahead of motorcycle M moving with the speed of 80 km h–1 in the same direction. What is the velocity of M relative to A?

Q11. How long does a car take to travel 30m, if it accelerates from rest at a rate of 2.0 m s2?

Q12. A motorcyclist covers half of the distance between two places at a speed of 30 km h–1 and the second half at the speed of 60 kmh–1. Compute the average speed of the motorcycle.

Q13. A duck, flying directly south for the winter, flies with a constant velocity of 20 km h–1 to a distance of 25 km. How long does it take for the duck to fly this distance?

Q14. Bangalore is 1200km from New Delhi by air (straight line distance) and 1500km by train. If it takes 2h by air and 20h by train, calculate the ratio of the average speeds.

Q15. A car accelerates along a straight road from rest to 50 kmh–1 in 5.0 s. What is the magnitude of its average acceleration?

Q18. A body with an initial velocity of 2.0 ms–1 is accelerated at 8.0 ms–2 for 3 seconds. (i) How far does the body travel during the period of acceleration? (ii) How far would the body travel if it were initially at rest?

Q19. A ball is released from rest from the top of a cliff. Taking the top of the cliff as the reference (zero) level and upwards as the positive direction, draw (i) the displacement-time graph, (ii) distance-time graph (iii) velocity-time graph, (iv) speed-time graph.

Q20. A ball thrown vertically upwards with a velocity v0 from the top of the cliff of height h, falls to the beach below. Taking beach as the reference (zero) level, upward as the positive direction, draw the motion graphs. i.e., the graphs between (i) distance-time, (ii) velocity-time, (iii) displacement-time, (iv) speed - time graphs.

Q21. A body is thrown vertically upward, with a velocity of 10m/s. What will be the value of the velocity and acceleration of the body at the highest point?

Q22. Two objects of different masses, one of 10g and other of 100g are dropped from the same height. Will they reach the ground at the same time? Explain your answer.

Q23. What happens to the uniform motion of a body when it is given an acceleration at right angle to its motion?

Q24. What does the slope of velocity-time graph at any instant represent?

Module II: Atomic Structure and Chemical Bonding - HW-04-02

Chapter Name - CHEMICAL BONDING - 4
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Q1. Define electrovalent bond.

Q2. Show the formation of a nitrogen molecule from two nitrogen atoms in terms of Lewis theory.

Q3. What do you understand by a polar covalent bond? Give two examples.

Q4. What is a coordinate covalent bond ? How is it different from a covalent bond?

Q5. What are the basic postulates of VSEPR theory?

Q6. Predict the shape of methane ( CH4 ) on the basis of VSEPR theory.

Q7. It is a molecule the difference between the electro-negativity of two atom is 1.7. How much % will be ionic and covalent character?

Q8. What do you understand by the term, ‘hybridisation’?

Q9. How would you explain the shape of ammonia molecule on the basis of hybridisation?

Q10. Draw the canonical structures of CO23 and SO2.

Q11. What is the basic difference between the valence bond and molecular orbital theories?

Q12. Calculate the bond orders for Li2 and Be2 molecules using the molecular orbital diagrams.

Q13. Predict the magnetic behaviour of O2

Q14. What do you understand by a chemical bond?

Q15. Explain the process of bond formation as a decrease in energy.

Q16. What do you understand by the term, ‘bond length’ ?

Q17. Describe the two possible ways in which the noble gas electronic configuration is achieved in the process of bond formation.

Q18. What are Lewis electron dot symbols ? Show the formation of MgCl2 in terms of Lewis symbols.

Q19. Define a coordinate bond and give some examples.

Q20. What is VSEPR theory ? predict the shape of SF6 molecule using this theory.

Q21. Why do we need the concept of hybridisation ? How does it help in explaining the shape of methane ?

Q22. Give the salient features of molecular orbital theory.

Q23. Be2 molecule does not exist. Explain on the basis of molecular orbital theory.

Q24. Write down the molecular orbital electronic configuration of the following species and compute their bond orders. O2 ; O2+; O2− ; O22−

Q25. BF3 is a polar molecule but it does not show dipole moment. Why?

Q26. Atom A and B combine to form AB molecule. If the difference in the electronegativity between A and B is 1.7. What type of bond do you expect in AB molecule?

Q27. Write down the resonating structures of N2O, SO42–, CO32– and BF3.

Module II: Atomic Structure and Chemical Bonding - HW-03-02

Chapter Name - PERIODIC TABLE AND PERIODICITY IN PROPERTIES - 3
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Q1. Classify the elements of group 14, 15 and 16 into metals, non-metals and metalloids.

Q2. Compare the metallic character of aluminium and potassium.

Q3. Name the group number for the following type of clements
(i) Alkaline earth metals
(ii) Alkali metals
(iii) Transition metals
(iv) Halogens
(v) Noble gases.

Q4. Name five man made elements

Q5. Write the names of the elements with atomic numbers 105, 109, 112, 115 according to IUPAC nomenclature.

Q6. Arrange the following in the order of increasing size Na+, Al3+, O2–, F–

Q7. How does the size of atoms vary from left to right in a period and on descending a group in the periodic table?

Q8. What is the correlation between atomic size and ionization enthalpy.

Q9. Which species, in each pair is expected to have higher ionization enthalpy.
(i) 3Li, 11Na (ii) 7N, 15P
(iii) 20Ca, 12Mg (iv) 13Al, 14Si
(v) 17Cl, 18Ar (vi) 18Ar, 19K
(vii) 13Al, 14C

Q10. Account for the fact that there is a decrease in first ionization enthalpy from Be to B and Mg to Al.

Q11. Why is the ionization enthalpy of the noble gases highest in their respective periods?

Q12. Name the most electronegative element.

Q13. Define modern periodic law.

Q14. Refer the periodic table given in Table 3.2 and answer the following questions.
(i) The elements placed in group number 18 are called ...............
(ii) Alkali and alkaline earth metals are collectively called ............... block metals.
(iii) The general configuration for halogens is ...............
(iv) Name a p-block element which is a gas other than a noble gas or a hologen.
(v) Name the groups that comprise the ‘s’ block of elements.
(vi) Element number 118 has not yet been established, to which block, will it belong?
(vii) How many elements should be there in total if all the 7s, 7p, 6d and 5f, blocks are to be full?

Q15. Describe the variation of electron affinity and ionization enthalpy in the periodic table.

Q16. Define the following:
(a) Electron gain enthalpy (b) Ionization enthalpy
(c) Ionic radius (d) Electronegativity.

Q17. What is electronegativity? How is it related to the type of bond formed?

Q18. Why is the electron gain enthalpy of Cl more in negative value as compared to that of F?

Module II: Atomic Structure and Chemical Bonding - HW-02-02

Chapter Name - ATOMIC STRUCTURE - 2
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Q1. Compare the mass of an electron with that of the proton.

Q2. What is a fundamental particle?

Q3. What is the name given to neutral particles in the atom?

Q5. List the three constituent particles of an atom.

Q6. What was the aim of Rutherford’s α-rays scattering experiment?

Q7. Briefly describe Rutherford’s model of an atom.

Q8. On what basis was the Rutherford’s model rejected?

Q9. What is an electromagnetic radiation?

Q10. List any three characteristics of electromagnetic radiation.

Q11. What is wave number? How is it related to wave length?

Q12. What is the difference between a ‘quantum’ and a ‘ photon’?

Q13. What is the difference between a line spectrum and a continuous spectrum?

Q14. What are the main postulates of Bohr’s model?

Q15. How does the energy of a Bohr orbit vary with the principle quantum number‘n’.

Q16. What do you understand by wave-particle duality?

Q17. Name the experiment that established the wave nature of electron.

Q18. Compute the de-Broglie wavelength associated with an electron moving with a velocity of 100 km /second? (me= 9.1 x 10-31kg)

Q19. State Heisenberg’s Uncertainty Principle?

Q20. What do you understand by a Wave Function?

Q21. What is the difference between an orbit and an orbital?

Q22. What are quantum numbers? List different quantum numbers obtained from Schrödinger Wave Equation?

Q23. Give the significance of the principal, azimuthal and magnetic quantum numbers?

Q24. What are the shapes of s,p and d orbitals?

Q25. Describe the shape of a 2s orbital. How is it different from that of a 1s orbital?

Q26. What do you understand by
(i) a spherical node?
(ii) a nodal plane?

Q27. How many spherical nodes will be there in 3s orbital ?

Q28. What do you understand by the electronic configuration of an atom?

Q29. What is Pauli’s exclusion principle?

Q30. What is Aufbau principle? What are ( n + l ) rules?

Q31. Which of the following orbitals will be filled first?
i) 2p or 3s ii) 3d or 4s

Q32. The electronic configuration of Cr is (Ar)3d5 4s1 not 3d4 4s2

Q33. (a) What are the three fundamental particles that constitute an atom? (b) Compare the charge and mass of an electron and of a proton.

Q34. What do you think is the most significant contribution of Rutherford to the development of atomic structure?

Q35. What experimental evidence shows the dual nature of light? (a) Compute the energy of a FM radio signal transmitted at a frequency of 100 MHz. (b) What is the energy of a wave of red light with l = 670 nm?

Q36. In what way was the Bohr’s model better than the Rutherford’s model?

Q37. What are the drawbacks of Bohr’s Model?

Q38. What led to the development of Wave Mechanical Model of the atom?

Q39. What do you understand by an orbital? Draw the shapes of s and p orbitals.

Q40. Explain the Hund’s rule of maximum multiplicity with the help of an example.

Module 2 Ecological Concepts and Issues - HW-04-02

Chapter Name - PRINCIPLES OF ECOLOGY - 4
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#. Answer the following Topics in 6 lines only or 6 points :

T1. DEFINITION OF ECOLOGY

T2. LEVELS OF ECOLOGICAL ORGANIZATION

T3. HABITAT AND ORGANISM
1.Factors leading to rise in population
2.Impact of population growth on environment

T4. NICHE AND ORGANISM

T5. ADAPTATION
1.What is a Species
2.Variation
3.Evolution

T6. SPECIES FORMATION: SPECIATION

T7. POPULATION
1.Pollution by agrochemicals

T8. POPULATION GROWTH
1. Density independent population growth

T9. COMMUNITIES AND THEIR CHARACTERISTICS
1.Organization of a biotic community
2.Stratification
3.Community Characteristics

T10. ECOLOGICAL SUCCESSION