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Volume has the units of (length)³ . So volume has units of m³ or cm³ ordm³. A common unit, litre (L) is not an SI unit, is used for measurement of volume of liquids. 1 L = 1000 mL, 1000 cm³ = 1 dm³.

Density of a substance is its amount of mass per unit volume.SI unit of density = SI unit of mass/SI unit of volume = kg/m³ or kg m^–3 This unit is quite large and a chemist often expresses density in g cm^–3.

There are three common scales to measure temperature—°C (degreecelsius), °F (degree Fahrenheit) and K (kelvin). Here, K is the SI unit. 

°F = 9/5(°C) + 32

K = °C + 273.15

Note—Temperature below 0 °C (i.e. negative values) are possible in Celsius scale but in Kelvin scale, negative temperature is not possible.

Characteristics of compound :

• Compounds always contain a definite proportion of the same elements by mass. 
• The properties of compounds are totally different from the elements from which they are formed. 
• Compounds are homogeneous.  
• Compounds are broadly classified into inorganic and organic compounds. Inorganic compounds are those, which are obtained from non-living sources such as minerals. For example, common salt, marble and limestone. Organic compounds are those, which occur in living sources such as plants and animals. They all contain carbon. Common organic compounds are oils, wax, fats etc.

A compound is a pure substance made up of two or more elements combined in a definite proportion by mass, which could be split by suitable chemical methods.

Unit factor:- Using dimensional analysis, one can obtain the unit factor to express a quantity in a new unit. In unit factor the nominators and denominators are expressed in different but represent the same or equivalent amounts multiplying by unit factor is the same as multiplying by one.

(a) 1 

(b) 2 and 

(c) 4

An element is the simplest form of matter that cannot be split into simpler substances or built from simpler substances by any ordinary chemical or physical method. There are 114 elements known to us, out of which 92 are naturally occurring while the rest have been prepared artificially. 

Elements are further classified into metals, non-metals and metalloids.

Molar mass:- The mass of 1 mole of any substance is called its molar mass.

Mixtures are of two types:

(I) Homogeneous – A mixture is said to be homogenous if its composition is uniform throughout.

(II) Heterogeneous – A mixture is said to be heterogeneous if its composition is not uniform throughout.

Elements: Silver, platinum

Compounds: Water, Carbon dioxide

Mixtures: Steel, Tea

Lattice points in face centred cubic or face centred tetragonal

= 8 (at corners) + 6 (at face centres) = 14

However, particles per unit cell

8 x 1/8 + 6 x 1/2 = 4

lattice points in body centred cube = 8 (at comers) + 1 (at body centre) = 9.

However, particles per unit cell = 8 x 1/8 + 1 = 2

(i) Schottky defect: It arises due to the missing of equal number of cations and anions from their normal positions leaving behind a pair of holes. This defect is usually observed in ionic compounds having high coordination number and ions of almost similar size. The density of the crystal is lowered due to the presence of vacancies in the crystal lattice. Schottky defect increases slightly the electrical conductance of the crystal. E.g. NaCl, KCl, CsCl. 

(ii) Frenkel defect: It arises w’hen an ion, usually cation, leaves its normal site and occupies an interstitial site. This defect is usually observed in ionic ‘ compounds having low co-ordination number , and crystals with anions are much larger in size than the cations. Since no ions are missing from the crystals as whole, it doesn’t affect the density of the crystal. E.g.: AgCl, ZnS, AgBr 

(iii) Interstitials: Atoms or ions which occupy the normally vacant interstitial positions in a crystal are called interstitials. 

(iv) F-centres: The electron occupying holes, created by missing of anions from the lattice sites are called F-centres. These F-centres are responsible for colour of compound.

Element – (iv), (ix); 

Compounds – (viii), (xii), (xiv), (xv); 

Mixtures – (i), (ii), (iii), (v), (vi), (vii), (x), (xi), (xiii).

Average atomic mass:- The average atomic mass of an element is the average relative mass of its atoms as compared with an atom of carbon taken as 12.

The atomic mass can be determined using the formula

\(M = \frac{\rho \times a^3 \times N_0}{Z}\)

Where

ρ is the density of the metal, a is the dimension of the unit cell, N₀ is Avogadro number and Z is the number of atoms in the unit cell.

(i) When a third layer is placed over the second layer in such a way that the spheres cover the tetrahedral or ‘C’ voids, a three dimensional closest packing is obtained where the spheres in every third or alternate layers are vertically aligned (i.e. the third layer is directly above the first, the fourth above the second layer or so on) calling the first layer as layer A and the second layer as layer B, the arrangement is ABAB pattern or hexagonal close packing. When the third layer is placed over the second layer in such a way that spheres cover the octahedral or ‘d’ voids, a layer different from layers A and B is produced. Let us call it as layer C, continuing further, a packing is obtained were the spheres in every fourth layer will be vertically aligned. This pattern of stocking spheres is called ABCABC… pattern or close cube packing (CCP). 

(ii) A regular three dimensional arrangement of points in space is called crystal lattice. Unit cell is the smallest portion of a crystal lattice which, when repeated in different directions, generates the entire lattice. 

(iii) Wherever a sphere of the second layer is above the void of the first layer (or vice versa) a tetrahedral void is formed. These voids are called tetrahedral voids because a tetrahedron is formed when the center of these four spheres are joined. The voids which are surrounded by six spheres is called octahedral voids.

Homogeneous: air, petrol, tap water, 22 carat gold, cloud and steel

Heterogeneous: milk, wood, iodized table salt.

1. Bohr magneton (μ_B)

2. Match :

• Paramagnetic – O₂ 
• Ferromagnetic – CrO₂
• Antiferromagnetic – MnO
• Ferrimagnetic – Fe₃O₄ 

(a) The smallest repeating unit of a crystal.

(b) Body centred unit cell, Face centred unit cell and End centred unit cell.

(c) Cubic crystal system.

(d) NaCI (Rock salt struãture)

(i) The coordination number of a constituent particles (atom, ion or molecule) in a crystal is the number of constituent particles which are the immediate neighbour of the particle in the crystal. In ionic crystals coordination number of an ion in the crystal is the number of oppositely charged ions surrounding that particular ion. 

(ii) (a) 12 

(b) 8.

(I) 0.281/5.006=0.0561 

(II) 0.58 + 324.65= 325.23

Ice has the weakest intermolecular force.

1. Names of A and B are :

• Hexagonal close packing.
• Square close packing,

2. shaded parts of A and B are :

• Is octahedral void; in
• Is tetrahedral void.  

Frenkel defect.

1. The number of atoms present per unit cell of ‘A’.

• Contribution from 8 corners = 1/8 × 8 = 1 
• Contribution from centres of 6 faces = 1/2 × 6 = 3
•  Total number of atoms of A’ per unit cell = 1 +3 = 4

2. Schottky defect 

(a) (ii) Isotropic nature
b) (i) Stoichiometric defects are those point defects which do not disturb the stoichiometry of the solid.

(ii)

As atoms Q are present at the 8 corners of the cube, therefore, number of atoms of in the unit cell = \(\frac{1}{8}\) × 8 = 1 

As atoms P are present at the body centre, therefore, number of atoms of x in the unit cell = 1 

∴ Ratio of atoms x:y =1:1 

Hence the formula of the compound is PQ coordination number of each of P and Q = 8.

Glass is an amorphous solid in which the constituent particles (SiO₄ tetrahedra) have only short range order and there is no long range order. On melting quartz and then cooling it rapidly, it is converted into glass.

Michael Jackson: 

Michael Jackson was born in 1958 in America. He was known as the King of Pop. He was famous as a singer, songwriter, dancer, and actor. His notable works are Bad, Dangerous, Off the Wall, etc. He passed away in 2009.

a. Lend a hand to life 

b. To help others, people are dying 

c. Love 

d. Live in harmony

a. comes 

b. heed 

c. together 

d. dying 

e. family 

f. love 

g. change

The love for mankind and the suffering people around the world have prompted the lyricists to sing a song like this.

lt is useless to think that someone will soon make a change somewhere.

According to lyrisicts a real change will come only when we stand together.

= \(\begin{pmatrix} 3&λ&2 \\ 1 &2&5\\2&1&1 \end{pmatrix}\)

│A│= 0 

1(2 x 1 – 5 x 1) - (1 x 1 – 5 x 2) + 2 ( 1 x 1 – 2 x 2) = 0 

-3 + 9 λ - 6 = 0 

9 λ = 9 

λ = 1

Matrix A is said to be nilpotent since there exist a positive integer k =1 such that Ak is zero matrix.

-1= 1|A| adj A 

adj A = |A| x A^-1

= 3 x \(\begin{pmatrix}3&-1 \\-5/ 3 &2/3\end{pmatrix}\)

= \(\begin{pmatrix}9&-3 \\-5 &2\end{pmatrix}\)

1 = 1/|A| adj A --------- 1 

|A|  = 3 x 2 – (1) x (-1) 

= 7 

C₁₁ = 3 C₁₂ = -1 

C₂₁ = 1 C₂₂ = 2

Cofactor matrix A = \( \begin{pmatrix} 2&1 \\ 4 &3 \end{pmatrix}\)

Adj A = \( \begin{pmatrix} 3&-1 \\ 1 &2 \end{pmatrix}\)

= \( \begin{pmatrix} 3&1 \\ -1 &2 \end{pmatrix}\)

Putting in 1

A^-1 = 1/|7| \( \begin{pmatrix} 3&1 \\ -1 &2 \end{pmatrix}\)

= \( \begin{pmatrix} 3/7&1/7 \\ -1/7 &2/7 \end{pmatrix}\)

When a given matrix is singular then the given matrix determinant is 0. 

|A| = 0

Given,

A = \(\begin{pmatrix} 1 & k & 3 \\ 3 & k &-2 \\ 2 & 3 & -4 \end{pmatrix}\)

|A| = 0

1(-4k + 6) –k(-12 + 4) +3 (9 -2k)= 0 

-4k + 6 +12k -4k + 27 -6k = 0 

-2k +33 = 0

k = 33/2,

To find adj A we will first find the cofactor matrix 

C₁₁ = d C₁₂ = -c 

C₂₁= -b C₂₂ = a 

Cofactor matrix A = \(​​​​\begin{pmatrix}d& -c \\-b & a \\\end{pmatrix}\)

Adj A = \(​​​​\begin{pmatrix}d& -c \\-b & a \\\end{pmatrix}\)

= \(​​​​\begin{pmatrix}d& -c \\-b & a \\\end{pmatrix}\)

2 – A + 2I = 0 

Multiplying by A^-1 

A^-1A² – A^-1A + 2I A^-1 = 0 

A^{- I} + 2 A^-1 = 0

A^-1 = 1/2(I - A)

2 + xI = yA

\( \begin{pmatrix}3&1 \\7 &5\end{pmatrix}\)\( \begin{pmatrix}3&1 \\7 &5\end{pmatrix}\) + x\( \begin{pmatrix}1&0 \\0 &1\end{pmatrix}\) = y \( \begin{pmatrix}3&1 \\7 &5\end{pmatrix}\)

\( \begin{pmatrix}16&8 \\56 &32\end{pmatrix}\) + x\( \begin{pmatrix}1&0 \\0 &1\end{pmatrix}\) = y\( \begin{pmatrix}3&1 \\7 &5\end{pmatrix}\)

8\( \begin{pmatrix}2&1 \\7 &4\end{pmatrix}\) + x\( \begin{pmatrix}1&0 \\0 &1\end{pmatrix}\) = y\( \begin{pmatrix}3&1 \\7 &5\end{pmatrix}\)

Comparing L.H.S and R.H.S 

x = 8 y = 8

s AB is a 0 matrix its determinant has to be 0. 

So │AB│=│A││B│= 0 

So │A│=│B│= 0

(AB)(AB)^-1 = I 

A -1(AB)(AB)^-1 = IA^-1 

(A^-1A)B (AB)^-1=A^-1 

IB(AB)^-1 = A^-1 

B(AB^)-1 = A-1 

B^-1B(AB)^-1 = B^-1 A^-1 

I (AB) -1 = B -1A -1 

(AB) -1 = B -1A -1

(AB)(AB)^-1 = I 

A^-1(AB)(AB)^-1 = IA^-1 

(A ^-1A)B (AB) ^-1=A^-1 

IB(AB)^-1 = A^-1 

B(AB)^-1 = A^-1 

B^-1B(AB)^-1 = B^-1 A^-1 

I (AB)^-1 = B^-1A^-1 

(AB)^-1 = B^-1A^-1

If the two matrices A and B are of same order it is not necessary that in every situation AB = BA 

AB = BA = I only when A = B^-1 

B = A^-1 

Other time AB ≠ BA

The property states that │adj A│= │A│^n-1 

Here n= 3 and │A│=5 

│adj A│= │5|^3-1 

= │5│² 

= 25