## Sunday, April 26, 2015

### Explanation Of S-Block (Group IA) Elements

S – Block
(Group IA Elements)

Group I A Elements:

Hydrogen (Atomic number-1)

Lithium (Atomic number-3)

Sodium (Atomic number-11)

Potassium (Atomic number-19)

Rubidium (Atomic number-37)

Strontium (Atomic number-55)

Francium(Atomic number-87)

•  Elements of group IA and IIA are known as s – block elements.
•  Except Hydrogen all elements are metals.
•  Group IA consists of Hydrogen, Lithium, Sodium, Potassium, Rubidium, Cesium, Francium.
•  Group IIA consists of Beryllium, Magnesium, Calcium, Stratium, Barium, Radium.
•   IA elements (except Hydrogen) – Alkali metals.
•   IIA Elements  $\to$ Alkaline earth metals.
•  Have one electron in the outermost orbit.
•  By loosing the electron they form stable + 1 ion.
•  They are univalent and show + 1 oxidation state.

General periodic trends:
•  As we move from up to down the atomic radius increases.
• Order of radius size = H < Li < Na < Rb < Cs < Fr
• Ionization potential coming from top to bottom decreases.
• Density generally increases with an increase in atomic mass.
• Exception  $\to$ Potassium is expected to have more density than sodium but actually has less than sodium.
• Electronegativity from top to bottom decreases.
• Alkali metals along their individual periods have largest atomic radius.
• Mobility of ions increases down the group
$\therefore R{{b}^{+}}>{{K}^{+}}>N{{a}^{+}}>L{{i}^{+}}>{{H}^{+}}$  is the order of mobility of ions.
• Hydration enthalpy decreases down the group.
$\therefore {{H}^{+}}>L{{i}^{+}}>N{{a}^{+}}>{{K}^{+}}>R{{b}^{+}}$  in case of hydration enthalpy.

Ø   Heating effects of carbonates of IA elements.

${{M}_{2}}C{{O}_{3}}\xrightarrow{\Delta }No\,\,\,effect.\,\,\,\left( M=Na,K,Rb,Cs \right)$

$\left( Exception\,\,\,L{{i}_{2}}C{{O}_{3}}\xrightarrow{{}}L{{i}_{2}}O+C{{O}_{2}} \right)$

Ø   All metal carbonates of IA except  $L{{i}_{2}}C{{O}_{3}}$  have no effect on heating.

Ø  Heating effect of nitrates of IA elements.
$MN{{O}_{3}}\xrightarrow{\Delta }MN{{O}_{2}}+\frac{1}{2}{{O}_{2}}\left( M=Na,K,Rb,Cs \right)$
$\left( LiN{{O}_{3}}\xrightarrow{{}}L{{i}_{2}}O+N{{O}_{2}} \right)$
Flame Test :

$Li\to$     Crimson Red
$Na\to$   Yellow
$K\to$     Violet
$Rb\to$    Red Violet
$Cs\to$    Blue

## Friday, April 24, 2015

### Indefinite Integrals

Explanation Of Indefinite Integrals.......
•  A function  $\phi \left( x \right)$ is called a primitive (or) an anti-derivative of a function f(x) of
•
$\phi '\left( x \right)=f\left( x \right)$
$\Rightarrow \,\,\,\,\,\,\int_{{}}^{{}}{f\left( x \right)}=\phi \left( x \right)+C$
Here  f(x) = integrand
$\frac{d\left( \int_{{}}^{{}}{f\left( x \right)}\,dx \right)}{dx}=f\left( x \right)$

Algorithm to solve integrals of form:

$\int_{{}}^{{}}{{{\sin }^{m}}x}\,{{\cos }^{n}}x\,dx ,$  $\int_{{}}^{{}}{{{\sin }^{m}}x\,dx}$
$\int_{{}}^{{}}{{{\cos }^{n}}x}dx,$          where  $m,n\in N$
Step 1:    Find m,n
Step 2:
*   If m is odd i.e. power or index of sin x is odd, put cos x = t and reduced the integrand in terms of it.
*  If n is odd then put sin x = t and reduce the integrand.
*  If both are odd you can use any of above two methods.

• To Evaluate integrals of form
$\int_{{}}^{{}}{{{\sin }^{m}}x}\,{{\cos }^{n}}x\,dx$    where     $m,n\in Q$ and m + n is negative even integer then put
tan x = t.
• To evaluate integrals of form $\int_{{}}^{{}}{{{\sin }^{m}}x}\,{{\cos }^{n}}x\,dx$   where m,n are positive even integers then.
Let Z = cos x + i sin x
$\frac{1}{Z}=cosx-i\text{ }sinx$
$Z+\frac{1}{Z}=2\,\cos x$
$Z-\frac{1}{Z}=2i\,\operatorname{sinx}$
$cos\,nx=\frac{1}{2}\left( {{Z}^{n}}+\frac{1}{{{Z}^{n}}} \right)$
$\sin nx=\frac{1}{2i}\left( {{Z}^{n}}-\frac{1}{{{Z}^{n}}} \right)$
• Some important standard integrals :
$\int_{{}}^{{}}{\frac{1}{{{x}^{2}}-{{a}^{2}}}dx=}\frac{1}{2a}\log \left| \frac{x-a}{x+a} \right|+C$

$\int_{{}}^{{}}{\frac{1}{{{a}^{2}}-{{x}^{2}}}dx=}\frac{1}{2a}\log \left| \frac{a+x}{a-x} \right|+C$

$\int_{{}}^{{}}{\frac{1}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}dx=\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C$
$\int_{{}}^{{}}{\frac{1}{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=}\log \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

Integrals of form

$\int_{{}}^{{}}{\frac{dx}{{{\left( a+b\cos x \right)}^{2}}}},$     $\int_{{}}^{{}}{\frac{dx}{{{\left( a+b\sin x \right)}^{2}}}}$    to evaluate this type of integrals

Define     $P=\frac{\sin x}{a+b\cos x}$   (or)      $\frac{\cos x}{a+b\sin x}$

Depending on integral then find   $\frac{dP}{dx}$  in terms of   $\frac{1}{a+b\cos x}$  (or) $\frac{1}{a+b\sin x}$

Now integrate both sides of expression to get required integral.

Integration by parts :

$\int_{{}}^{{}}{u\nu }dx=u\left( \int_{{}}^{{}}{\nu dx} \right)-\int_{{}}^{{}}{\left\{ \frac{du}{dx}.\int_{{}}^{{}}{\nu dx} \right\}}dx$

Very IMP Note:
Choose first function as the function which come first in word “ILATE”
(1)  I – Inverse trigonometric
L – logarithmic functions
A – algebraic functions
T for Trigonometric functions
E for exponential functions.

è   Integrals of the form
$\int_{{}}^{{}}{{{e}^{x}}}\left( f\left( x \right)+f'\left( x \right) \right)dx={{e}^{x}}f\left( x \right)+C$
Note:
$\int_{{}}^{{}}{{{e}^{kx}}}\left\{ k\,f\left( x \right)+f'\left( x \right) \right\}={{e}^{kx}}f\left( x \right)+C$
è    $\int_{{}}^{{}}{{{e}^{ax}}}\sin bx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}\left( a\sin \,bx+b\cos \,bx \right)+C$
è    $\int_{{}}^{{}}{{{e}^{ax}}}\,\cos \,bx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}\left( a\,cos\,bx+b\sin \,bx \right)+C$

Some Very Important Integrals :

$\int_{{}}^{{}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{1}{2}{{a}^{2}}{{\sin }^{-1}}\left( \frac{x}{a} \right)+C$
$\int_{{}}^{{}}{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{{{a}^{2}}}{2}\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C$
$\int_{{}}^{{}}{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\frac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{{{a}^{2}}}{2}\log \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

Reduction Formulae :
$\int_{{}}^{{}}{{{\sin }^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{-{{\sin }^{n-1}}x\,\cos \,x}{n}+\frac{n-1}{n}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\cos }^{n}}x\,dx={{I}_{n}}}$
${{I}_{n}}=\frac{{{\cos }^{n-1}}x\,\sin \,x}{n}+\frac{n-1}{n}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\tan }^{n}}x\,dx={{I}_{n}}}$
${{I}_{n}}=\frac{{{\tan }^{n-1}}x}{n-1}-{{I}_{n-2}}$

$\int_{{}}^{{}}{\cos e{{c}^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{-\cos e{{c}^{n-2}}\,\cot x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\sec }^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{{{\sec }^{n-2}}x\,\tan \,x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\cot }^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{-{{\cot }^{n-1}}x}{n-1}-{{I}_{n-2}}$