also check

http://en.wikipedia.org/wiki/Avogadro's_number

The Mole Concept (Avogadro's Number)

Molecules and atoms are extremely small objects - both in size and mass. Consequently, working with them in the laboratory requires a large collection of them. How large does this collection need to be? A standard needs to be introduced. This standard is the "mole". The mole is based upon the carbon-12 isotope. We ask the following question: How many carbon-12 atoms are needed to have a mass of exactly 12 g. That number is NA - Avogadro's number. Thus, NA is defined by 



 

 

A convenient name is given when there is an Avogadro's number of objects - it is called a "mole". Thus in the above example there was a mole of pennies.

The mole concept is no more complicated than the more familiar concept of a dozen : 1 dozen = 12 objects. From the penny example above one might suspect that the mass of a mole of objects is huge. Well, that is true if we're considering a mole of pennies, however a mole of atoms or molecules is a different story. Recall that the atomic mass unit (amu) is defined as 1/12 the mass of a carbon-12 atom. Consequently we have the relation 

Thus, a mole of carbon-12 atoms has a mass of just 12 g. What about other atoms? In the periodic table the atomic mass of the elements is given. For example the atomic mass of magnesium is 24.305 amu. This is the average isotopic mass of naturally occurring magnesium. What is the molar mass of magnesium in grams? From the equation above we get 1 amu = 1g/NA or 1 amu = 1.66054x10^-24 g. Thus, a mole of magnesium atoms has a mass of NA x 24.305 amu x (1.66054x10^-24 g/amu) = 24.305 g. A mole of magnesium atoms has a mass of 24.305 g. This example demonstrates that the atomic mass of magnesium can be interpreted in one of two ways: (1) the average mass of a single magnesium atom is 24.305 amu or (2) the average mass of a mole of magnesium atoms is 24.305 g;

 A similar conclusion follows for all of the other elements.

Crash course (11tn and 12th both) won't be of any help for u are merely in class 11th.



Better u opt the same when u are gone through roughly the syllabus for 12th, i.e. nect year.







The course for 11 and 12 is designed in such a manner that it takes a span of 2 years to get it completed fairly.



Thu dont rush, but build your concepts, fundamentals and knowledge steadily and strongly.



Gud luck and all the best for ur future endeavors.

Use the following approach



Sin ( a + h ) = Sin a Cos h + Cos a Sin h

or Sin ( a + h ) = Sin a ( 1 + h^2/ 2! + ...) + Cos a (h + h^3 / 3! +...)



Thus,

Difference D is given by

D = Sin ( a + h ) - (Sin a + h.Cos a)

=Sin a ( 1 + h^2 / 2 ! +...) + Cos a. (h + h^3 / 3! +...) - Sin a - h.Cos a

= Sin a ( h^2 / 2 ! +...) + Cos a . ( h^3 / 3! +...)

 

What is X-ray diffraction (XRD)

Bragg's Law

or

A real 3-dimensional crystal contains many sets of planes. For diffraction, crystal must have the correct orientation with respect to the incoming beam.

Perfect, infinite crystal and perfectly collimated beam: diffraction condition must be satisfied ``exactly.''

Strains, defects, finite size effects, instrumental resolution: diffraction peaks are broadened.

More formally, the scattered intensity is proportional to the square of the Fourier transform of the charge density:

where is the charge density.

For perfect crystals, I(q) consists of delta functions (perfectly sharp scattering). For imperfect crystals, the peaks are broadened. For liquids and glasses, it is a continuous, slowly varying function.

Features of Electron, X-ray, or Neutron Diffraction

• For a known structure, pattern can be calculated exactly.


• Symmetry of the diffraction pattern given by symmetry of the lattice.


• Intensities of spots determined by basis of atoms at each lattice point.


• Sharpness and shape of spots determined by perfection of crystal.


• Liquids, glasses, and other disordered materials produce broad fuzzy rings instead of sharp spots.


• Defects and disorder in crystals also result in diffuse scattering.

The ``Ultimate'' (Technically Challenging) Experiment

• Sample is tiny (micron-sized).


• The effect is weak (light elements, small modulations, subtle modifications of the long-range order).


• Instrumental resolution (angle and energy) is ``perfect'' allowing detailed measure- ments of structural disorder.


• Measurement is time-resolved (nanosecond time scale).

To achieve all of the above, will need lots of intensity in the primary beam together with sensitive detection systems.

Powder vs. Single Crystal X-ray Diffraction

SINGLE CRYSTAL

Put a crystal in the beam, observe what reflections come out at what angles for what orientations of the crystal with what intensities.

Advantages

In principle you can learn everything there is to know about the structure.

Disadvantages

You may not have a single crystal. It is time-consuming and difficult to orient the crystal. If more than one phase is present, you will not necessarily realize that there is more than one set of reflections.

POWDER

Samples consists of a collection of many small crystallites with random orientations. Average over crystal orientations and measure the scattered intensity as a function of outgoing angle.

Disadvantage

Inversion of the measured intensities to find the structure is more difficult and less reliable.

Advantages

It is usually much easier to prepare a powder sample. You are guaranteed to see all reflections. The best way to follow phase changes as a function of temperature, pressure, or some other variable.

To prove    2^sinx+2^cosx > 2^(1-1/)

Let y = 2^sinx+2^cosx

Differentiating both the sides we obtain

dy/dx = Sin x . Cos x . 2^{sinx -1} - Cos x . Sin x . 2^{cosx - 1} 

For maxima or minima

dy/dx = 0

or  Sin x . Cos x . 2^{sinx -1} - Cos x . Sin x . 2^{cosx - 1}  = 0

or Sin x . Cos x . 2^{sinx -1} = Cos x . Sin x . 2^{cosx - 1} 

or 2^{sinx -1} =  2^{cosx - 1} 

or Sin x - 1= Cos x  - 1

or Sin x = Cos x

which is possible when x = pi / 4

or Sin x = Cos x  =1 / 

Thus for Sin x = Cos x  =1 / 

 y  =  2^sinx+2^cosx  = 2.2^1/

 This value is either maxima or minima, but second derivative test shows that it is minima,

(another way is that when x = 0, y = 3 which is greater than 2.2^1/  Hence later is minima)

Thus y  > or = 2.2^1/ > 2^(1-1/)

Hence Proved

To prove    2^sinx+2^cosx > 2^(1-1/)

Let y = 2^sinx+2^cosx

Differentiating both the sides we obtain

dy/dx = Sin x . Cos x . 2^{sinx -1} - Cos x . Sin x . 2^{cosx - 1} 

For maxima or minima

dy/dx = 0

or  Sin x . Cos x . 2^{sinx -1} - Cos x . Sin x . 2^{cosx - 1}  = 0

or Sin x . Cos x . 2^{sinx -1} = Cos x . Sin x . 2^{cosx - 1} 

or 2^{sinx -1} =  2^{cosx - 1} 

or Sin x - 1= Cos x  - 1

or Sin x = Cos x

which is possible when x = pi / 4

or Sin x = Cos x  =1 / 

Thus for Sin x = Cos x  =1 / 

 y  =  2^sinx+2^cosx  = 2.2^1/

 This value is either maxima or minima, but second derivative test shows that it is minima,

(another way is that when x = 0, y = 3 which is greater than 2.2^1/  Hence later is minima)

Thus y  > or = 2.2^1/ > 2^(1-1/)

Hence Proved

Maxima and Minima from Calculus

One of the great powers of calculus is in the determination of the maximum or minimum value of a function. Take f(x) to be a function of x. Then the value of x for which the derivative of f(x) with respect to x is equal to zero corresponds to a maximum, a minimum or an inflexion point of the function f(x).

For example, the height of a projectile that is fired straight up is given by the motion equation:

Taking y₀ = 0, a graph of the height y(t) is shown below.

The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function y(t) plotted as a function of t. The derivative is positive when a function is increasing toward a maximum, zero (horizontal) at the maximum, and negative just after the maximum. The second derivative is the rate of change of the derivative, and it is negative for the process described above since the first derivative (slope) is always getting smaller. The second derivative is always negative for a "hump" in the function, corresponding to a maximum

Thus from above theory

I = (V₀ / R) * exp (-t/RC)

For the two capacitors C is different but when

t= 0

I = (V₀ / R) and is independent of capacitance.

Thus

(b)current in each discharging cicuit at t=0 are equal but not 0

is the correct option

Capacitor Discharge Calculation

For circuit parameters:

 

However if opposite end of the rod is not fixed and free.

Then impuse will make rotate the rod about its center.

The approach will exactly be like the one suggested above, but moment of inertia will be calulated around center of the rod.

Here we apply the conservation of momentum as follows

Angular momentum imparted is given by

I x w = L x P

where P = impulse imparted

I = moment of inertia of the rod across one end  or I = mL²/3

w = angular velocity

L = length of the rod

Thus w = LP/I = 3LP/mL²

or w = 3P/mL  ...(1)

Now let time taken by the rod to become perpendicular =  t (here we assume that another end is fixed i.e rod is movinf about it)

thus w t = pi /2   ...(2)

From (1) and (2) we obtain the following relation

or t = pi/ 2w =  pi * mL/6P

Thus correct option is B) pi *mL/6p

How effective a dielectric is at allowing a capacitor to store more charge depends on the material the dielectric is made from. Every material has a dielectric constant k. This is the ratio of the field without the dielectric (E_o) to the net field (E) with the dielectric:

k = E_o/E

E is always less than or equal to E_o, so the dielectric constant is greater than or equal to 1. The larger the dielectric constant, the more charge can be stored.

Completely filling the space between capacitor plates with a dielectric increases the capacitance by a factor of the dielectric constant:

C = k C_o, where C_o is the capacitance with no dielectric between the plates.

For a parallel-plate capacitor containing a dielectric that completely fills the space between the plates, the capacitance is given by:

C = k e_o A / d

The capacitance is maximized if the dielectric constant is maximized, and the capacitor plates have large area and are placed as close together as possible.

If a metal was used for the dielectric instead of an insulator the field inside the metal would be zero, corresponding to an infinite dielectric constant. The dielectric usually fills the entire space between the capacitor plates, however, and if a metal did that it would short out the capacitor - that's why insulators are used instead.

Material	Dielectric constant	Dielectric Strength (kV/mm)
Vacuum	1.00000	-
Air (dry)	1.00059	3
Polystyrene	2.6	24
Paper	3.6	16
Water	80	-