Orbital Speed

The orbital speed of a celestial body measures its speed around another object’s center of gravity. This can be its speed at a given time and place in its path or may be its average speed. Depending upon the eccentricity of heavenly body's orbit the orbital speed changes as a satellite or moon gets closer to its center of gravity or further away. The two regions where speeds can change the most are pericenter and apocenter.

A satellite in orbit moves faster (pericenter) when it is close to the planet or other body that it orbits and slower (apocenter) when it is farther away. A satellite moving in a circular orbit has a constant speed which depends only on the mass of the planet and the distance between the satellite and the center of the planet.

Calculating the Orbital Speed

The speed (v) of a satellite in circular orbit is:

`v = sqrt((GM)/r)`

Where, `G` is the universal gravitational constant and the value is `6.6726 xx 10^-11 N m^2 kg^-2`,
`M`  is the mass of the combined planet-satellite system, in case Earth's mass is `5.972 xx 10^24 kg`, and we can ignore the satellite's mass, in case for smaller man made satellites.
and `r` is the radius of the orbit measured from the planet's center.
The period `P` of a satellite in circular orbit is the orbit's circumference divided by the satellite's speed:

`P = (2*pi*r)/v`

Kepler's Law for Orbital Speed:

Kepler's second law is illustrates that the line joining the Sun and planet sweeps out equal areas in equal times, this means the planet moves faster when it is nearer the Sun (perihelion). Henceforth, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. The point of nearest distance of the planet to the Sun is known as perihelion; the point of greatest separation is known as aphelion. Therefore, by Kepler's second law, the planet moves fastest when it is near perihelion and slowest when it is near aphelion.

Arrhenius theory acid

Introduction :
According to Arrhenius an acid  is any substance that dissociates   to give  a H+ ion  in aqueous solution.

Thus an aqueous solution of hydrochloric acid will  show the presence of H+ ions and Cl- ions.
 Later it was found that the H+ ions have no free existence in water or in aqueous solution but exist in the solvated state as hydronium ion as H₃O^+.

  Thus according to Arrhenius definition of an acid,  any substance that will increase the concentration of H+ ions in water is said to be an acid.

  General Representation:  An Arrhenius acid  is generally represented by the formula with a H in the begining as HCl , H₂SO4 , HNO_3. All  acids that can donate a proton in aqueous solution or  can increase the concentration of H₃O⁺ ions in solution  are called Arrhenius acids.The dissociation of an acid in water can be represented by the equation a follows:
 If HCl be the acid then the dissociation is given as
HCl((g)  +  H₂O(l)    ---> H₃O⁺  + Cl^-

The advantage of the Arrhenius definition of an acid.

1) All protic acids that can dissociate in aqueous solution to increase the H+ ione concentration show similar properties such as :

Reaction with bases result in neutralisation of the acid and formation of water and a salt
example: HCl + NaOH----------> NaCl + H₂O
                  HNO₃ + KOH --------> KNO₃ + H₂O      

2)  The basicity of the acids depends upon number of H+ ions, an acid can releas in aqueous solution

3) The pH of any substance depends upon the number of H+ ions the substance can release in solution

Limitation of Arrhenius definition of acid>

1) Arrhenius definition of acid holds good only for acids in aqueous solution. For example
HCl (aq) ----------> H⁺(aq)  + Cl^- (aq)
      H⁺ (aq) + H₂O (l) ---------> H₃O⁺ (aq)
    But HCl in gaseous state is neither acidic nor basic.

2) Acid not dissolved in (aq) solution cannot dissociate into H+ ions. For example
     HNO3 (l) + 2H2SO4 (l) -----------> NO⁺⁺ (l) + H₃O⁺ (l) + 2HSO₄^-   (l)
Here HNO3 acts as a base

3) Arrhenius definition of acid cannot account for the acidic character of AlCl₃

Learn definition of sec

Introduction :
Learn definition of Sec is defined as the function which is used to calculate the ratio of sides of the triangle. It is also known as inverse of cos function. Sec is a one kind of trigonometric functions. Sec of an angle is the ratio of hypotenuse and the length of the adjacent side. In other words, the learn definition of Sec is the reciprocal of cos.

In a right angle triangle,

                  Sec(A)=hypotenuse       
                              adjacent side        
               Here A is a angle, Sec (A) = 1/ (cos A)        

Learn definition of sec:

Learn properties of sec angle:
Learn definition of sect is intervallic and repeats itself every 2 radians. An essential property is sec(0)=1. Few other main properties are

By the definition of sec,
           sec x               = 1/cos x
           sec( x + 2 )   = sec (x) sec ( /2)    = ∞ (infinity)
           sec(-x)            = sec(x)
           sec(x)             = i sec h(ix)

Learn important calculus relations of sec:
             d/dx sec(x) = sec(x) tan(x)    (differentiation)
            ∫ sec(x) dx   = ln sec(x) + tan(x) = ln ( /4 + /2)  (integral)

Learn series expansion of the function of sec:
           sec(x)  = 1 + x²/2 + 5x⁴/24 + 61z⁶/ 720 + ..... + (-1)ⁿ E_2n/(2n)! X_2n + ......
Here the E's are the Euler numbers of secant.

Learn domain of sec:
          Every numbers are real except /2 + k , k is an integer.

Learn range of sec:
           (-∞ , -1] U [1 , +∞)

Learn Period of sec:
                 2 π

Learn y intercepts of sec:
           y = 1

Learn of sec symmetry:
           sec(-x) = sec (x).   Because sec (x) is an even function and graph is symmetric.

Learn of sec intervals of increase/decrease:
From 0 to 2, sec (x) is increasing on (0 to /2) U ( /2  to  ) and decreasing on ( to 3 /2) U (3 /2 to 2 ).

Learn vertical asymptotes of sec:
           Vertical asymptote = /2 + k π , where k is an integer.

Learn co function for sec : 
           sec x  = cosec (90^o - x).

Practice problem for learn definition of sec:

Practice problems using learn of secant function:
1. Find the function value of sec 45^o.

Solution:
         Use the Sec's co task identity to solve the problem.
By the definition of sec,
        function for sec is  sec x   = cosec (90^o - x)
                                         sec 30^o= cosec (90^o – 30^o )
                                                      = cosec (60^o)
                                          sec 30^o = 0.866
         The solution of sec of 30^o is 0.866.

2. Find the angle of a right triangle where hypotenuse = 2, length of the adjacent side = 1 using secant function?

  Given:
         In a right angle triangle length of the hypotenuse = 2, length of the adjacent side = 1

  Solution:         
By the definition of sec,
           Sec x  =  hypotenuse/adjacent side
                      = 2 / 1
                   x = sec^-1 (2)
                   x = 60^o 
The secant angle of triangle is 60 degrees.

Electricity sources

Introduction :

Electricity is the main branch of physics, which deals with the motion of the charges. The device which produce electricity or which can convert one form of energy into another form is called the sources of electricity.The sources of elecricity can be had from nuclear plant,  fossil fuels like coal, natural gas etc. Even the solar radiation can be used to obtain electricity.Here we discuss the sources of electricity.

About Electricity sources

Electricity is the very important form of energy, which we use, in our daily life. Now these days we even do not imagine the life in the absence of electricity. The sources of electricity are cells, which are very common and very convenient in use. There are two types of the cells one are called the primary cells and t0he other are called the secondary cells. The primary cells are those, which cannot be recharged, and the secondary cells are recharged several times up to some limit. Voltaic cell is the most important type of the source of electricity.

It is the simplest form of the electrochemical cell. It consists of a glass vessel containing dil. sulphuric acid. Copper and the zinc rod are dipped in the dil. Sulphuric acid. These copper and zinc rods are called the electrodes. When the bulb of a torch is connected across the electrodes through a conducting wire, it glows that means the electricity is produced due to some chemical reaction. When sulphuric acid dissolved in water, it decomposes into ions as shown in the reaction given below

H2SO4       `->`         2H+ + SO4 2-

Zinc atoms from zinc plate begin to dissolve slowly in H2SO4 in form of Zn2+. Each zinc atom dissolved in H2SO4 from the Zn plates leaves two electrons on the Zn plate. This zinc plate becomes negatively charged.

Zn (in solution)        `->`        Zn 2+   + 2e- (on Zn plate)

The potential acquired by the Zn plate is – 0.62 Volt. When Zn ions enter the solution from the Zn plate, an equal number of H+ leaves the solution and deposit on the copper plate. The hydrogen ions get electrons from the copper plate to become neutral hydrogen atoms.

H+ + e -    `->`            H

A pair of hydrogen atom combines to form the hydrogen molecule and the bubbles of the hydrogen gas are formed. Copper plate becomes positively charged and the potential is 0.46 Volt. So the potential difference of the voltaic cell is 0.46 – (-0.62) = 1.08 Volts.

Conclusion of the electricity sources

Thus, we observed that the same potential difference is developed across the plates of the cell. That means there are some free charges, which are in motion. As we connect a resistance in the path of the free charges, it will move and the electricity is produced so that the current flows through the bulb and it will glow.

How does electricity works

Introduction:

The basic electrical circuits consists of the following ;

The Source which generates power such as a generator,

the load which utilizes the power,

and two wires to carry the electrons act as conductor from the power source and back to power source.This is how the electricity will works.

                                        how does electricity works

Measure of electricity or power is Watts and Kilowatts.

Watts = (Volts X Amps).

Electron: In order to understand electrons, we need to have a understanding of the atom. Atom consists of  a nucleus(having protons and neutrons)  and electrons revolving around it in circular orbits.

How does electricity works

Current: Movement of charge carriers (electrons) from one place to another place is called as current.

One amp is defined as one coulomb of charge carriers passed through a cross section of conductor per unit time.That is  6.28 x 10^18 electrons per second.

Voltage: It is the electrical force that gives the energy to free electrons to move from one atom to another. Just as water needs some pressure to force it through a pipe, electrical current needs some force to make electron flow. "Volts" is the unit of measurement of "electrical pressure" that causes current flow. Voltage is also called as potential difference between two points along a conductor.

Load: Load is the one which consumes the generated energy

 The power source generates the voltage which force the electrons to flow through the conductors when load is connected and it is a closed loop. The load utilizes the power to convert electrical energy to some other form.

Importance of electricity:

Electricity is  life blood that flows in our society. Our survival based on electricity. Electricity generates in several forms. Electricity most basic generated form is lightning. Portable devices like torches,mobiles use batteries which is a form of static electricity.Sun is also one of the major source of electricity if we convert the radiated light energy to electricity with the help of photocells.

The very fundamentals of electricity starts with electrons. Electron flow depends upon the type of material . Some materials do not allow electrons to move through it freely from one atom to another, those type of materials called as Insulators. Some materials allow free flow of electrons those are good conductors of electricity called as conductors. The movement of electrons referred as current.

Electrons can freely move in conductors then why we need voltage ?

 Conductors having the movement of electrons in a random direction in order to make the the electrons to flow in a particular direction we need to impart energy for the electrons that energy is called as Voltage. We can compare this two terms with a dam having potential head as voltage and the stream of water flow can be compared with electron flow. The more the potential head in the dam the larger will be the flow of water.

A battery works in the similar way it is having two terminals, a positive terminal and a negative terminal. The source, whether may be generator or battery, will push the electrons to the negative terminus. The rate at which it pushes the electrons is the voltage. The equipment, electronics appliances you that will consume electricity  is called the load. The electrons will leave the negative side of the power source, energize the equipment, and flow to the positive side of the power source.

Resistance: It is the force that resists the flow of electrons.. The units of resistance Ohms..

OHM’S LAW:Ohm’s law  state that voltage  (V) in a circuit is directly proportional to the , current or amps are (I) .

 V=IR.

In  a light bulb. The thin wire inside  the bulb is called filament. When power is applied to the bulb, the tungsten filament resists the flow of electrons. We can calculate that resistance by Making Resistance as subject of formula, r=V/I. So a 60 Watt light bulb’s resistance would be 240 Ohms.

There are basically two types of electrical currents,
    Direct current (DC)

   Alternating current (AC).

Direct current :Here the magnitude of the current is constant through out the wave form .Example battery  produces the DC current that flows in one direction only that is moving directly from the negative terminal to the positive terminal of the battery.

Alternating current: The magnitude of the current continuously varies with time.Example of AC power sources Generator.The magnitude is varying with time so frequency comes into picture. Frequency can be defined as the number of cycles produced in a given unit of time.This is called Hertz (Hz) or Cycle.

How does electricity works

Advantages:

The advantage of alternating current is that from  power  generating stations send millions of volts from their power plants through small conductors to transformers that will step down to required voltages in the distribution end.

Fcc unit cell

Introduction to unit cell
A regular three dimensional arrangement of points in space is called a crystal lattice. A unit cell is the smallest of a crystal lattice which, when repeated in different directions generates the entire lattice

Face centered cubic unit cell

A face centered cubic unit cell contains atoms at all the corners and at the centre of all the faces of the cube. Each atom located at the face centre is shared between two adjacent unit cells and only half of each atom belongs to a unit cell. Thus, in a face centered cubic unit cell:

1. 8 corners atoms × 1/8 atoms per unit cell=1 atom
2. 6 face centered atoms × 1/2 atoms per cell=3 atoms

Therefore, total no. of atoms per unit cell = 4 atoms

Packing efficiency of fcc unit cell

Packing efficiency is the percentage of total space filled by the particles. Let us calculate the packing efficiency of fcc unit cell. Let the unit cell edge be ‘a’  and face diagonal be ‘b’.
We know that b=√2a
If r is the radius of the sphere, we find
    b = 4r =√2a or a = 2√2r
we know that each unit cell in fcc structure, has effectively 4 spheres. Total volume of four spheres is equal to 4×(4/3)πr³ and the volume of the cube is a³ or (2√2r)³.
Therefore,
Packing efficiency of fcc unit cell
=volume of 4 spheres ×100/volume of unit cell  %
=4×(4/3)πr³×100/(2√2r)³
=74%

Density of unit cell

Volume of unit cell = a³
Mass of the unit cell =number of atoms in unit cell×mass of each atom
=z × m
Where z is the number of atoms in unit cell and m is the mass of single atom.
Mass of an atom present in a unit cell:
    m = M/N_a   (M is molar mass)
therefore, density of the unit cell =mass/volume
         =z×m/a³ = z×M/a³×N_a
     d   =    zM/a³N_a

Summay

The number of atoms in a fcc unit cell is four and these are present at all corners as well as at the centre of all faces of the cube

Make a fuel cell

Introduction :
A Fuel Cell is a device that converts the energy of the chemical reaction between a fuel and an oxidant into electricity and heat.  It is easy to make a fuel cell working, by using an anode, cathode, catalysts and most often an electrolyte.  Fuels and oxidants are also necessary to make a fuel cell.  Fuel cells are combined into groups to obtain a usable voltage and power output, called stacks. Fuel cells generate electricity electrochemically, rather than mechanically, so they are more efficient over a wider load factor and can cut greenhouse gases by over 50 percent.  Fuel cells are very much different from batteries. Fuel cells consume reactant from an external source, which must be replenished.  Fuel cells produce electricity with an efficiency of about 70 % compared to thermal plants whose efficiency is about 40%.

How to make a fuel cell

We can make a fuel cell (Hydrogen Fuel Cell) in our kitchen in just 10 minutes, and demonstrate how hydrogen and oxygen combines to give clean electricity.
To make a fuel cell, we would need:

• One foot of platinum coated nickel wire along with small piece of wood or Popsicle stick.  
• A 9 volt battery clip and a 9 Volt battery.
• A little transparent sticky tape.
• One glass full of water.
• Volt meter device.

Hydrogen-oxygen Fuel cell

One of the most successful fuel cells uses the reaction of hydrogen as fuel and oxygen as oxidant to form water (Hydrogen oxygen fuel cell).  The cell was used for providing electrical power in the Apollo space programme.  The water vapors produced during the reaction were condensed and added to the drinking water supply for the astronauts.  In the cell, hydrogen and oxygen are bubbled through porous carbon electrodes into concentrated aqueous solutions of sodium hydroxide.  Catalysts like finely divided platinum or palladium metal are incorporated into the electrodes for increasing the rate of electrode reactions.

Catalysis plays a very important role in Hydrogen oxygen fuel cells, separating the electrons and protons of the reactant fuel (at the anode), and forcing the electrons to travel though a circuit, generating electrical power.  At the cathode, another catalytic process takes the electrons back, combining them with the protons, which have traveled across the electrolyte and the oxidant to form waste products like carbon dioxide and water.
The electrode reactions of Hydrogen oxygen fuel cell are given below:
Cathode reaction:     O₂ (g) + 2H₂O (l) + 4e⁺    4OH^–(aq)
Anode reactions:       2H₂ (g) + 4OH^–(aq)  4H₂O (l) + 4e^–
Overall reactions:     2H₂ (g) + O₂ (g)   2 H₂O (l)
The cell can run continuously as long as the reactants are supplied.