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Tuesday, March 28, 2017

Brain Teaser 9 [With Answers]

Brain Teasers Logo
Puzzle 1: Fill in the blanks with a word which makes two different words if you read from both sides:

Tim's father has kept his ______ in the ______ for scoring good marks in math.

Answer: Reward and Drawer.

Puzzle 2: There are 17 pigeons sitting in a row on a wall. A boy shoots the fifth pigeon. How many pigeons remain?

Answer: None.

Explanation: They all fly away at the sound of the gunshot.

Puzzle 3: What is the largest number you can get using only 2 digits?

Answer: 387420489 (i.e. 99)

Puzzle 4: Complete the sequence:

Z   O   T   T   F   F   S   S   __

Answer: E

Explanation: The sequence is Zero, One, Two, Three, Four, Five, Six, Seven, Eight.

Puzzle 5: Using four 7s and one 1 make 100. You can use (), /, X, +, -

Answer: 177 - 77 = 100.

Equation of Normal

Normal: The normal at any point P of the centre is the line which passes through P and is perpendicular to the tangent at P.
Equation of Normal
Equation of Normal:

i. The equation of the normal at p(x₁, y₁) of the circle


S = 0


x² + y² + 2gx + 2fy + c = 0 is (x - x₁) (y₁ + f) - (y - y₁) (x₁ + g) = 0

ii. The equation of the normal to the circle x² + y² = r² at p(x₁, y₁) is xy₁ - yx₁ = 0.

iii. The line lx + my + n = 0 is a normal to the circle x² + y² + 2gx + 2fy + c = 0 if and only if gl + mf = n.

Monday, March 27, 2017

Brain Teaser 9

Puzzle 1: Fill in the blanks with a word which makes two different words if you read from both sides:

Tim's father has kept his ______ in the ______ for scoring good marks in math.

Puzzle 2: There are 17 pigeons sitting in a row on a wall. A boy shoots the fifth pigeon. How many pigeons remain?

Puzzle 3: What is the largest number you can get using only 2 digits?

Puzzle 4: Complete the sequence:

Z   O   T   T   F   F   S   S   __

Puzzle 5: Using four 7s and one 1 make 100. You can use (), /, X, +, -.

Adsorption II

Adsorption isotherms: Plot of the amount of gas adsorbed on the surface of the adsorbent and pressure at constant temperature.
a. Freundlich adsorption isotherm:
 (n > 1) … 1
x = mass of the gas adsorbed
m = mass of the adsorbent
P = pressure
k, n = constants. Depend on the nature of the adsorbent and the gas.
Taking logarithm of eqn 1 → 
The plot of log(x/m) and log P is a straight line. If the plot is not a straight line then Freundlich isotherm is not valid. The slope of the straight line is 1/n and the y intercept is equal to k.
If 1/n = 0, then x/m is constant and doesn’t depend on pressure
If 1/n = 1, then x/m α P

b. Langmuir adsorption isotherm:

Postulates:
Gases undergoing adsorption behave ideally
The surface containing the adsorbing sites is perfectly flat plane with no corrugations (assume the surface is homogeneous).
The adsorbing gas adsorbs into an immobile state.
All sites are equivalent.
Each site can hold at most one molecule of gas (mono-layer coverage only).
There are no interactions between adsorbate molecules on adjacent sites.
Dynamic equilibrium exists between adsorbed gaseous molecules and the free gaseous molecules.
Where A(g) is unabsorbed gaseous molecule, B(s) is unoccupied metal surface and AB is Adsorbed gaseous molecule. He gave the following relation:

, kad = adsorption rate constant, kd = description rate constant
K’ = kK
The plot of  and P is a straight line, whose slope is (K/K’) and y intercept is (1/K’)
 
When pressure is very high, then 1 + KP = KP
Thus, at high pressures the degree of adsorption approaches a limiting value.

When pressure is low then 1 + KP = 1
The degree of adsorption is directly proportional to pressure.

Adsorption from solution phase:
The adsorption decreases with increase in temperature.
Adsorption increases with increase in surface area.
The extent of adsorption depends on concentration of solute in solution.
The extent of adsorption depends on nature of adsorbent and adsorb ate.
The Freundlich equation is modified as 
C - Equilibrium concentration

Positive adsorption: The concentration of adsorbate is more on the surface of adsorbent than in the bulk of the solution

Negative adsorption: The concentration of adsorbate is more in the bulk of the solution than in the surface of the adsorbent.

Sunday, March 26, 2017

Riddles 8 [With Answers]

Puzzle 1: Imagine you’re in a room that is filling with water. There are no windows or doors. How do you get out?
Answer: Stop imagining.
Puzzle 2: The more you take, the more you leave behind. What are they?
Answer: Footprints.
Puzzle 3: What two keys can’t open any door?
Answer: A monkey and a donkey.
Puzzle 4: What invention lets you look right through a wall?
Answer: A window.

Growth of Current in an L - R Circuit

Consider a circuit containing an inductance L and a resistance R connected in series with a d.c. source of steady e.nif E volt and a tap key K (Morse key). When the key is pressed K is in contact with the point A, the current grows from zero to a maximum value of I₀. Let the current in the circuit at any instant t be I.


As the current increases, the magnetic field surrounding the inductor increases. The flux passing through the inductor changes and an e.m.f. is induced in it. This e.m.f. opposes the growth of current through the inductor. So the e.m.f. is called back e.m.f. and is equal to - L dl/dt. The negative sign shows that induced e.m.f. opposes the applied e.m.f. E.

The net e.m.f. in the circuit is E - L. dl/dt.

The resultant potential difference across the resistance R, according to Kirchhoff's voltage law is,

IR = E - L dl/dt

L dl/dt + IR = E

This is the equation for the e.m.f. of a circuit containing an inductance L and a resistance R in series. When the current grows to a steady maximum value I₀, dI = 0, hence E = I₀R.



Integrating, log (I₀ - I) = (- R/L) t + K

Where K is the constant of integration. When t = 0, I = 0. Substituting in equation (ii)

log I₀ = K



I = I – 0 [1 - e-Rt/L]

I is the current flowing through the L - R series circuit at any instant t and I₀ is the maximum value of the current. Equation (iii) represents the equation for the growth of current through an L - R circuit, when it is connected to d.c source of e.m.f. This equation is called Helmholtz, equation.

Equation (iii) shows that the growth of current is exponential. With increase in time t, e-Rt/L approaches zero and the current approaches the final steady value I₀ i.e. when t = ∞ (infinity), I = I₀ (1 - e- ∞) = I₀. This means the current reaches its maximum  value after infinite time. But in practice the current reaches its maximum value (approximately) after a short time depending on the value of L and R.

Time Constant:

From equation (v) it can be seen that LIR has the dimensions of time. L/R is called the inductive time constant or the time constant of an L - R circuit.

Time constant = L/R = λ

When = L/R,



Hence time constant of an L - R circuit may be defined as the time taken by the current to grow from zero to 0.63214 I₀ or 63.2% of its final steady value.