MaxWell's Speed Distribution Law
It is an equation giving the distribution of molecules in different speeds in a gas at a temperature.
If dN represents the number of molecules with speeds between v and v+dv then
dN = 4πN[m/2πkT]3/2v²e-mv²/2kTdv
where
dN represents the number of molecules with speeds between v and v+dv
N = total number of molecules in the gas
m = mass of a molecule
T = absolute temperature of the gas
v = velocity of the molecules
The speed vp at which dN/dv is maximum is called the most probable speed.
Its value is given by
vp = √(2kT/m)
(Topic: Kinetic theory of gases)
Avogadro's Law
Under similar conditions of temperature and pressure, equal volumes of all gases contain equal number of molecules.
(Basic concepts of chemistry)
Vectors
The sum of the resolved parts of two forces acting through a point along any direction is equal to the resolved part of the resultant of the two forces along that direction.
Explanation
If you find the resolved parts in some direction initially and then find their resultant, it will be equal to the resolved part of the resultant of the two forces in the same direction.
(Topic: Vectors)
Tuesday, July 8, 2008
Revision - Laws and Theories - 1
Laws of Thermodynamics
Zeroth law: If two bodies A and B are in thermal equilibrium and A and C are also in thermal equilibrium then B and C are also in thermal equilibrium.
First law of thermodynamics
∆U = ∆Q -∆W
Where
∆U = change in internal energy of a thermodynamic system
∆Q = Heat given to the system
∆W = work done by the system
Change in internal energy of a thermodynamic system is equal to the heat given to the system minus the work done by the system on surroundings or environment.
Second law of thermodynamics
Kelvin-Planck statement
It is not possible to design a heat engine which works in cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work.
(Topic: Laws of Thermodynamics)
1. Law of conservation of mass
During any physical or chemical change, the total mass of the products is equal to the total mass of reactants.
2. Law of constant proportions
A pure chemical compound always contains same elements combined together in the same definite proportion by weight.
3. Law of multiple proportions
When two elements combine to form two or more than two compounds, the weights of one of the elements which combine with a fixed weight of the other, bear a simple whole number ratio.
4. Law of reciprocal proportions
When two different elements combine separately with the same weight of a third element, the ratio in which they do so will be the same or some simple multiple of the ratio in which they combine with each other.
5. Gay Lussac’s law of combing volumes
Under similar conditions of temperature and pressure, whenever gases react together, the volumes of the reacting gases as well as products (if gases) bear a simple whole number ratio.
Lagrange's Identity
(a × b)² = a²b² - (a.b)
a and b are vectors
a and b are magnitudes of a and b respectively.
Lagrange's identity is a relation between the cross product and the dot product.
(Topic: Vectors)
Zeroth law: If two bodies A and B are in thermal equilibrium and A and C are also in thermal equilibrium then B and C are also in thermal equilibrium.
First law of thermodynamics
∆U = ∆Q -∆W
Where
∆U = change in internal energy of a thermodynamic system
∆Q = Heat given to the system
∆W = work done by the system
Change in internal energy of a thermodynamic system is equal to the heat given to the system minus the work done by the system on surroundings or environment.
Second law of thermodynamics
Kelvin-Planck statement
It is not possible to design a heat engine which works in cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work.
(Topic: Laws of Thermodynamics)
1. Law of conservation of mass
During any physical or chemical change, the total mass of the products is equal to the total mass of reactants.
2. Law of constant proportions
A pure chemical compound always contains same elements combined together in the same definite proportion by weight.
3. Law of multiple proportions
When two elements combine to form two or more than two compounds, the weights of one of the elements which combine with a fixed weight of the other, bear a simple whole number ratio.
4. Law of reciprocal proportions
When two different elements combine separately with the same weight of a third element, the ratio in which they do so will be the same or some simple multiple of the ratio in which they combine with each other.
5. Gay Lussac’s law of combing volumes
Under similar conditions of temperature and pressure, whenever gases react together, the volumes of the reacting gases as well as products (if gases) bear a simple whole number ratio.
Lagrange's Identity
(a × b)² = a²b² - (a.b)
a and b are vectors
a and b are magnitudes of a and b respectively.
Lagrange's identity is a relation between the cross product and the dot product.
(Topic: Vectors)
Thursday, June 19, 2008
Electricty questions from physicsgoeasy blog
Electricty questions from physicsgoeasy blog
http://physicsgoeasy.blogspot.com/2008/05/iit-jee-test-serieselectricity-1.html
dated 18 May 2008
IIT JEE Test series (ELectricty) solutions
dated 18 June 2008
http://physicsgoeasy.blogspot.com/2008/06/iit-jee-test-series-electricty.html
http://physicsgoeasy.blogspot.com/2008/05/iit-jee-test-serieselectricity-1.html
dated 18 May 2008
IIT JEE Test series (ELectricty) solutions
dated 18 June 2008
http://physicsgoeasy.blogspot.com/2008/06/iit-jee-test-series-electricty.html
Wednesday, June 18, 2008
Answers to 8 June 2008 Practice paper
Mathematics
1. if |z-1| /|z-4| = 2, the locus of z = x+iy in Argand plane will be:
a. parabola
b. hyperbola
c. circle
d. straight line
Answer (c)
z = x+iy
| (x+iy-1) |/| (x+iy-4) | = 2
(x-1) ² + y² = 4[(x-4) ² +y²]
x² -2x+1+y² = 4{x-8x+16+y²]
x² -2x+1+y² = 4x²-32x+64+4y²
3x²-30x+63+3y² = 0
x²+y²-10x+21 = 0
Equation represents a circle with (5,0) as centre and SQRT(5²-21) as radius.
Radius is 2
2.
Let a,b,c be three non-coplanar vectors and p,q,r are vectors defined by the relations
p = b×c/[abc]
q = c×a/[abc]
r = a×b/[abc]
then the value of the expression
[(a+b).p +(b+c).q +(c+a).r] is equal to
a. 3
b. 1
c. 0
d. 2
Answer: (a)
[(a+b).p +(b+c).q +(c+a).r] = [1/(abc)][( [(a+b).(b×c)+ (b+c) .( c×a ) +(c+a).( a×b)]
The property of distributivity of scalar product over vector addition is
a.(b+c) = a.b+a.c and
(b+c).a = b.a+b.c
[1/(abc)][( [(a+b).(b×c)+ (b+c) .( c×a ) +(c+a).( a×b)] =
[1/(abc)][( a .(b×c)+ b.(b×c)+b .( c×a ) + b.(b×c)+c.( a×b) +a.( a×b)]
If a┴b, a.b = 0
Hence b.(b×c), and b.(b×c) and a.( a×b) are zero as .(b×c) will be a vector perpendicular to vectors b and c.
Hence the expression reduces to [1/(abc)][( a .(b×c)+ b .( c×a ) + c.( a×b)]
As a .(b×c) = b .( c×a ) = c.( a×b) = (abc)
The expression reduces to 3/(abc) /(abc) = 3
Physics
3. A particle moving eastwards with a velocity 5 m/s. In 10 seconds the velocity changes to 5 m/s northwards. The average acceleration in this time is
a. zero
b. 1/SQRT(2) m/s² towards north-west
c. ½ m/s² toward north-west
d. ½ m/s² towards north
Answer (b)
The average acceleration must have an westward component and northward component to make the velocity towards east as zero and to make velocity towards north as 5 m/s
Hence it will act in the direction of northwest.
Westward component of average acceleration = (0-5)/10 = -½ m/s²
Northward component of average of acceleration = (5-0)/10 = ½ m/s²
Hence magnitude of average acceleration = Sqrt( (-1/2) ²+(1/2) ²) = 1/SQRT(2)
Chemistry
4. Which of the following has maximum number of atoms?
a. 24 g of C (M=12 g mol^-1
b. 23 g of Na (M = 23 g mol^-1)
c. 48 g of S (M = 32 b mol^-1)
d. 108 g of Ag (M = 108 mol ^-1)
Answer: (a)
Number of moles (N) = m/M
Where m = given mass
M = molar mass
Hence Number of moles of carbon = 2
Number of moles of Na = 1
Number of moles of S = 1.5
Number of moles of Ag = 1
So 24 g of C having 2 moles has the maximum number of atoms.
5. The wave number of the shortest wavelength transition in the Balmer series of atomic hydrogen is.
a. 27420 cm^-1
b. 28420 cm^-1
c. 29420 cm^-1
d. 12186 cm^-1
Answer: (a)
Wave number = 1/ λ = R(1/nf² - 1/ni²)
Wave length is shortest, when energy emitted is highest. This will happen when the electron jumps from n = ∞ to n=2.
So wave number = 109677 cm^-1(1/2² - 1/∞²)
= 109677(1/4)
= 27419.25 cm^-1
1. if |z-1| /|z-4| = 2, the locus of z = x+iy in Argand plane will be:
a. parabola
b. hyperbola
c. circle
d. straight line
Answer (c)
z = x+iy
| (x+iy-1) |/| (x+iy-4) | = 2
(x-1) ² + y² = 4[(x-4) ² +y²]
x² -2x+1+y² = 4{x-8x+16+y²]
x² -2x+1+y² = 4x²-32x+64+4y²
3x²-30x+63+3y² = 0
x²+y²-10x+21 = 0
Equation represents a circle with (5,0) as centre and SQRT(5²-21) as radius.
Radius is 2
2.
Let a,b,c be three non-coplanar vectors and p,q,r are vectors defined by the relations
p = b×c/[abc]
q = c×a/[abc]
r = a×b/[abc]
then the value of the expression
[(a+b).p +(b+c).q +(c+a).r] is equal to
a. 3
b. 1
c. 0
d. 2
Answer: (a)
[(a+b).p +(b+c).q +(c+a).r] = [1/(abc)][( [(a+b).(b×c)+ (b+c) .( c×a ) +(c+a).( a×b)]
The property of distributivity of scalar product over vector addition is
a.(b+c) = a.b+a.c and
(b+c).a = b.a+b.c
[1/(abc)][( [(a+b).(b×c)+ (b+c) .( c×a ) +(c+a).( a×b)] =
[1/(abc)][( a .(b×c)+ b.(b×c)+b .( c×a ) + b.(b×c)+c.( a×b) +a.( a×b)]
If a┴b, a.b = 0
Hence b.(b×c), and b.(b×c) and a.( a×b) are zero as .(b×c) will be a vector perpendicular to vectors b and c.
Hence the expression reduces to [1/(abc)][( a .(b×c)+ b .( c×a ) + c.( a×b)]
As a .(b×c) = b .( c×a ) = c.( a×b) = (abc)
The expression reduces to 3/(abc) /(abc) = 3
Physics
3. A particle moving eastwards with a velocity 5 m/s. In 10 seconds the velocity changes to 5 m/s northwards. The average acceleration in this time is
a. zero
b. 1/SQRT(2) m/s² towards north-west
c. ½ m/s² toward north-west
d. ½ m/s² towards north
Answer (b)
The average acceleration must have an westward component and northward component to make the velocity towards east as zero and to make velocity towards north as 5 m/s
Hence it will act in the direction of northwest.
Westward component of average acceleration = (0-5)/10 = -½ m/s²
Northward component of average of acceleration = (5-0)/10 = ½ m/s²
Hence magnitude of average acceleration = Sqrt( (-1/2) ²+(1/2) ²) = 1/SQRT(2)
Chemistry
4. Which of the following has maximum number of atoms?
a. 24 g of C (M=12 g mol^-1
b. 23 g of Na (M = 23 g mol^-1)
c. 48 g of S (M = 32 b mol^-1)
d. 108 g of Ag (M = 108 mol ^-1)
Answer: (a)
Number of moles (N) = m/M
Where m = given mass
M = molar mass
Hence Number of moles of carbon = 2
Number of moles of Na = 1
Number of moles of S = 1.5
Number of moles of Ag = 1
So 24 g of C having 2 moles has the maximum number of atoms.
5. The wave number of the shortest wavelength transition in the Balmer series of atomic hydrogen is.
a. 27420 cm^-1
b. 28420 cm^-1
c. 29420 cm^-1
d. 12186 cm^-1
Answer: (a)
Wave number = 1/ λ = R(1/nf² - 1/ni²)
Wave length is shortest, when energy emitted is highest. This will happen when the electron jumps from n = ∞ to n=2.
So wave number = 109677 cm^-1(1/2² - 1/∞²)
= 109677(1/4)
= 27419.25 cm^-1
Saturday, June 7, 2008
IIT JEE June 8 2008 Practice paper
Mathematics
1. if |z-1| /|z-4| = 2, the locus of z = x+iy in Argand plane will be:
a. parabola
b. hyperbola
c. circle
d. straight line
2.
Let a,b,c be three non-coplanar vectors and p,q,r are vectors defined by the relations
p = b×c/[abc]
q = c×a/[abc]
r = a×b/[abc]
then the value of the expression
[(a+b).p +(b+c).q +(c+a).r] is equal to
a. 3
b. 1
c. 0
d. 2
Physics
3. A particle moving eastwards with a velocity 5 m/s. In 10 seconds the velocity changes to 5 m/s northwards. The average acceleration in this time is
a. zero
b. 1/SQRT(2) m/s² towards north-west
c. ½ m/s² toward north-west
d. ½ m/s² towards north
Chemistry
4. Which of the following has maximum number of atoms?
a. 24 g of C (M=12 g mol^-1
b. 23 g of Na (M = 23 g mol^-1)
c. 48 g of S (M = 32 b mol^-1)
d. 108 g of Ag (M = 108 mol ^-1)
5. The wave number of the shortest wavelength transition in the Balmer series of atomic hydrogen is.
a. 27420 cm^-1
b. 28420 cm^-1
c. 29420 cm^-1
d. 12186 cm^-1
1. if |z-1| /|z-4| = 2, the locus of z = x+iy in Argand plane will be:
a. parabola
b. hyperbola
c. circle
d. straight line
2.
Let a,b,c be three non-coplanar vectors and p,q,r are vectors defined by the relations
p = b×c/[abc]
q = c×a/[abc]
r = a×b/[abc]
then the value of the expression
[(a+b).p +(b+c).q +(c+a).r] is equal to
a. 3
b. 1
c. 0
d. 2
Physics
3. A particle moving eastwards with a velocity 5 m/s. In 10 seconds the velocity changes to 5 m/s northwards. The average acceleration in this time is
a. zero
b. 1/SQRT(2) m/s² towards north-west
c. ½ m/s² toward north-west
d. ½ m/s² towards north
Chemistry
4. Which of the following has maximum number of atoms?
a. 24 g of C (M=12 g mol^-1
b. 23 g of Na (M = 23 g mol^-1)
c. 48 g of S (M = 32 b mol^-1)
d. 108 g of Ag (M = 108 mol ^-1)
5. The wave number of the shortest wavelength transition in the Balmer series of atomic hydrogen is.
a. 27420 cm^-1
b. 28420 cm^-1
c. 29420 cm^-1
d. 12186 cm^-1
Solutions to June 1 2008 practice set
1. State whether the statement is true or false
A projectile fired form the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.
Answer: true
At the top of the path, the vertical component of velocity is zero and the particle has only horizontal component. Hence the speed (the magnitude of the velocity) is minimum at the top of the path.
2. The number of vectors of unit length perpendicular to vectors a = (1,1,0) and b = (0,1,1) is
a. three
b. two
c. one
d. infinite
e. none of these
Answer (b)
If vector (x,y,z) is a unit vector perpendicular to (1,1,0) their scalar product is zero
=> x +y = 0
=> y = -x
If vector (x,y,z) is a unit vector perpendicular to (0,1,1) their scalar product is zero
=> y+z = 0
=> y = -z
As it is a unit vector x²+y²+z² = 1
=> 3y² = 1
=> y = ±1/SQRT(3)
This gives two value of y and hence there are two vectors
3. Let z1 and z2 be complex numbers such that z1≠z2 and |z1| = |z2|. If z1 has positive real part and z2 has negative imaginary part, then (z1+z2)/(z1-z2) may be
a. real and positive
b. zero
c. real and negative
d. purely imaginary
e. none of these
Answer: (b) and (d)
Assume z1 = a+ib, a is positive means a>0,
z2 = c+id, d<0
|z1| = |z2|
=> a²+b² = c²+d²
(z1+z2)/(z1-z2) = [(a+c) + i(b+d)]/ [(a-c) + i(b-d)]
Taking the multiplicative inverse of denominator and multiplying the numerator
= [(a+c) + i(b+d)]* [(a-c) - i(b-d)]/[(a-c) ² + (b-d) ²]
= [(a²-c²)+(b²-d²)]+i[(b+d)(a-c)-(a+c)(b-d)]/ [(a-c) ² + (b-d) ²]
=[{(a²+b²)-(c²+d²)}+i{(b+d)(a-c)-(a+c)(b-d)}]/ [(a-c) ² + (b-d) ²]
As a²+b² = c²+d² the first term of the numerator is zero.
Hence
(z1+z2)/(z1-z2)
= i[(b+d)(a-c)-(a+c)(b-d)]/ [(a-c) ² + (b-d) ²]
So this is a pure imaginary number.
If b+d as well as a+c are equal zero, the expression can be zero also.
4. If two compounds have the same empirical formula but different molecular formulae they must have ------
a. different percentage composition
b. different molecular weights
c. same viscosity
d. same vapour density
answer (b)
Different molecular formulae but same empirical formula imply same empirical mass but different molecular masses
A projectile fired form the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.
Answer: true
At the top of the path, the vertical component of velocity is zero and the particle has only horizontal component. Hence the speed (the magnitude of the velocity) is minimum at the top of the path.
2. The number of vectors of unit length perpendicular to vectors a = (1,1,0) and b = (0,1,1) is
a. three
b. two
c. one
d. infinite
e. none of these
Answer (b)
If vector (x,y,z) is a unit vector perpendicular to (1,1,0) their scalar product is zero
=> x +y = 0
=> y = -x
If vector (x,y,z) is a unit vector perpendicular to (0,1,1) their scalar product is zero
=> y+z = 0
=> y = -z
As it is a unit vector x²+y²+z² = 1
=> 3y² = 1
=> y = ±1/SQRT(3)
This gives two value of y and hence there are two vectors
3. Let z1 and z2 be complex numbers such that z1≠z2 and |z1| = |z2|. If z1 has positive real part and z2 has negative imaginary part, then (z1+z2)/(z1-z2) may be
a. real and positive
b. zero
c. real and negative
d. purely imaginary
e. none of these
Answer: (b) and (d)
Assume z1 = a+ib, a is positive means a>0,
z2 = c+id, d<0
|z1| = |z2|
=> a²+b² = c²+d²
(z1+z2)/(z1-z2) = [(a+c) + i(b+d)]/ [(a-c) + i(b-d)]
Taking the multiplicative inverse of denominator and multiplying the numerator
= [(a+c) + i(b+d)]* [(a-c) - i(b-d)]/[(a-c) ² + (b-d) ²]
= [(a²-c²)+(b²-d²)]+i[(b+d)(a-c)-(a+c)(b-d)]/ [(a-c) ² + (b-d) ²]
=[{(a²+b²)-(c²+d²)}+i{(b+d)(a-c)-(a+c)(b-d)}]/ [(a-c) ² + (b-d) ²]
As a²+b² = c²+d² the first term of the numerator is zero.
Hence
(z1+z2)/(z1-z2)
= i[(b+d)(a-c)-(a+c)(b-d)]/ [(a-c) ² + (b-d) ²]
So this is a pure imaginary number.
If b+d as well as a+c are equal zero, the expression can be zero also.
4. If two compounds have the same empirical formula but different molecular formulae they must have ------
a. different percentage composition
b. different molecular weights
c. same viscosity
d. same vapour density
answer (b)
Different molecular formulae but same empirical formula imply same empirical mass but different molecular masses
Saturday, May 31, 2008
IIT JEE June 1 st Weekly Cumulative Practice Test
1. State whether the statement is true or false
A projectile fired form the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.
2. The number of vectors of unit length perpendicular to vectors a = (1,1,0) and b = (0,1,1) is
a. three
b. two
c. one
d. infinite
e. none of these
3. Let z1 and z2 be complex numbers such that z1≠z2 and |z1| = |z2|. If z1 has positive real part and z2 has negative imaginary part, then (z1+z2)/(z1-z2) may be
a. real and positive
b. zero
c. real and negative
d. purely imaginary
e. none of these
4. If two compounds have the same empirical formula but different molecular formulae they must have ------
a. different percentage composition
b. different molecular weight
c. same viscosity
d. same vapour density
A projectile fired form the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.
2. The number of vectors of unit length perpendicular to vectors a = (1,1,0) and b = (0,1,1) is
a. three
b. two
c. one
d. infinite
e. none of these
3. Let z1 and z2 be complex numbers such that z1≠z2 and |z1| = |z2|. If z1 has positive real part and z2 has negative imaginary part, then (z1+z2)/(z1-z2) may be
a. real and positive
b. zero
c. real and negative
d. purely imaginary
e. none of these
4. If two compounds have the same empirical formula but different molecular formulae they must have ------
a. different percentage composition
b. different molecular weight
c. same viscosity
d. same vapour density
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