Q5. The relative motion between the adjacent layers of a fluid in motion is called viscous force. Which of the flowing suits the best in the sentence.
Anonymous Quiz
36%
Effects at some conditions
21%
Never affects
29%
Opposes
14%
Facilitates
Q6. Which of the following is the correct definition of solubility?
Anonymous Quiz
29%
The rate at which a solute dissolve into a solvent.
7%
The ability of a solid particle to disperse throughout a liquid continuous phase
14%
The rate at which the immiscible liquid droplets disperse within a second in the liquid phase
50%
The ability of a solute to dissolve into a solvent
Q7. Which among the following statement is the definition of a pseudoplasticity liquid?
Anonymous Quiz
14%
It is a liquid that becomes less viscous over time when constant shear stress is applied.
29%
As the rate of shear increases, it becomes less viscous
57%
When constant shear stress is applied theliquid becomes more viscous over time
0%
A liquid which becomes more viscous as the rate of shear increases
Q8.Which of the following represents kinematic viscosity?
Anonymous Quiz
20%
Viscosity/temperature
30%
Viscosity/area
40%
Viscosity/density
10%
Viscosity/mass
Q9.What happens to the viscosity of liquid with the increase in temperature ?
Anonymous Quiz
22%
It increases
56%
It decreases
0%
It may increases or decreases
22%
No change
Q10.Which of these ratio gives the value of viscosity of fluid?
Anonymous Quiz
0%
Ratio of the shearing stress to the weight of the fluid
33%
Ratio of the shearing stress to the density of the fluid
33%
Ratio of the velocity gradient and shearing stress
33%
The Ratio of the shearing stress to the velocity gradient
🔰 Notes on Kinetic Theory of Gases 🔰
Kinetic Theory of Matter:-
(a) Solids:- It is the type of matter which has got fixed shape and volume. The force of attraction between any two molecules of a solid is very large.
(b) Liquids:- It is the type of matter which has got fixed volume but no fixed shape. Force of attraction between any two molecules is not that large as in case od solids.
(c) Gases:- It is the type of matter does not have any fixed shape or any fixed volume.
Ideal Gas:- A ideal gas is one which has a zero size of molecule and zero force of interaction between its molecules.
Ideal Gas Equation:- A relation between the pressure, volume and temperature of an ideal gas is called ideal gas equation.
PV/T = Constant or PV = nRT
Here, n is the number of moles and R is the universal gas constant.
Gas Constant:-
(a) Universal gas constant (R):-
R= P0 V0/T0
=8.311 J mol-1K-1
(b) Specific gas constant (r):-
PV= (R/M) T = rT,
Here, r = R/M
Real Gas:-The gases which show deviation from the ideal gas behavior are called real gas.
Vander wall’s equation of state for a real gas:-
[P+(na/V)2?][V-nb] = nRT
Here n is the number of moles of gas.
Avogadro’s number (N):- Avogadro’s number (N), is the number of carbon atoms contained in 12 gram of carbon-12.
N = 6.023×10^23
(a) To calculate the mass of an atom/molecule:-
Mass of one atom = atomic weight (in gram)/N
Mass of one molecule = molecular weight (in gram)/N
(b) To calculate the number of atoms/molecules in a certain amount of substance:-
Number of atoms in m gram = (N/atomic weight)×m
Number of molecules in m gram = (N/molecular weight)×m
(c) Size of an atom:-
Volume of the atom, V = (4/3)πr3
Mass of the atom, m = A/N
Here, A is the atomic weight and N is the Avogadro’s number.
Radius, r =[3A/4πNρ]1/3\
Here ρ is the density.
Gas laws:-
Graph Between Pressure and Volume for Boyle's Law(a) Boyle’s law:- It states that the volume of a given amount of gas varies inversely as its pressure, provided its temperature is kept constant.
PV = Constant
(b) Charlers law or Gey Lussac’s law:- It states that volume of a given mass of a gas varies directly as its absolute temperature, provided its pressure is kept constant.
Graph Between Volume and Temperature for Charles LawV/T= Constant
V–V0/V0t = 1/273 = γp
Here γp (=1/273) is called volume coefficient of gas at constant pressure.
Volume coefficient of a gas, at constant pressure, is defined as the change in volume per unit volume per degree centigrade rise of temperature.
(c) Gay Lussac’s law of pressure:- It states that pressure of a given mass of a gas varies directly as its absolute temperature provided the volume of the gas is kept constant.
P/T = P0/T0 or P – P0/P0t = 1/273 = γp
Here γp (=1/273) is called pressure coefficient of the gas at constant volume.
Pressure coefficient of a gas, at constant volume, is defined as the change in pressure per unit pressure per degree centigrade rise of temperature.
(d) Dalton’s law of partial pressures:-
Partial pressure of a gas or of saturated vapors is the pressure which it would exert if contained alone in the entire confined given space.
P= p1+p2+p3+……..
nRT/V = p1+p2+p3+……..
(e) Grahm’s law of diffusion:- Grahm’s law of diffusion states that the rate of diffusion of gases varies inversely as the square root of the density of gases.
R∝1/√ρ or R1/R2 =√ρ2/ ρ1
So, a lighter gas gets diffused quickly.
(f) Avogadro’s law:- It states that under similar conditions of pressure and temperature, equal volume of all gases contain equal number of molecules.
For m gram of gas, PV/T = nR = (m/M) R
Kinetic Theory of Matter:-
(a) Solids:- It is the type of matter which has got fixed shape and volume. The force of attraction between any two molecules of a solid is very large.
(b) Liquids:- It is the type of matter which has got fixed volume but no fixed shape. Force of attraction between any two molecules is not that large as in case od solids.
(c) Gases:- It is the type of matter does not have any fixed shape or any fixed volume.
Ideal Gas:- A ideal gas is one which has a zero size of molecule and zero force of interaction between its molecules.
Ideal Gas Equation:- A relation between the pressure, volume and temperature of an ideal gas is called ideal gas equation.
PV/T = Constant or PV = nRT
Here, n is the number of moles and R is the universal gas constant.
Gas Constant:-
(a) Universal gas constant (R):-
R= P0 V0/T0
=8.311 J mol-1K-1
(b) Specific gas constant (r):-
PV= (R/M) T = rT,
Here, r = R/M
Real Gas:-The gases which show deviation from the ideal gas behavior are called real gas.
Vander wall’s equation of state for a real gas:-
[P+(na/V)2?][V-nb] = nRT
Here n is the number of moles of gas.
Avogadro’s number (N):- Avogadro’s number (N), is the number of carbon atoms contained in 12 gram of carbon-12.
N = 6.023×10^23
(a) To calculate the mass of an atom/molecule:-
Mass of one atom = atomic weight (in gram)/N
Mass of one molecule = molecular weight (in gram)/N
(b) To calculate the number of atoms/molecules in a certain amount of substance:-
Number of atoms in m gram = (N/atomic weight)×m
Number of molecules in m gram = (N/molecular weight)×m
(c) Size of an atom:-
Volume of the atom, V = (4/3)πr3
Mass of the atom, m = A/N
Here, A is the atomic weight and N is the Avogadro’s number.
Radius, r =[3A/4πNρ]1/3\
Here ρ is the density.
Gas laws:-
Graph Between Pressure and Volume for Boyle's Law(a) Boyle’s law:- It states that the volume of a given amount of gas varies inversely as its pressure, provided its temperature is kept constant.
PV = Constant
(b) Charlers law or Gey Lussac’s law:- It states that volume of a given mass of a gas varies directly as its absolute temperature, provided its pressure is kept constant.
Graph Between Volume and Temperature for Charles LawV/T= Constant
V–V0/V0t = 1/273 = γp
Here γp (=1/273) is called volume coefficient of gas at constant pressure.
Volume coefficient of a gas, at constant pressure, is defined as the change in volume per unit volume per degree centigrade rise of temperature.
(c) Gay Lussac’s law of pressure:- It states that pressure of a given mass of a gas varies directly as its absolute temperature provided the volume of the gas is kept constant.
P/T = P0/T0 or P – P0/P0t = 1/273 = γp
Here γp (=1/273) is called pressure coefficient of the gas at constant volume.
Pressure coefficient of a gas, at constant volume, is defined as the change in pressure per unit pressure per degree centigrade rise of temperature.
(d) Dalton’s law of partial pressures:-
Partial pressure of a gas or of saturated vapors is the pressure which it would exert if contained alone in the entire confined given space.
P= p1+p2+p3+……..
nRT/V = p1+p2+p3+……..
(e) Grahm’s law of diffusion:- Grahm’s law of diffusion states that the rate of diffusion of gases varies inversely as the square root of the density of gases.
R∝1/√ρ or R1/R2 =√ρ2/ ρ1
So, a lighter gas gets diffused quickly.
(f) Avogadro’s law:- It states that under similar conditions of pressure and temperature, equal volume of all gases contain equal number of molecules.
For m gram of gas, PV/T = nR = (m/M) R
♦️Revision Notes on Arithmetic Progression♦️
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d
The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d]
If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l)
If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d]
If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.
If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.
If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.
Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P.
If nth term of a series is tn = An + B, then the series is in A.P.
If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.
In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d
Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.
In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.
The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.
The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P.
Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d
Hence, this gives ‘d’ = (b-a)/(n+1)
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d
The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d]
If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l)
If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d]
If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.
If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.
If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.
Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P.
If nth term of a series is tn = An + B, then the series is in A.P.
If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.
In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d
Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.
In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.
The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.
The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P.
Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d
Hence, this gives ‘d’ = (b-a)/(n+1)
💠Properties of Solids and Liquids - Revision Notes on Liquids at Rest:💠
⛔️Force of cohesion:- It is force between two molecules of similar nature.
⛔️Force of adhesion:- It is the force between two molecules of different nature.
⛔️Molecular range:- The maximum distance between two molecules so that the force of attraction between them remains effective is called molecular range.
⛔️Sphere of influence:- Sphere of influence of any molecule is the sphere with molecule as its center and having a radius equal to molecular range (=10-7 cm).
⛔️Surface film:- Surface film of a liquid is defined as the portion of liquid lying on the surface and caught between two parallel planes situated molecular range apart.
⛔️Surface Tension
Surface tension is the property of a liquid by virtue of which its free surface behaves like a stretched membrane and supports, comparatively heavier objects placed over it. It is measured in terms of force of surface tension.
⛔️Force of surface tension:- It is defined as the amount of force acting per unit length on either side of an imaginary line drawn over the liquid surface.
(a) T = Force/length = F/l
(b) T = Surface energy/Surface area = W/A
Units:- S.I – Nm-1
C.G.S- dyn cm-1
⛔️Additional force:-
(a) For a cylindrical rod:- F = T×2πr (Here r is the radius of cylindrical rod)
(b) For a rectangular block:- F = T×2(l+d) (Here l is the length and d is the thickness of the rectangular block)
(c) For a ring:- F = T×2×2πr (Here r is the radius of cylindrical rod)
⛔️Surface energy:-
Potential energy per unit area of the surface is called surface energy.
(a) Expansion under isothermal condition:-
To do work against forces of surface tension:-
W= T×A (Here A is the total increase in surface area)
To supply energy for maintaining the temperature of the film:-
E = T+H
(b) Expansion under adiabatic conditions:-
E = T
Force of surface tension is numerically equal to the surface energy under adiabatic conditions.
⛔️Drops and Bubbles:-
(a) Drop:- Area of surface film of a spherical drop of radius R is given by, A = 4πR2
(b) Bubble:- The surface area of the surface films of a bubble of radius R is, A = 2×4πR2
⛔️Combination of n drops into one big drop:-
(a) R = n1/3r
(b) Ei = n (4πr2T), Ef =4πR2T
(c) Ef/ Ei = n -1/3
(d) ΔE/Ei = [1-(1/n1/3)]
(e) ΔE = 4πR2T (n1/3-1) = 4πR3T (1/r – 1/R)
⛔️Angle of contact:- Angle of contact, for a pair of solid and liquid, is defined as the angle between tangent to the liquid surface drawn at the point of contact and the solid surface inside the liquid.
(a) When θ < 90º (acute):-
Fa >Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) Liquid molecules will stick with the solid, thus making solid wet.
(iii) Such liquid is put in the solid tube; it will have meniscus concave upwards.
(b) When θ > 90º (obtuse):-Fa<Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) In this case, liquids do not wet the solids.
(iii) Such liquids when put in the solid tube will have a meniscus convex upwards.
(c) When θ = 90º:-?
Fa=Fc/√2
The surface of liquid at the point of contact is plane. In this case force of cohesion and adhesion are comparable to each other.
(d) cosθc = Tsa – Tsl/Tla
Here, Tsa,Tsl and Tla represent solid-air, solid-liquid and liquid-air surface tension respectively). Here θc is acute if Tsl < Tsa while θc is obtuse if Tsl >Tsa.
⛔️Capillarity:-
?Rise of Liquid in a Capillary Tube?Capillarity is the phenomenon, by virtue of which the level of liquid in a capillary tube is different from that outside it, is called capillarity.
Weight of liquid, W = Vρg = πr2[h+(r/3)]ρg (Here r is the radius meniscus)
If weight of meniscus is taken into account, the force of surface tension will be,
T = [r(h+(r/3)) ρg]/2 cosθ
For fine capillary, force of surface tension, T = rhρg/2 cosθ
So height, h = 2T cosθ/ rρg
⛔️Force of cohesion:- It is force between two molecules of similar nature.
⛔️Force of adhesion:- It is the force between two molecules of different nature.
⛔️Molecular range:- The maximum distance between two molecules so that the force of attraction between them remains effective is called molecular range.
⛔️Sphere of influence:- Sphere of influence of any molecule is the sphere with molecule as its center and having a radius equal to molecular range (=10-7 cm).
⛔️Surface film:- Surface film of a liquid is defined as the portion of liquid lying on the surface and caught between two parallel planes situated molecular range apart.
⛔️Surface Tension
Surface tension is the property of a liquid by virtue of which its free surface behaves like a stretched membrane and supports, comparatively heavier objects placed over it. It is measured in terms of force of surface tension.
⛔️Force of surface tension:- It is defined as the amount of force acting per unit length on either side of an imaginary line drawn over the liquid surface.
(a) T = Force/length = F/l
(b) T = Surface energy/Surface area = W/A
Units:- S.I – Nm-1
C.G.S- dyn cm-1
⛔️Additional force:-
(a) For a cylindrical rod:- F = T×2πr (Here r is the radius of cylindrical rod)
(b) For a rectangular block:- F = T×2(l+d) (Here l is the length and d is the thickness of the rectangular block)
(c) For a ring:- F = T×2×2πr (Here r is the radius of cylindrical rod)
⛔️Surface energy:-
Potential energy per unit area of the surface is called surface energy.
(a) Expansion under isothermal condition:-
To do work against forces of surface tension:-
W= T×A (Here A is the total increase in surface area)
To supply energy for maintaining the temperature of the film:-
E = T+H
(b) Expansion under adiabatic conditions:-
E = T
Force of surface tension is numerically equal to the surface energy under adiabatic conditions.
⛔️Drops and Bubbles:-
(a) Drop:- Area of surface film of a spherical drop of radius R is given by, A = 4πR2
(b) Bubble:- The surface area of the surface films of a bubble of radius R is, A = 2×4πR2
⛔️Combination of n drops into one big drop:-
(a) R = n1/3r
(b) Ei = n (4πr2T), Ef =4πR2T
(c) Ef/ Ei = n -1/3
(d) ΔE/Ei = [1-(1/n1/3)]
(e) ΔE = 4πR2T (n1/3-1) = 4πR3T (1/r – 1/R)
⛔️Angle of contact:- Angle of contact, for a pair of solid and liquid, is defined as the angle between tangent to the liquid surface drawn at the point of contact and the solid surface inside the liquid.
(a) When θ < 90º (acute):-
Fa >Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) Liquid molecules will stick with the solid, thus making solid wet.
(iii) Such liquid is put in the solid tube; it will have meniscus concave upwards.
(b) When θ > 90º (obtuse):-Fa<Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) In this case, liquids do not wet the solids.
(iii) Such liquids when put in the solid tube will have a meniscus convex upwards.
(c) When θ = 90º:-?
Fa=Fc/√2
The surface of liquid at the point of contact is plane. In this case force of cohesion and adhesion are comparable to each other.
(d) cosθc = Tsa – Tsl/Tla
Here, Tsa,Tsl and Tla represent solid-air, solid-liquid and liquid-air surface tension respectively). Here θc is acute if Tsl < Tsa while θc is obtuse if Tsl >Tsa.
⛔️Capillarity:-
?Rise of Liquid in a Capillary Tube?Capillarity is the phenomenon, by virtue of which the level of liquid in a capillary tube is different from that outside it, is called capillarity.
Weight of liquid, W = Vρg = πr2[h+(r/3)]ρg (Here r is the radius meniscus)
If weight of meniscus is taken into account, the force of surface tension will be,
T = [r(h+(r/3)) ρg]/2 cosθ
For fine capillary, force of surface tension, T = rhρg/2 cosθ
So height, h = 2T cosθ/ rρg
A solid has a definite size and shape which does not change its state without any external condition. In solids, particles are very close to each other due to strong intermolecular force which keeps their particles in a fixed position. Due to this factor solids are hard and rigid and cannot be compressed. When the atoms are forced to come closer, the intermolecular force becomes repulsive. With an increase in thermal energy, the kinetic energy of the molecules increases resulting in the conversion of solid into a liquid.
📚Notes on Chemical Coordination and Regulation📚
Properties of hormones
(a) These are secreted by endocrine gland (biogenic in origin).
(b) Their secretions is released directly into blood (except local hormones e.g. gastrin).
(c) These are carried to distantly locate specific organs, called target organ.
(d) These have specific physiological action (excitatory or inhibatory). These co-ordinate different physical, mental and metabolic activities and maintain homeostasis.
(e) The hormones have low molecular weight e.g. ADH has a molecular weight of 600–2000 daltons.
(f) These act in very low concentration e.g. around10–10 molar.
(g) Hormones are non antigenic.
(h) These are mostly short-lived. So have a no camulative effect.
(i) Some hormones are quick acting e.g. adrenalin, while some acting slowly e.g. ostrogen of ovary.
(j) Some hormones secreted in inactive form called Prohormone e.g. Pro-insulin.
(k) Hormones are specific. They are carriers of specific information to their specific target organ. Only those target cell respond to a particular hormone for which they have receptors.
@arvind_arora_vedantu_chemistry
Properties of hormones
(a) These are secreted by endocrine gland (biogenic in origin).
(b) Their secretions is released directly into blood (except local hormones e.g. gastrin).
(c) These are carried to distantly locate specific organs, called target organ.
(d) These have specific physiological action (excitatory or inhibatory). These co-ordinate different physical, mental and metabolic activities and maintain homeostasis.
(e) The hormones have low molecular weight e.g. ADH has a molecular weight of 600–2000 daltons.
(f) These act in very low concentration e.g. around10–10 molar.
(g) Hormones are non antigenic.
(h) These are mostly short-lived. So have a no camulative effect.
(i) Some hormones are quick acting e.g. adrenalin, while some acting slowly e.g. ostrogen of ovary.
(j) Some hormones secreted in inactive form called Prohormone e.g. Pro-insulin.
(k) Hormones are specific. They are carriers of specific information to their specific target organ. Only those target cell respond to a particular hormone for which they have receptors.
@arvind_arora_vedantu_chemistry
✍️Revision Notes on Vectors
➖➖➖➖➖➖➖➖➖➖➖➖
The length or the magnitude of the vector = (a, b, c) is defined by w = √a2+b2+c2
A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)
If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.
Some key points of vectors:
1) The magnitude of a vector is a scalar quantity
2) Vectors can be multiplied by a scalar. The result is another vector.
3) Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv = c (a, b) = (ca, cb). Hence each component of vector is multiplied by the scalar.
4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.
If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)
2) u + 0 = u
3) u + (-u) = 0 (existence of additive inverses)
4) c (du) = (cd)u
5) (c + d)u = cu + d u
6) c(u + v) = cu + cv
7) 1u = u
8) u + (v + w) = (u + v) + w (the associative law of addition)
Some Basic Rules of Algebra of Vectors:
1) a.a = |a|2 = a2
2) a.b = b.a
3) a.0 = 0
4) a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a
5) a.(b + c) = a.b + a.c (This is also termed as the distributive law)
6) (la).(mb) = lm (a.b)
7) (a ± b)2 = (a ± b) . (a ± b) = a2 + b2 ± 2a.b
8) If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|
9) a x a = 0
10) a x b = - (b x a)
11) a x (b + c) = a x b + a x c
Any vector perpendicular to the plane of a and b is l(a x b) where l is a real number.
Unit vector perpendicular to a and b is ± (a x b)/ |a x b|
The position of dot and cross can be interchanged without altering the product. Hence it is also represented by [a b c]
1) [a b c] = [b c a] = [c a b]
2) [a b c] = - [b a c]
3) [ka b c] = k[a b c]
4) [a+b c d] = [a c d] + [b c d]
5) a x (b x c) = (a x b) x c, if some or all of a, b and c are zero vectors or a and c are collinear.
Methods to prove collinearity of vectors:
1) Two vectors a and b are said to be collinear if there exists k ? R such that a = kb.
2) If p x q = 0, then p and q are collinear.
3) Three points A(a), B(b) and C(c) are collinear if there exists k ? R such that AB = kBC i.e. b-a = k (c-b).
4) If (b-a) x (c-b) = 0, then A, B and C are collinear.
5) A(a), B(b) and C(c) are collinear if there exists scalars l, m and n (not all zero) such that la + mb+ nc = 0, where l + m + n = o
Three vectors p, q and r are coplanar if there exists l, m ? R such that r = lp + mq i.e., one can be expressed as a linear combination of the other two.
If [p q r] = 0, then p, q and r are coplanar.
Four points A(a), B(b), C(c) and D(d) lie in the same plane if there exist l, m ? R such that b-a = l(c-b) + m(d-c).
If [b-a c-b d-c] = 0 then A, B, C, D are coplanar.
Two lines in space can be parallel, intersecting or neither (called skew lines). Let r = a1 + μb1 and r = a2 + μb2 be two lines.
They intersect if (b1 x b2)(a2 - a1) = 0
The two lines are parallel if b1 and b2 are collinear.
The angle between two planes is the angle between their normal unit vectors i.e. cos q = n1 . n2
If a, b and c are three coplanar vectors, then the system of vectors a', b' and c' is said to be the reciprocal system of vectors if aa' = bb' = cc' = 1 where a' = (b xc) /[a b c] , b' = (c xa)/ [a b c] and c' = (a x b)/[a b c] Also, [a' b' c'] = 1/ [a b c]
Dot Product of two vectors a and b defined by a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is given by a1b1 + a2b2 + ..., + anbn .
➖➖➖➖➖➖➖➖➖➖➖➖
The length or the magnitude of the vector = (a, b, c) is defined by w = √a2+b2+c2
A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)
If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.
Some key points of vectors:
1) The magnitude of a vector is a scalar quantity
2) Vectors can be multiplied by a scalar. The result is another vector.
3) Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv = c (a, b) = (ca, cb). Hence each component of vector is multiplied by the scalar.
4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.
If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)
2) u + 0 = u
3) u + (-u) = 0 (existence of additive inverses)
4) c (du) = (cd)u
5) (c + d)u = cu + d u
6) c(u + v) = cu + cv
7) 1u = u
8) u + (v + w) = (u + v) + w (the associative law of addition)
Some Basic Rules of Algebra of Vectors:
1) a.a = |a|2 = a2
2) a.b = b.a
3) a.0 = 0
4) a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a
5) a.(b + c) = a.b + a.c (This is also termed as the distributive law)
6) (la).(mb) = lm (a.b)
7) (a ± b)2 = (a ± b) . (a ± b) = a2 + b2 ± 2a.b
8) If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|
9) a x a = 0
10) a x b = - (b x a)
11) a x (b + c) = a x b + a x c
Any vector perpendicular to the plane of a and b is l(a x b) where l is a real number.
Unit vector perpendicular to a and b is ± (a x b)/ |a x b|
The position of dot and cross can be interchanged without altering the product. Hence it is also represented by [a b c]
1) [a b c] = [b c a] = [c a b]
2) [a b c] = - [b a c]
3) [ka b c] = k[a b c]
4) [a+b c d] = [a c d] + [b c d]
5) a x (b x c) = (a x b) x c, if some or all of a, b and c are zero vectors or a and c are collinear.
Methods to prove collinearity of vectors:
1) Two vectors a and b are said to be collinear if there exists k ? R such that a = kb.
2) If p x q = 0, then p and q are collinear.
3) Three points A(a), B(b) and C(c) are collinear if there exists k ? R such that AB = kBC i.e. b-a = k (c-b).
4) If (b-a) x (c-b) = 0, then A, B and C are collinear.
5) A(a), B(b) and C(c) are collinear if there exists scalars l, m and n (not all zero) such that la + mb+ nc = 0, where l + m + n = o
Three vectors p, q and r are coplanar if there exists l, m ? R such that r = lp + mq i.e., one can be expressed as a linear combination of the other two.
If [p q r] = 0, then p, q and r are coplanar.
Four points A(a), B(b), C(c) and D(d) lie in the same plane if there exist l, m ? R such that b-a = l(c-b) + m(d-c).
If [b-a c-b d-c] = 0 then A, B, C, D are coplanar.
Two lines in space can be parallel, intersecting or neither (called skew lines). Let r = a1 + μb1 and r = a2 + μb2 be two lines.
They intersect if (b1 x b2)(a2 - a1) = 0
The two lines are parallel if b1 and b2 are collinear.
The angle between two planes is the angle between their normal unit vectors i.e. cos q = n1 . n2
If a, b and c are three coplanar vectors, then the system of vectors a', b' and c' is said to be the reciprocal system of vectors if aa' = bb' = cc' = 1 where a' = (b xc) /[a b c] , b' = (c xa)/ [a b c] and c' = (a x b)/[a b c] Also, [a' b' c'] = 1/ [a b c]
Dot Product of two vectors a and b defined by a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is given by a1b1 + a2b2 + ..., + anbn .
Tricks for Conversions in Organic Chemistry
✅Haloalkanes can give you every possible functional group. They can also help you in increasing the chain size. So you might want to convert your starting material to haloalkanes and then go to the desired product
✅Remember the series of oxidation/reduction: Hydrocarbons (with various substituents) can be oxidised to alcohols, then appropriate carbonyl compounds and lastly carboxylic acids (or their derivatives)
✅Strong oxidizing agents are KMnO4 and K2Cr2O7
✅Mild oxidising agents depending on the situation are CrO4, Ammoniacal AgNO3, Benedicts Solution, Fehling’s solution, Cu or CuO at 573K, Bromine water etc.
✅Any carboxylic acid derivative can be converted back to the carboxylic acid by hydrolysis.
✅There are possibilities of hydride and methyl to get the most stable intermediate (carbocation/free radical)
✅A most common reaction, if your starting material is an alkane, is free-radical halogenation.
✅Saytzeff/Markovnikov's rules must be kept in mind while dealing with alkenes (they are based on the electron displacement effects only)
✅Decarboxylation and ozonolysis could go-to methods for decreasing the number of carbon atoms
✅Grignard reagent gives you the much-needed Alkyl nucleophile, which can be used at appropriate places.
✅Haloalkanes can give you every possible functional group. They can also help you in increasing the chain size. So you might want to convert your starting material to haloalkanes and then go to the desired product
✅Remember the series of oxidation/reduction: Hydrocarbons (with various substituents) can be oxidised to alcohols, then appropriate carbonyl compounds and lastly carboxylic acids (or their derivatives)
✅Strong oxidizing agents are KMnO4 and K2Cr2O7
✅Mild oxidising agents depending on the situation are CrO4, Ammoniacal AgNO3, Benedicts Solution, Fehling’s solution, Cu or CuO at 573K, Bromine water etc.
✅Any carboxylic acid derivative can be converted back to the carboxylic acid by hydrolysis.
✅There are possibilities of hydride and methyl to get the most stable intermediate (carbocation/free radical)
✅A most common reaction, if your starting material is an alkane, is free-radical halogenation.
✅Saytzeff/Markovnikov's rules must be kept in mind while dealing with alkenes (they are based on the electron displacement effects only)
✅Decarboxylation and ozonolysis could go-to methods for decreasing the number of carbon atoms
✅Grignard reagent gives you the much-needed Alkyl nucleophile, which can be used at appropriate places.
💠Properties of Solids and Liquids - Revision Notes on Liquids at Rest:💠
⛔️Force of cohesion:- It is force between two molecules of similar nature.
⛔️Force of adhesion:- It is the force between two molecules of different nature.
⛔️Molecular range:- The maximum distance between two molecules so that the force of attraction between them remains effective is called molecular range.
⛔️Sphere of influence:- Sphere of influence of any molecule is the sphere with molecule as its center and having a radius equal to molecular range (=10-7 cm).
⛔️Surface film:- Surface film of a liquid is defined as the portion of liquid lying on the surface and caught between two parallel planes situated molecular range apart.
⛔️Surface Tension
Surface tension is the property of a liquid by virtue of which its free surface behaves like a stretched membrane and supports, comparatively heavier objects placed over it. It is measured in terms of force of surface tension.
⛔️Force of surface tension:- It is defined as the amount of force acting per unit length on either side of an imaginary line drawn over the liquid surface.
(a) T = Force/length = F/l
(b) T = Surface energy/Surface area = W/A
Units:- S.I – Nm-1
C.G.S- dyn cm-1
⛔️Additional force:-
(a) For a cylindrical rod:- F = T×2πr (Here r is the radius of cylindrical rod)
(b) For a rectangular block:- F = T×2(l+d) (Here l is the length and d is the thickness of the rectangular block)
(c) For a ring:- F = T×2×2πr (Here r is the radius of cylindrical rod)
⛔️Surface energy:-
Potential energy per unit area of the surface is called surface energy.
(a) Expansion under isothermal condition:-
To do work against forces of surface tension:-
W= T×A (Here A is the total increase in surface area)
To supply energy for maintaining the temperature of the film:-
E = T+H
(b) Expansion under adiabatic conditions:-
E = T
Force of surface tension is numerically equal to the surface energy under adiabatic conditions.
⛔️Drops and Bubbles:-
(a) Drop:- Area of surface film of a spherical drop of radius R is given by, A = 4πR2
(b) Bubble:- The surface area of the surface films of a bubble of radius R is, A = 2×4πR2
⛔️Combination of n drops into one big drop:-
(a) R = n1/3r
(b) Ei = n (4πr2T), Ef =4πR2T
(c) Ef/ Ei = n -1/3
(d) ΔE/Ei = [1-(1/n1/3)]
(e) ΔE = 4πR2T (n1/3-1) = 4πR3T (1/r – 1/R)
⛔️Angle of contact:- Angle of contact, for a pair of solid and liquid, is defined as the angle between tangent to the liquid surface drawn at the point of contact and the solid surface inside the liquid.
(a) When θ < 90º (acute):-
Fa >Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) Liquid molecules will stick with the solid, thus making solid wet.
(iii) Such liquid is put in the solid tube; it will have meniscus concave upwards.
(b) When θ > 90º (obtuse):-Fa<Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) In this case, liquids do not wet the solids.
(iii) Such liquids when put in the solid tube will have a meniscus convex upwards.
(c) When θ = 90º:-?
Fa=Fc/√2
The surface of liquid at the point of contact is plane. In this case force of cohesion and adhesion are comparable to each other.
(d) cosθc = Tsa – Tsl/Tla
Here, Tsa,Tsl and Tla represent solid-air, solid-liquid and liquid-air surface tension respectively). Here θc is acute if Tsl < Tsa while θc is obtuse if Tsl >Tsa.
⛔️Capillarity:-
?Rise of Liquid in a Capillary Tube?Capillarity is the phenomenon, by virtue of which the level of liquid in a capillary tube is different from that outside it, is called capillarity.
Weight of liquid, W = Vρg = πr2[h+(r/3)]ρg (Here r is the radius meniscus)
If weight of meniscus is taken into account, the force of surface tension will be,
T = [r(h+(r/3)) ρg]/2 cosθ
For fine capillary, force of surface tension, T = rhρg/2 cosθ
So height, h = 2T cosθ/ rρg
⛔️Force of cohesion:- It is force between two molecules of similar nature.
⛔️Force of adhesion:- It is the force between two molecules of different nature.
⛔️Molecular range:- The maximum distance between two molecules so that the force of attraction between them remains effective is called molecular range.
⛔️Sphere of influence:- Sphere of influence of any molecule is the sphere with molecule as its center and having a radius equal to molecular range (=10-7 cm).
⛔️Surface film:- Surface film of a liquid is defined as the portion of liquid lying on the surface and caught between two parallel planes situated molecular range apart.
⛔️Surface Tension
Surface tension is the property of a liquid by virtue of which its free surface behaves like a stretched membrane and supports, comparatively heavier objects placed over it. It is measured in terms of force of surface tension.
⛔️Force of surface tension:- It is defined as the amount of force acting per unit length on either side of an imaginary line drawn over the liquid surface.
(a) T = Force/length = F/l
(b) T = Surface energy/Surface area = W/A
Units:- S.I – Nm-1
C.G.S- dyn cm-1
⛔️Additional force:-
(a) For a cylindrical rod:- F = T×2πr (Here r is the radius of cylindrical rod)
(b) For a rectangular block:- F = T×2(l+d) (Here l is the length and d is the thickness of the rectangular block)
(c) For a ring:- F = T×2×2πr (Here r is the radius of cylindrical rod)
⛔️Surface energy:-
Potential energy per unit area of the surface is called surface energy.
(a) Expansion under isothermal condition:-
To do work against forces of surface tension:-
W= T×A (Here A is the total increase in surface area)
To supply energy for maintaining the temperature of the film:-
E = T+H
(b) Expansion under adiabatic conditions:-
E = T
Force of surface tension is numerically equal to the surface energy under adiabatic conditions.
⛔️Drops and Bubbles:-
(a) Drop:- Area of surface film of a spherical drop of radius R is given by, A = 4πR2
(b) Bubble:- The surface area of the surface films of a bubble of radius R is, A = 2×4πR2
⛔️Combination of n drops into one big drop:-
(a) R = n1/3r
(b) Ei = n (4πr2T), Ef =4πR2T
(c) Ef/ Ei = n -1/3
(d) ΔE/Ei = [1-(1/n1/3)]
(e) ΔE = 4πR2T (n1/3-1) = 4πR3T (1/r – 1/R)
⛔️Angle of contact:- Angle of contact, for a pair of solid and liquid, is defined as the angle between tangent to the liquid surface drawn at the point of contact and the solid surface inside the liquid.
(a) When θ < 90º (acute):-
Fa >Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) Liquid molecules will stick with the solid, thus making solid wet.
(iii) Such liquid is put in the solid tube; it will have meniscus concave upwards.
(b) When θ > 90º (obtuse):-Fa<Fc/√2
(i) Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.
(ii) In this case, liquids do not wet the solids.
(iii) Such liquids when put in the solid tube will have a meniscus convex upwards.
(c) When θ = 90º:-?
Fa=Fc/√2
The surface of liquid at the point of contact is plane. In this case force of cohesion and adhesion are comparable to each other.
(d) cosθc = Tsa – Tsl/Tla
Here, Tsa,Tsl and Tla represent solid-air, solid-liquid and liquid-air surface tension respectively). Here θc is acute if Tsl < Tsa while θc is obtuse if Tsl >Tsa.
⛔️Capillarity:-
?Rise of Liquid in a Capillary Tube?Capillarity is the phenomenon, by virtue of which the level of liquid in a capillary tube is different from that outside it, is called capillarity.
Weight of liquid, W = Vρg = πr2[h+(r/3)]ρg (Here r is the radius meniscus)
If weight of meniscus is taken into account, the force of surface tension will be,
T = [r(h+(r/3)) ρg]/2 cosθ
For fine capillary, force of surface tension, T = rhρg/2 cosθ
So height, h = 2T cosθ/ rρg
♦️Trigonometry- Solution of Triangles♦️
Sine rule: Sides of a triangle are proportional to the sine of the angles opposite to them. So, in ΔABC,
sin A/a = sin B/b = sin C/c = 2Δ/abc.
This may also be written as (a/sin A) = (b/sin B) = (c/sin C)
Cosine rule: In any ΔABC,
cos A = (b2 + c2 – a2) /2bc
cos B = (a2 + c2 – b2)/2ac
cos C = (a2 + b2 - c2)/2ab
Trigonometric ratios of half-angles:
sin A/2 = √[(s-b)(s-c)/bc]
sin B/2 = √[(s-c) (s-a)/ac]
sin C/2 = √[(s-a) (s-b)/ab]
cos A/2 = √s(s - a)/bc
cos B/2 = √s(s - b)/ac
cos C/2 = √s(s - c)/ab
tan A/2 = √[(s - b) (s - c)/s(s - a)]
tan B/2 = √[(s - c) (s - a)/s(s - b)]
tan C/2 = √[(s - a) (s - b)/s(s - c)]
Projection rule: In any ΔABC,
a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A
Area of a triangle
If Δ denotes the area of the triangle ABC, then it can be calculated in any of the following forms:
Δ = 1/2 bc sin A = 1/2 ca sin B = 1/2 ab sin C
Δ = √s(s - a)(s – b)(s - c)
Δ = 1/2. (a2 sin B sin C)/ sin(B + C)
= 1/2. (b2 sin C sin A)/ sin (C + A)
= 1/2. (c2 sin A sin B)/ sin (A + B)
Semi-perimeter of the triangle
If S denotes the perimeter of the triangle ABC, then s = (a + b + c)/2
Napier’s analogy
In any ΔABC,
tan [(B – C)/2] = (b – c)/(b + c) cot A/2
tan [(C – A) /2] = (c – a)/(c + a) cot B/2
tan [(A – B) /2] = (a – b)/(a + b) cot C/2
m-n theorem
Consider a triangle ABC where D is a point on side BC such that it divides the side BC in the ratio m: n, then as shown in the figure, the following results hold good:
Triangle ABC(m + n) cot θ = m cot α – n cot ß.
(m + n) cot θ = n cot B – m cot C.
Apollonius theorem
In a triangle ABC, if AD is the median through A, then
AB2 + AC2 = 2(AD2 + BD2).
If the three sides say a, b and c of a triangle are given, then angle A is obtained with the help of the formula
tan A/2 = √(s - b) (s - c) / s(s - a) or cos A = b2 + c2 - a2 / 2bc.
Angles B and C can also be obtained in the same way.
If two sides b and c and the included angle A are given, then
tan (B – C) /2 = (b – c)/ (b + c) cot A/2
This gives the value of (B- C)/2.
Hence, using (B + C)/2 = 90o - A/2 along with the last equation both B and C can be evaluated. Now, the sides can be evaluated using the formula
a = b sin A/sin B or a2 = b2 + c2 – 2bc cosA.
If two sides b and c and the angle B (opposite to side b) are given, then using the following results, we can easily obtain the remaining elements
sin C = c/b sinB, A = 180o – (B + C) and b = b sin A/sinB
Sine rule: Sides of a triangle are proportional to the sine of the angles opposite to them. So, in ΔABC,
sin A/a = sin B/b = sin C/c = 2Δ/abc.
This may also be written as (a/sin A) = (b/sin B) = (c/sin C)
Cosine rule: In any ΔABC,
cos A = (b2 + c2 – a2) /2bc
cos B = (a2 + c2 – b2)/2ac
cos C = (a2 + b2 - c2)/2ab
Trigonometric ratios of half-angles:
sin A/2 = √[(s-b)(s-c)/bc]
sin B/2 = √[(s-c) (s-a)/ac]
sin C/2 = √[(s-a) (s-b)/ab]
cos A/2 = √s(s - a)/bc
cos B/2 = √s(s - b)/ac
cos C/2 = √s(s - c)/ab
tan A/2 = √[(s - b) (s - c)/s(s - a)]
tan B/2 = √[(s - c) (s - a)/s(s - b)]
tan C/2 = √[(s - a) (s - b)/s(s - c)]
Projection rule: In any ΔABC,
a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A
Area of a triangle
If Δ denotes the area of the triangle ABC, then it can be calculated in any of the following forms:
Δ = 1/2 bc sin A = 1/2 ca sin B = 1/2 ab sin C
Δ = √s(s - a)(s – b)(s - c)
Δ = 1/2. (a2 sin B sin C)/ sin(B + C)
= 1/2. (b2 sin C sin A)/ sin (C + A)
= 1/2. (c2 sin A sin B)/ sin (A + B)
Semi-perimeter of the triangle
If S denotes the perimeter of the triangle ABC, then s = (a + b + c)/2
Napier’s analogy
In any ΔABC,
tan [(B – C)/2] = (b – c)/(b + c) cot A/2
tan [(C – A) /2] = (c – a)/(c + a) cot B/2
tan [(A – B) /2] = (a – b)/(a + b) cot C/2
m-n theorem
Consider a triangle ABC where D is a point on side BC such that it divides the side BC in the ratio m: n, then as shown in the figure, the following results hold good:
Triangle ABC(m + n) cot θ = m cot α – n cot ß.
(m + n) cot θ = n cot B – m cot C.
Apollonius theorem
In a triangle ABC, if AD is the median through A, then
AB2 + AC2 = 2(AD2 + BD2).
If the three sides say a, b and c of a triangle are given, then angle A is obtained with the help of the formula
tan A/2 = √(s - b) (s - c) / s(s - a) or cos A = b2 + c2 - a2 / 2bc.
Angles B and C can also be obtained in the same way.
If two sides b and c and the included angle A are given, then
tan (B – C) /2 = (b – c)/ (b + c) cot A/2
This gives the value of (B- C)/2.
Hence, using (B + C)/2 = 90o - A/2 along with the last equation both B and C can be evaluated. Now, the sides can be evaluated using the formula
a = b sin A/sin B or a2 = b2 + c2 – 2bc cosA.
If two sides b and c and the angle B (opposite to side b) are given, then using the following results, we can easily obtain the remaining elements
sin C = c/b sinB, A = 180o – (B + C) and b = b sin A/sinB
📖Notes on Thermal Properties of Matter
Expansion in two dimensions (Superficial expansion):-
When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.
Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.
β = St-S0/S0t
Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.
Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .
γ = Vt-V0/V0t
Relation between expansion coefficients:-
(a) Relation between α and β: β = 2α
(b) Relation between α and γ: γ= 3α
(c) Relation between β and γ: γ = 3/2 β
(d) α : β : γ = 1:2:3
Thermal expansion of liquids:-
(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.
γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)
(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.
γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)
Work and Heat:-
Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.
W∝H or W = JH
Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.
J = W/H
Value of J:- J = 4.2×107 erg cal-1 = 4.2 J cal-1
Specific heat capacity or specific heat (c):-
Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.
c = Q/mΔT
Unit:- kcal kg-1K-1 or J kg-1K-1
Dimension:- M0L2T-2K-1
Molar specific heat capacity(C):-
Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.
(a) C = Mc (Here M is the molecular weight of the substance)
(b) C = 1/n (dQ/dT)
Heat Capacity or Thermal Capacity:-
It is defined as the amount of heat required to raise the temperature of body through 1ºC.
Q = mcΔT
If ΔT = 1ºC, Q = heat capacity = mc
Unit:- kcal K-1 or JK-1
Water Equivalent:-
Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.
w = mc
Water equivalent of a body is equal to the product of its mass and its specific heat.
Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.
(a)Specific latent heat of fusion (Lf):-
Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.
(b) Specific latent heat of vaporization (Lv):-
Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.
Dimensional formula:- M0L2T-2
Unit:- kg cal kg-1 or J kg-1
Triple point of water = 273.16 K
Absolute zero = 0 K = -273.15ºC
For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)
For a resistance thermometer, Re = R0[1+αθ]
Expansion in two dimensions (Superficial expansion):-
When the thermal expansion of a body is confined to a plane, it is to be two dimensional expansion or superficial expansion.
Coefficient of superficial expansion (β):- It is defined as the change in area of the surface per unit area at 0ºC, per degree centigrade rise of temperature.
β = St-S0/S0t
Expansion in three dimensions (Cubical expansion/volume expansion):- When thermal expansion of the body takes place in space, it is said to be three dimensional expansion or cubical expansion.
Coefficient of cubical expansion (γ):- Coefficient of cubical expansion is defined as the change in volume per unit volume, at 0ºC, per degree celsius rise of temperature .
γ = Vt-V0/V0t
Relation between expansion coefficients:-
(a) Relation between α and β: β = 2α
(b) Relation between α and γ: γ= 3α
(c) Relation between β and γ: γ = 3/2 β
(d) α : β : γ = 1:2:3
Thermal expansion of liquids:-
(a) Co-efficient of apparent expansion (γa):- The coefficient of apparent expansion of a liquid is defined as the apparent (or observed) increase in volume, per unit volume of the liquid at 0ºC per degree celcius rise of temperature.
γa = apparent increase in volume/(original volume at 0ºC) × (rise of temperature)
(b) Co-efficient of real expansion (γr):- The coefficient of real expansion of a liquid is defined as the real increase in volume, per unit volume of the liquid at 0ºC per degree centigrade rise of temperature.
γa= real increase in volume/(original volume at 0ºC) × (rise of temperature)
Work and Heat:-
Whenever heat is conserved into work or work into heat, the quantity of energy disappearing in one form is equivalent to the quantity of energy appearing in the order.
W∝H or W = JH
Joule’s mechanical equivalent of heat is defined as the amount of work required to produce a unit quantity of heat.
J = W/H
Value of J:- J = 4.2×107 erg cal-1 = 4.2 J cal-1
Specific heat capacity or specific heat (c):-
Specific heat capacity of a material is defined as the amount of heat required to raise the temperature of a unit mass of material through 1ºC.
c = Q/mΔT
Unit:- kcal kg-1K-1 or J kg-1K-1
Dimension:- M0L2T-2K-1
Molar specific heat capacity(C):-
Molar specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of one gram molecule of the substance through one degree centigrade.
(a) C = Mc (Here M is the molecular weight of the substance)
(b) C = 1/n (dQ/dT)
Heat Capacity or Thermal Capacity:-
It is defined as the amount of heat required to raise the temperature of body through 1ºC.
Q = mcΔT
If ΔT = 1ºC, Q = heat capacity = mc
Unit:- kcal K-1 or JK-1
Water Equivalent:-
Water equivalent of a body is defined as the mass of water which gets heated through certain range of temperature by the amount of heat required to raise the temperature of body through same range of temperature.
w = mc
Water equivalent of a body is equal to the product of its mass and its specific heat.
Latent Heat:- When the state of matter changes, the heat absorbed or evolved is given by: Q = mL. Here L is called the latent heat.
(a)Specific latent heat of fusion (Lf):-
Specific latent heat of fusion of a substance is defined as the amount of heat required to convert 1 gram of substance from solid to liquid state, at the melting point, without any change of temperature.
(b) Specific latent heat of vaporization (Lv):-
Specific latent heat of vaporization of a substance is defined as the amount of heat required to convert 1 gram of liquid into its vapours at its boiling point without any rise of temperature.
Dimensional formula:- M0L2T-2
Unit:- kg cal kg-1 or J kg-1
Triple point of water = 273.16 K
Absolute zero = 0 K = -273.15ºC
For a gas thermometer, T = (273.15) P/Ptriple (Kelvin)
For a resistance thermometer, Re = R0[1+αθ]