Thermodynamics notes class 11

•  CBSE Class 11 Chemistry Notes : Thermodynamics. The branch of science which deals with the quantitative relationship between heat and other forms of energies is called thermodynamics. (i) System It refers to the part of universe in which observations are carried out.

The branch of physics which deals with the study of transformation of heat energy into other forms of energy and vice-versa.

A thermodynamical system is said to be in thermal equilibrium when macroscopic variables (like pressure, volume, temperature, mass, composition etc) that characterise the system do not change with time.

Thermodynamical System

An assembly of an extremely large number of particles whose state can be expressed in terms of pressure, volume and temperature, is called thermodynamic system.

Thermodynamic system is classified into the following three systems

(i) Open System It exchange both energy and matter with surrounding.
(ii) Closed System It exchanges only energy (not matter) with surroundings.
(iii) Isolated System It exchanges neither energy nor matter with the surrounding.

A thermodynamic system is not always in equilibrium. For example, a gas allowed to expand freely against vacuum. Similary, a mixture of petrol vapour and air, when ignited by a spark is not an equilibrium state. Equilibrium is acquired eventually with time.

Thermodynamic Parameters or Coordinates or Variables

The state of thermodynamic system can be described by specifying pressure, volume, temperature, internal energy and number of moles, etc. These are called thermodynamic parameters or coordinates or variables.

Work done by a thermodynamic system is given by
W = p * ΔV

where p = pressure and ΔV = change in volume.

Work done by a thermodynamic system is equal to the area enclosed between the p-V curve and the volume axis

Work done in process A-B = area ABCDA

Work done by a thermodynamic system depends not only upon the initial and final states of the system but also depend upon the path followed in the process.

Work done by the Thermodynamic System is taken as

Positive → 4 as volume increases.
Negative → 4 as volume decreases.

Internal Energy (U)

The total energy possessed by any system due to molecular motion and molecular configuration, is called its internal energy.

Internal energy of a thermodynamic system depends on temperature. It is the characteristic property of the state of the system.

Zeroth Law of Thermodynamics

According to this law, two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other. Thus, if A and B are separately in equilibrium with C, that is if T_A = T_C and T_B = T_{C, then this implies that TA = TB i.e., the systems A and B are also in thermal equilibrium.}

First Law of Thermodynamics

Heat given to a thermodynamic system (ΔQ) is partially utilized in doing work (ΔW) against the surrounding and the remaining part increases the internal energy (ΔU) of the system.

Therefore, ΔQ = ΔU + ΔW

First law of thermodynamics is a restatement of the principle conservation of energy.

In isothermal process, change in internal energy is zero (ΔU = 0).

Therefore, ΔQ = ΔW

In adiabatic process, no exchange of heat takes place, i.e., Δθ = O.

Therefore, ΔU = – ΔW

In adiabatic process, if gas expands, its internal energy and hence, temperature decreases and vice-versa.

In isochoric process, work done is zero, i.e., ΔW = 0, therefore

ΔQ = ΔU

Thermodynamic Processes

A thermodynamical process is said to take place when some changes’ occur in the state of a thermodynamic system i.e., the therrnodynamie parameters of the system change with time.

(i) Isothermal Process A process taking place in a thermodynamic system at constant temperature is called an isothermal process.

Isothermal processes are very slow processes.

These process follows Boyle’s law, according to which

p_V = constant

From dU = nC_vdT< as dT= 0 so dU = 0, i.e., internal energy is constant.

From first law of thermodynamic dQ = dW, i.e., heat given to the system is equal to the work done by system surroundings.

Work done W = 2.3026μRT l0g₁₀(V_f / V_i) = 2.3026μRT l0g₁₀(p_i / p_f)

where, μ = number of moles, R = ideal gas constant, T = absolute temperature and V_i V_f and P_i, P_f are initial volumes and pressures.

After differentiating P V = constant, we have

i.e., bulk modulus of gas in isothermal process, β = p.

P – V curve for this persons is a rectangular hyperbola

Examples

(a) Melting process is an isothermal change, because temperature of a substance remains constant during melting.
(b) Boiling process is also an isothermal operation.

(ii) Adiabatic Process A process taking place in a thermodynamic system for which there is no exchange of heat between the system and its surroundings.

Adiabatic processes are very fast processes.

These process follows Poisson’s law, according to which

From dQ = nCdT, C_adi = 0 as dQ = 0, i.e., molar heat capacity for adiabatic process is zero.

From first law, dU = – dW, i.e., work done by the system is equal to decrease in internal energy. When a system expands adiabatically, work done is positive and hence internal energy decrease, i.e., the system cools down and vice-versa.

Work done in an adiabatic process is

where Ti and Tf are initial and final temperatures. Examples

(a) Sudden compression or expansion of a gas in a container with perfectly non-conducting wall.
(b) Sudden bursting of the tube of a bicycle tyre.
(c) Propagation of sound waves in air and other gases.

(iii) Isobaric Process A process taking place in a thermodynamic system at constant pressure is called an isobaric process.

Molar heat capacity of the process is C_p and dQ = nC_pdT.

Internal energy dU = nC_v dT

From the first law of thermodynamics

dQ = dU + dW
dW = pdV = nRdT

Process equation is V / T = constant.
p- V curve is a straight line parallel to volume axis.

(iv) Isochoric Process A process taking place in a tlaermodynars system at constant volume is
called an isochoric process.

dQ = nC_vdT, molar heat capacity for isochoric process is C_v. Volume is constant, so dW = 0, Process equation is p / T = constant p- V curve is a straight line parallel to pressure axis.

(v) Cyclic Process When a thermodynamic system returns to . initial state after passing through several states, then it is called cyclic process.

Efficiency of the cycle is given by

Work done by the cycle can be computed from area enclosed cycle on p- V curve.

Isothermal and Adiabatic Curves

The graph drawn between the pressure p and the volume V of a given mass of a gas for an isothermal process is called isothermal curve and for an adiabatic process it is called adiabatic curve .

The slope of the adiabatic curve

= γ x the slope of the isothermal curve

Volume Elasticities of Gases

There are two types of volume elasticities of gases

(i) Isothermal modulus of elasticity E_S = p
(ii) Adiabatic modulus of elasticity E_T = γ p

Ratio between isothermal and adiabatic modulus

E_S / E_T = γ = C_p / C_V

where Cp and Cv are specific heats of gas at constant pressure and at constant volume.

For an isothermal process Δt = 0, therefore specific heat,

c = Δ θ / m Δt = &infi;

For an adiabatic process 119= 0, therefore specific heat,

c = 0 / m Δt = 0

Second Law of Thermodynamics

The second law of thermodynamics gives a fundamental limitation to the efficiency of a heat engine and the coefficient of performance of a refrigerator. It says that efficiency of a heat engine can never be unity (or 100%). This implies that heat released to the cold reservoir can never be made zero.

Kelvin’s Statement

It is impossible to obtain a continuous supply of work from a body by cooling it to a temperature below the coldest of its surroundings.

Clausius’ Statement

It is impossible to transfer heat from a lower temperature body to a higher temperature body without use of an extemal agency.

Planck’s Statement

It is impossible to construct a heat engine that will convert heat completely into work.

All these statements are equivalent as one can be obtained from the other.

Entropy

Entropy is a physical quantity that remains constant during a reversible adiabatic change. Change in entropy is given by dS = δQ / T

Where, δQ = heat supplied to the system and T = absolute temperature.

Entropy of a system never decreases, i.e., dS ≥ o.

Entropy of a system increases in an irreversible process

Heat Engine

A heat energy engine is a device which converts heat energy into mechanical energy.

A heat engine consists of three parts

(i) Source of heat at higher temperature
(ii) Working substance
(iii) Sink of heat at lower temperature

Thermal efficiency of a heat engine is given by

where Q1 is heat absorbed from the source,

Q2 is heat rejected to the sink and T1 and T2 are temperatures of source and sink. Heat engine are of two types

(i) External Combustion Engine In this engine fuel is burnt a chamber outside the main body of the engine. e.g., steam engine. In practical life thermal efficiency of a steam engine varies from 12% to 16%.

(ii) Internal Combustion Engine In this engine. fuel is burnt inside the main body of the engine. e.g., petrol and diesel engine. In practical life thermal efficiency of a petrol engine is 26% and a diesel engine is 40%.

Carnot’s Cycle

Carnot devised an ideal cycle of operation for a heat engine, called Carnot’s cycle.

A Carnot’s cycle contains the following four processes

(i) Isothermal expansion (AB)
(ii) Adiabatic expansion (BO)
(iii) Isothermal compression (CD)
(iv) Adiabatic compression (DA)

The net work done per cycle by the engine is numerically equal to the area of the loop representing the Carnot’s cycle .

After doing the calculations for different processes we can show that

[Efficiency of Carnot engine is maximum (not 1000/0) for given temperatures T₁ and T₂. But still Carnot engine is not a practical engine because many ideal situations have been assumed while designing this engine which can practically not be obtained.]

Refrigerator or Heat Pump

A refrigerator or heat pump is a device used for cooling things. It absorb heat from sink at lower temperature and reject a larger amount of heat to source at higher temperature.

Coefficient of performance of refrigerator is given by

where Q₂ is heat absorbed from the sink, Q₁ is heat rejected to source and T₁ and T₂ are
temperatures of source and sink.

Revision Notes on Oscillations class 11

Revision Notes on Oscillations

Types of Motion:-


(a) Periodic motion:- When a body or a moving particle repeats its motion along a definite path after regular intervals of time, its motion is said to be Periodic Motion and interval of time is called time or harmonic motion period (T). The path of periodic motion may be linear, circular, elliptical or any other curve.

(b) Oscillatory motion:-‘ To and Fro' type of motion is called an Oscillatory Motion. It need not be periodic and need not have fixed extreme positions.The force/torque acting in oscillatory motion (directed towards equilibrium point) is called restoring force/torque.

(c) Simple harmonic motion (SHM):- Simple harmonic motion is the motion in which the restoring force is proportional to displacement from the mean position and opposes its increase.


Simple harmonic motion (SHM):- A particle is said to move in SHM, if its acceleration is proportional to the displacement and is always directed towards the mean position.


Conditions of Simple Harmonic Motion




For SHM is to occur, three conditions must be satisfied.

(a) There must be a position of stable equilibrium

At the stable equilibrium potential energy is minimum.

So, dU/dy= 0 and d2U/dy2> 0

(b) There must be no dissipation of energy

(c) The acceleration is proportional to the displacement and opposite in direction.

That is,a = -ω2y          


Equation of SHM:-



(a)   F = -ky (b) d2y/dt2 +ω2y = 0

Here ω = √k/m (k is force constant)  


Displacement (y) :- Displacement of a particle vibrating in SHM, at any instant , is defined as its distance from the mean position at that instant.





y = r sin (ωt+?) 

Here ? is the phase and r is the radius of the circle.

Condition:

When, ? = 0, then, y = r sin ωt

and

When, ? = π/2, then,y = rcosωt


Amplitude (r):-



Amplitude of a particle, vibrating in SHM, is defined as its maximum displacement on either side of mean position. 

As the extreme value of value of ωt = ± 1, thus, y = ±r  


Velocity (V):-V= dy/dt = rωcos(ωt+?)= vcos(ωt+?) = ω√r2-y2



Here v is the linear velocity of the particle.

Condition:- When, y = 0, then, V = v = rω

and

When, y = ±r, then, V=0

A particle vibrating in SHM, passes with maximum velocity through the mean position and is at rest at the extreme positions.


y2/r2 + y2/ω2r2 = 1




Acceleration (a): a = dV/dt = (-v2/r) sinωt = -ω2y



Condition:-

When, y = 0, then, a = 0

And,

When, y = ±r, then, a = ±ω2r

A particle vibrating in SHM, has zero acceleration while passing through mean position and has maximum acceleration while at extreme positions.

(i) Acceleration is directly proportional to y (displacement).

(ii) Acceleration is always directed towards the mean position.  


Time period (T):- It is the time taken by the particle to complete one vibration.



(a) T = 2π/ω

(b) T =2π√(displacement/acceleration)

(c) T = 2π√m/k  


Frequency (f):-It is the number of vibrations made by the body in one second.



(a) f=1/T 

(b) f=1/2π√k/m  


Angular frequency (ω):-



(a) ω = 2π/T 

(b) ω =√(acceleration /displacement)  


Relation betweenAngular frequency (ω) and Frequency (f):- ω = 2πf=√k/m  




Phase:-



(a) Phase of a particle is defined as its state as regards its position and direction of motion.

(b) It is measured by the fraction of time period that has elapsed since the particle crossed its mean position, last, in the positive direction.

(c) Phase can also be measured in terms of the angle, expressed as a fraction of 2π radian, traversed by the radius vector of the circle of reference while the initial position of the radius vector is taken to be that which corresponds to the instant when the particle in SHM is about to cross mean position in positive direction.  


Energy in SHM:-



(a) Kinetic Energy (Ek):-

Ek = ½ mω2(r2-y2) = ½ mω2r2cos2ωt

When, y = 0, then, (Ek)max = ½ mω2r2    (maximum)

And

When, y = ±r, then, (Ek)min =0    (minimum)  

(b) Potential Energy (Ep):-

Ep = ½ mω2r2 = ½ mω2r2sin2ωt

(Ep)max = ½ mω2r2  

(c) Total Energy (E):-

E = Ek +Ep=½ mω2r2 = consereved

E = (Ek)max =(Ep)max  


Average Kinetic Energy:-<Ek> = (¼) mω2r2




Average Potential Energy:-<Ep> = (¼) mω2r2




<E/2> = <Ek> = <Ep>




Spring-mass system:-



(a) mg=kx0

(b) Time period, T = 2π√m/k = 2π√x0/g  


Massive spring:-Time period, T = 2π√[m+(ms/3)]/k




Cutting a spring:-



(a) Time period, T'= T0/√n

(b) Frequency,f' =√(n) f0

(c) Spring constant,k' =nk

(d) If spring is cut into two pieces of length l1 and l2 such that, l1= nl2, then,

k1 = (n+1/n)k,

k2 = (n+1)k

and

k1l1 = k2l2


Spring in parallel connection:-





?(a) Total  spring constant, k =k1+k2

(b) Time period, T = 2π√[m/(k1+k2)]

(c) If T1 = 2π√m/k1 and T2 = 2π√m/k2, then,

T= T1T2/√ T12+ T12 and ω2=ω12+ω22    

Spring in series connection:- 




(a) Total  spring constant, 1/k = 1/k1+1/k2 or, k = k1k2/ k1+k2

(b) Time period, T2 =T12+ T22

(c) T = 2π√[m(k1+k2)/k1k2]

(d) 1/ω2= 1/ω12+1/ω22

(e) f = 1/2π √[k1k2/m(k1+k2)]  


Law’s of simple pendulum:-?



Laws of isochronisms:- Its states that (≤4°), the time period of a simple pendulum is independent of its amplitude.

Laws of length:- It states that time period of a simple pendulum varies directly as the square root of its length.

T∝√l

Law of acceleration due to gravity:- It states that, the time period of a simple pendulum varies inversely as the square root of acceleration due to gravity at that place.

T∝1/√g

So, Time period of simple pendulum, T = 2π√l/g 

(a) When placed inside a lift being accelerated upwards, the effective value of g increases. Thus the time period of pendulum decreases.

(b) When placed inside a lift being accelerated downwards, the effective value of g decreases. Thus the time period of pendulum increases.

(c) Time period of the pendulum increases at higher altitudes due to decrease in g.

(d) Time period of the pendulum at a place below the surface of earth decrease due to increase in g.

(e) At the center of earth (g=0). So the time period is infinite.

(f) Time period is greater at equator than at poles.

(g) Due to decrease in the value of g due to rotation of earth, the time period of the pendulum increases as the earth rotates about its axis.

(h) Equation of motion:-d2θ/dt2+(g/l)θ = 0

(i)  Frequency, f =1/2π √(g/l)

(j) Angular frequency, ω =√(g/l)


Second Pendulum:-A second’spendulumis thatpendulum whose time perios is two second.



(a) T = 2 sec

(b) l = 0.9925 m  


Mass-less loaded spring in the horizontal alignment:-



Force, F = -kx

Acceleration, a =-kx/m

Time period, T = 2π√m/k

Frequency,f = 1/2π√k/m  


Time period of mass-less loaded spring in the vertical alignment:-



T = 2π√m/k  and T = 2π√l/g  


Time period of bar pendulum:-





T = 2π√I/mgl

Here I is the rotational inertia of the pendulum.

and

T = 2π√L/g

Here, L = (k2/l)+l  


Time period of torsion pendulum:-





(a) T=2π√I/C

Here I is therotational inertia of the pendulum and C is the restoring couple per unit angular twist. 

(b) Equation of motion:-d2θ/dt2+(C/I)θ = 0

Here, θ =θ0 sin (ωt+?)

(c) Angular frequency, ω = √C/I

(d) Frequency, f = 1/2π√C/I  


Conical Pendulum:-



Time period, T = 2π√(Lcosθ/g)

Velocity, v = √(gRtanθ)  


Restoring couple (τ):-



?=Cθ

Here C is the restoring couple per unit angular twist and θ is the twist produced in the wire.


Liquid contained in a U-tube:-



Time period, T =2π√l/g  


Electrical oscillating circuit:-



Time period, T =2π√LC

Here, L is the inductance and C is the capacitance.

Angular frequency, ω = 1/√LC


Ball in a bowl:-



Time period, T = 2π√[(R-r)/g]  


Free vibrations:- Vibrationsof a body are termed as free vibrations if it vibrates in the absence of any constraint.


Damped Vibrations:-?



Equation:d2y/dt2 + 2µdy/dt+ω2y = 0

Here amplitude, R = Ae-µt

And

ω' = √ω2-µ2

(a)   µ<<ω signifies the body will show oscillatory behavior with gradually decreasing amplitude.

(b)   µ>>ω signifies the amplitude may decrease from maximum to zero without showing the oscillatory behavior.

(c)    In between the above two cases, the body is in the state of critically damped.

(d)   Time period of oscillation, T' = 2π/ω' = 2π/√ω2-µ2. Thus, presence of damping factor µ in the denominator indicates an increase of time period due to damping.  


Forced vibrations:- Forced vibrations is the phenomenon of setting a body into vibrations by a strong periodic force whose frequency is different from natural frequency of body.



Equation: d2y/dt2+2µdy/dt+ω2y = (F0/m) cospt

Here,µ = r/2m and ω=√k/m

Solution: y =Acos [pt-?]

Amplitude:- A = F0/m√4µ2p2+(p2-ω2)2 and Amax = F0/2µm√ω2-µ2

This state of forced vibrations in which the amplitude reaches a maximum value is known as amplitude resonance.

Amplitude vibration depends upon value of ω = √k/m. Greater the value of stiffness (k), smaller is the amplitude.  


Resonance:- Resonance is the phenomenon of setting a body into vibrations by a strong periodic force whose frequency coincides with the natural frequency of the body.