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# Thermodynamics Theory

THERMODYNAMICS

Thermodynamics: The branch of physics which deals with the energy transformations from one form to another. It is a macroscopic phenomenon where we deal with group of large number of particles rather than the individual particles. Thermodynamics deals with only energy changes and not how the energy changes are brought about [mechanism]

Thermodynamic System: A group of extremely large number of particles having certain value of pressure, volume and temperature is called thermodynamic system. For example large collection of gas molecules is a thermodynamic system.

Surroundings: Anything outside the system which can exchange energy with it and has direct effect on its behavior is called surrounding.

Thermodynamic Variable: The parameters [pressure, volume and temperature] which determines the state of the system are called variables. Other thermodynamic variables like internal energy [U], or entropy etc can be expressed in terms of P, V and T.

Equation Of State: A relation between pressure, volume and temperature for a system is called the equation of state. For example ideal gas equation PV =nRT is an example of equation of state for n moles of ideal gas.

Thermodynamic process: A thermodynamic process is said of taking place if the thermodynamic variables change with time

[a] Isothermal Process: The process in which the temperature of the system remains constant is called isothermal process. DT =0

[b] Isobaric process: the thermodynamic process in which the pressure of the system remains constant with time is called isobaric process. DP =0

[c] Isochoric Process: The thermodynamic process in which the volume of the system remains constant is called isochoric process. D V =0

[d]
Adiabatic Process: The thermodynamic process in which heat content of the system remains constant is called an adiabatic process. D Q =0

Conditions for perfect Isothermal and adiabatic Change:

Isothermal Change: When the cylinder containing some gas is compressed the temperature of the system increases, but if the temperature of the system is to remain constant then heat produced in compression must be equal to the heat lost to the surroundings. Similarly if the gas expands some heat is taken from the gas and gas cools. But if temperature is to be kept constant then heat lost should be equal to the heat gained from the surroundings. To acquire such a state

[a] The walls of the container in which the thermodynamic phenomenon takes place should be perfectly conducting so that heat can be exchanged with the surroundings.

[b] The process of expansion and compression should be slow so that time is available for heat exchange with the surroundings.

Adiabatic Change: When gas is compressed or expands its temperature can either rise or fall respectively. In both the cases the system has tendency to exchange heat with the surroundings so that the thermal equilibrium with surroundings could be established. But if the heat content of the system has to remain constant as in adiabatic process two conditions are required

[a] The walls of the container in which the thermodynamic process takes place should be perfectly insulating so that no heat is exchanged with the surroundings.

[b] The process of compression or expansion should take place fast so that no time is available for heat exchange with the surroundings.

Though perfectly adiabatic process are not realized in practice some examples are

[1] the sudden bursting of cycle tyre.

[2] the propagation of sound wave through the medium as it is fast phenomenon.

# ZEROTH LAW OF THERMODYNAMICS:

When two systems A and B are put into actual contact with each other or are separated by a diathermy wall, their state coordinates may or may not change. Eventually a stage is reached where no further change in the state coordinates of A and B takes place. The joint state of both the systems that exists when all changes in the coordinates ceases is called thermal equilibrium.

Now suppose two systems a and B are separated from each other by an adiabatic wall but each is in contact with the system C through diathermic walls. The whole system is surrounded by an adiabatic wall. Experiment shows that the two system will be in thermal equilibrium with the third system C and no further change takes place if the adiabatic wall separated them is removed, which implies that the two system are in thermal equilibrium with each other also. This can be stated as ‘ two system in thermal equilibrium with third system are in thermal equilibrium with each other also.’ This principle is known as zeroth law of thermodynamics. The temperature is the property that determines whether or not it will be in equilibrium with other systems. If two systems are in thermal equilibrium they always have the same temperature.

# FIRST LAW OF THERMODYNAMICS:

First law of thermodynamics is the law of conservation of energy that ‘energy can neither be created nor destroyed it can only be transferred from one form to another’ or ‘when the mechanical work is spent in producing heat, a definite quantity of heat is produced for every unit of work spent; and conversely when heat is employed to do work, the same definite quantity of heat disappears for every unit of work obtained’

The heat energy supplied to the system is partially used for increasing the internal energy and the remaining heat energy is used for doing the pressure volume work. If dQ is the heat energy supplied , then dU will be the change in internal energy and dW is the external work done. Then according to the first law of thermodynamics

dQ = dU + dW

Thus first law of thermodynamics is the relationship between the heat energy supplied and the external work done.

Sign Conventions: [a] The heat energy added to the system is taken as positive and heat energy removed from the system is taken as negative

[b] If internal energy of the system increases then dU is positive and if the temperature of the system decreases or internal energy decreases then dU is negative.

[c] Work done on the system is negative and work done by the system is positive i.e in the case of expanding gas work done is positive and in case of compressed gas work done is negative.

## SPECIFIC HEAT:

The amount of heat required to raise the temperature of given body through a given amount varies from body to body. If the heat DQ given to a body can increase the temperature of a given body through DT then heat capacity of the body is

Heat Capacity =

The heat capacity per unit mass of the body is called specific heat capacity.

C =

If m=1,DT=10C then C=DQ.

Thus, specific heat can also be defined as the amount of heat required to raise the temperature of unit mass of the substance through unit amount.

Units of Specific Heat: the units of specific heat are J/ kg0C (in SI) or ergl/gm0C. But the commonly used units are cal/gm0C. The specific heat of water is 1cal/gm0C or 4186 J/kg0C.

The above definition of the specific heat is valid for solids or liquids but the specific heat of gases can vary from zero to infinity. For e.g. if the gas is compressed the temperature of the gas rises although no heat energy is supplied to it, thus specific heat for such a gas is zero. Similarly if the gas is supplied heat and the gas is allowed to expand so that their is no rise in the temperature then the specific heat of the gas becomes infinite as DT=0. Thus to find the specific heat of the gas either the pressure or the volume of the gas is kept constant.

## Specific Heat Of Gases:

[a]Specific heat at constant volume: It is defined as the amount of heat energy required to raise the temperature of unit mass of the gas through 10C when the volume of the gas is constant. In is denoted by cv

[b]Specific heat at constant pressure: it is defined as the amount of heat energy required to raise the temperature of unit mass of the substance through 10C if the pressure of the gas is constant. It is denoted by cp.

## Molar Specific Heat:

Molar specific heat is defined as the amount of heat required to raise the temperature of one mole of the substance through 10C. Molar specific heat is different if the gas is heated at constant pressure or if the gas is heated at constant volume. Molar specific heat at constant volume Cv and the molar specific heat at constant pressure is Cp.

If m denotes the molecular weigh of a gas then

Cp = M cp and Cv = M cv

Cp is greater than Cv: When gas is heated at constant volume the heat energy supplied is used for raising the temperature of the gas alone as no work is done for expansion. But if the gas is heated at constant pressure and its temperature is to be raised then some amount of heat supplied is wasted in doing the pressure volume work. Thus, heating a gas at constant pressure requires larger amount of heat as compared to constant volume.

Relation Between Cp and Cv [Cp – Cv = R]

Let us take one mole of a gas and the gas under isochoric conditions so that temperature of the gas increases from T to T + dT. As the gas is heated at constant volume therefore the pressure volume work is zero. Therefore whole of the heat energy supplied should be used for increasing the internal energy of the gas.

dU = CvdT

If the same gas is heated at constant p[pressure through a temperature dT. If dQ is the amount of heat supplied to the gas in this case,

dQ = CpdT

The heat energy supplied at the constant volume will be used for increasing the internal energy of the gas and the remaining energy is used for doing the pressure volume work. The pressure volume work done by the gas is

dW = PdV

According to first law of thermodynamics,

dQ = dU + dW

CpdT = CvdT +PdV

CpdT = CvdT + RdT

Cp – Cv = R

Boiling Process: Let us heat a liquid of mass m and at its boling point it starts changing into vapors. Let V1 be the volume of the liquid and V2 the volume of vapours formed. The pressure volume work done is

dW= PdV= P ( V2 – V1)

If the latent heat of vaporization is Lv, then heat energy absorbed in the boiling process is

dQ = mLv

If Ui denoted internal energy in liquid phase and Uf denotes final internal energy then

dU = Uf – Ui

From first law of thermodynamics , we get

mLv = (Uf – Ui) + P (V2 – V1)

Melting Process: If heat energy is supplied to mass m of solid so that it begins to melt in liquid state, the process is referred as melting. In melting process when solid changes into liquid the volume of the system remains constant. Thus dV = 0. Thus, no work is done in the melting phenomenon by the heat supplied. Using first law of thermodynamics we get

dW = dU + PdV

mLf = dU

here Lf denotes the latent heat of fusion of the solid.

According to first law of thermodynamics the total heat energy supplied to the system is used for changing the internal energy and doing the pressure volume work

dQ = dU +PdV    …….[1]

If one mole of is heated at constant volume so that its temperature increases by dT then the total heat energy supplied is equal to dU such that

dU = CvdT     ………. [2]

From [1] and [2], we have

dQ = CvdT + PdV    ….[3]

In adiabatic change the heat content of the system remains constant, dQ=0.

CvdT + PdV =0    …..[4]

According to the ideal gas equation for 1 mole of a gas PV=RT. Differentiating this equation, we get

PdV + VdP= RdT

dT =     …….[5]

Substitute [5] in [4] we obtain

CvPdV + CvVdP + RPdV=0

dividing both sides by CvVP, we get

where is the ratio of two specific heats of a gas i.e. g=

Integrating both sides,

logeP + glogeV = C

logePVg= C

taking antilog

PVg = K

Adiabatic Relation between volume and temperature: the pressure volume relation for adiabatic process is

PVg = constant

Also for one mole of ideal gas P =

T Vg-1 = constant

Or T1

Adiabatic Relation Between pressure and temperature: as for ideal gas V =

P

Work done in isothermal expansion: Isothermal phenomena is one in which the temperature of the system remains constant. Thus the change in internal energy for an isothermal process is zero. From first law of thermodynamics

dQ = dU + dW

As the change in internal energy is zero, therefore dQ = dW

dW = PdV

dW =

Total work done to change the volume from V1 to V2 is

W = RT

W = RT [ln V2 – ln V1]

W = RT ln

Thus for n moles of an ideal gas,

W = 2.303 nRT log

Work Done id adiabatic process: According first law of thermodynamics

dQ = dW + dU

In an adiabatic process, there is no transfer of heat between system and surroundings, thus dQ = 0

dW = -dU

dW = -nCvdT

the total work done to change the temperature of the system from T1 to T2 is

W =

W = -nCv ( T2 – T1)

For one mole of ideal gas,

W = -Cv (T2-T1)

Also, we know that Cv =

W =

Limitations of first Law of thermodynamics: First law of thermodynamics is the law of conservation of energy, but it has two major limitations

[1] It doe not indicate the direction of heat transfer.

[b] It does not indicate the extent to which the heat change takes place.

For e.g.

[1] This law explains the heating of bullet when it strikes a target but fails to explain why this heat can’t be converted back into kinetic energy.

[2] Similarly, if the brakes are applied and cycle stops, the kinetic energy getting converted into heat energy. Why this heat energy can’t be converted back into kinetic energy of rotation of wheel.

[3] The first law gives no idea how much or what percentage of heat energy supplied can be converted into work and if there is some limitation in this conversion.

Reversible Process: Reversible process is one in which the process can be retraced in opposite direction passing through the same intermediate process for e.g slow expansion or contraction of the spring. The conditions for reversible process are;

[a]the process should take place slowly so that the system should remain in thermal, chemical and mechanical equilibrium at all stages of the process

[b]there should be no frictional losses. This is so because energy spend against such dissipative forces can’t be recovered back. There is loss of energy due to friction.

Some more examples of reversible process are

[1] the working substance taken along complete cannot cycle

[2] All thermal processes taking place at infinitely slow rate.

[3] All mechanical processes taking place under conservative forces.

Irreversible Process: It is process in which the system cannot be made to proceed in the reverse direction through the same intermediate steps as in the case of direct process. A part of energy of the system does work against dissipative forces which can’t be recovered back. In the irreversible process there is always some loss of energy due to heat energy generated in friction and the fast thermodynamics process. For e.g.

[1] Dissolving of sugar into water is an irreversible process

[2] diffusion of gases is an irreversible process.

[3] Adiabatic expansion of compression of gas is an irreversible process.

Second law of thermodynamics:

Lord Kelvin statement: it is impossible to get a continuous supply of work form a body by cooling it to temperature lower than the temperature of the surroundings. This law applies to the heat engine, which absorbs heat from the source does some work and reject remaining heat to the sink at low temperature. It’s not possible to make a heat engine, which converts all of the heat energy absorbed into useful work.

Clausius Statement: It is impossible to make heat flow from the body at lower temperature to the body at higher temperature without doing any external work on the working substance. This applies to the refrigerator, if heat energy is to be removed from interiors of refrigerator at lower temperature and released into the surrounding atmosphere t higher temperature then some external work must be done.

Heat Engine:

Heat engine is device used for converting heat energy into mechanical heat engine.heat engines are of two main types:

External combustion engine: these engines are those in which the burning of fuel is done outside the main body of the cylinder as in the case of the steam engine.

Internal combustion engine: internal combustion engine is one in which the burning of fuel takes place inside the body of the engine as in the case of petrol engine or diesel engine.

For any heat engine there are three essential requirements:

[a]Source: a hot body at fixed higher temperature T1 from which the heat engine can draw heat, is called source.

[b]Sink: A cold body at lower temperature T2 to which any amount of heat can be rejected is called sink.

[c]Working substance: The material which on being supplied heat performs work is called the working substance.

Efficiency of Heat engine: It is defined as the ratio of external work done to the amount of heat energy absorbed from the heat source.

If Q1 is the heat energy absorbed from the source and Q2 is the heat energy released into sink after doing external work then the work done in the process is given by

W = Q1 – Q2

The efficiency is thus given by

Carnot Heat Engine: heat engine is a practical arrangement to convert heat into mechanical work. Sadi Carnot devised an ideal heat engine free from all imperfectness of the actual heat engines and hence it is not possible to generate such an engine in actual practice. Carnot’s heat engine consists of four main parts:

[a]Source : the source is maintained at a fixed higher temperature and has infinite thermal capacity. By infinite thermal capacity we mean that any amount of heat can be taken out of it without changing the temperature of the source.

[b]Sink: It is the reservoir at lower temperature T2 and it also has infinite thermal capacity i.e. any amount of heat can be added to it without changing its temperature.

[c]Working substance: the working substance in the Carnot engine is the ideal gas which absorbs heat from the source does some mechanical work and rejects the remaining amount of heat into sink. It is placed in a cylinder with insulating base but perfectly conducting bottom.

[d]Insulating pad: The pad is used in Carnot cycle for adiabatic expansion and contraction of the gas.

# Carnot Cycle

Carnot cycle consists of four main steps:

[a]ISOTHERMAL EXPANSION: For isothermal expansion of the gas the cylinder is placed in contact with the source so that acquires the temperature of the source T1. The gas allowed to expand slowly by the outward motion of the piston. The expansion results in cooling and the decrease in temperature is compensated by gaining the required amount of heat from the source. Thus overall temperature of the gas remains constant during the expansion. The pressure and volume of the gas are (P1, V1) and the final pressure and volume are (P2,V2). The work done in the process is given by:

Q1= W1= RT1 loge
= Area ABMKA

[b]ADIABATIC EXPANSION: For adiabatic expansion of the gas, the cylinder is removed from the source and placed on the insulating pad. The gas is allowed to expand further from (P2, V2) to (P3,V3). The process is adiabatic because the cylinder is thermally insulated from all the sides. Thus, because of the expansion of the gas temperature of the gas falls from T1 to T2.the work done in the process is given by:

W2 = = Area BCNMB

[c]ISOTHERMAL COMPRESSION: the cylinder is removed from the insulating pad and placed on the sink. The gas is then compressed by the inward motion of the piston. Compression of the gas results in generation of heat and the temperature of the gas is kept constant by releasing the heat generated to the sink. The pressure and volume of the gas changes from (P3, V3) to (P4, V4). The work done on the gas is given by:

Q2= W3= RT2 loge
= Area CNLDC

[d]ADIABATIC COMPRESSION: The cylinder containing the gas is placed on the insulating pad and compressed so that the pressure and the volume of the gas returns to the initial value (P1,V1). The temperature of the gas returns increases to T1. The work done in the process is given by:

In the first two steps the work done by the gas is positive as the gas is expanding whereas the work done by the gas in the compression is negative. Thus total work done during Carnot cycle is:

W= W1 + W2 -W3 -W4

W= W1 – W3

As W1 is equal to the heat energy absorbed Q1 and W3 is the heat energy released in the sink Q2 thus net work done is

W= Q1 – Q2

# EFFICIENCY OF CARNOT ENGINE:

The efficiency of Carnot engine is the ratio of amount of work done by the gas to the total heat energy absorbed by the gas

h =

also as, thus efficiency of the Carnot engine is given by:

h =

thus we can conclude that the efficiency of the Carnot engine:

[a]depends upon the temperature of the source and the sink

[b]it is always less than 100% as the temperature of source can’t be infinite and temperature of the sink can’t be 00K

[c]is directly proportional to the temperature difference between the source and sink.

# Refrigerator:

Refrigerator works in a way opposite to the heat engine. It absorbs heat q2 from the sink does some work W on iy and release the heat Q1 to the source at higher temperature. An amount of work W is done on it by some external means. In actual refrigerator the vapors of Freon (dichlorodifluoromethane) gas acts as the working substance. The interior of the refrigerator acts as the sink. A certain amount of substance W is performed by the compressor of the refrigerator on the gas. The source in this case is the atmosphere or surrounding air at room temperature T1 to which the heat Q1 is rejected by the radiator fixed at the back of the refrigerator. thus work done in the process is the difference of heat rejected to the source to the heat absorbed from sink.

W= Q1 – Q2

# Coefficient of performance:

It is defined as the ratio of quantity of heat removed per cycle from the contents of the refrigerator to the energy spent per cycle to remove this heat.

b = Q2/W

b = Q2/(Q1 – Q2)

also as Q2/Q1 = T2/T1

thus,

b = T2/ (T1 – T2)

Greater the temperature difference between the source and the sink smaller is the coefficient of performance. The value of the coefficient of performance is the measure of the efficiency of the refrigerator and smaller the value of temperature difference smaller will be the efficiency.