Second law of Thermodynamics , Carnot Heat Engine & Refrigerator

Second law of thermodynamics:
• Clausius statement: It is impossible for a self acting machine unaided by any external agency to transfer heat from a cold reservoir to a hot reservoir. In other words heat can’t flow by itself from a colder to a hotter body.

• Kelvin-Planck Statement: It is impossible for any heat engine to convert all the heat absorbed from a reservoir completely into useful work. In other words 100% conversion of heat into work is impossible.

• These two statements of the second law are equivalent to each other. Because, if one is violated, the other is also automatically violated.

Reversible process:
• A process which can be retraced back in such a way that the system passes through the states as in direct process and finally the system acquires the initial conditions, leaving no change anywhere else, is called reversible process. Any quasi-static process can be reversible.

Conditions for a process to be reversible:

(a) There should be no loss of energy due to conduction, convection or dissipation of energy against any resistance, like friction, viscosity etc.

(b) No heat should be converted into magnetic or electric energy.

The system must always be in thermal, mechanical and chemical equilibrium with the surroundings. (i.e the process must be quasi-static)

Examples: In practice, there is no reversible process. But approximately we can give the following examples.

(i) The process of change of static from ice into water is a reversible process.

(ii) The process of change of state from water to steam.

(iii) The gradual extension and compression of an elastic spring is approximately reversible.

(iv) The electrolysis process is reversible if internal resistance is negligibly small.

(v) Slow compression and expansion of an ideal gas at constant temperature.

Irreversible process:
• In this process the system does not pass through the same intermediate states as in the direct process.

• Most of the processes occurring in nature are irreversible.

Examples: (i) Diffusion of gas
(ii) Dissolving of salt in water
(iii) Sudden expansion or compression of gas

Heat engine:

• The device, used to covert heat energy into mechanical energy is called a heat engine.

• For conversion of heat into work with the help of a heat engine the following conditions required.

(i) There should be a reservoir at constant higher temperature ‘T1’ from which heat is extracted. It is called the source.

(ii) Working substance which undergoes thermodynamic cyclic changes (ex: ideal gas).

(iii) There should be a reservoir at constant lower temperature ‘T2’ to which heat can be rejected. This is called the sink.

• The source and sink should have very high thermal capacity.

Working of heat engine:

(a) Schematic diagram of heat engine

(b) Engine derives  amount ‘Q1’ of heat from the source.

(c) A part of this heat is converted into work ‘W’.

(d) Remaining heat ‘Q2’ is rejected to the sink. Thus Q1 = W + Q2

or, the work done by the engine is given by W = Q1 – Q2

(e) The energy Q2 is unavailable in the universe, which causes increase in entropy of universe.

Efficiency of heat engine:

Efficiency of heat engine (η) is defined as the fraction of total heat supplied to the engine which is converted into work.

Mathematically ,

$\large \eta = \frac{W}{Q_1} = \frac{Q_1 – Q_2}{Q_1} $

$\large \eta = 1 – \frac{Q_2}{Q_1}  $

According to this, efficiency is 100% if Q2 = 0, that is no heat is rejected to the cold reservoir or sink. That is the entire heat absorbed must be converted to mechanical work, which is impossible according to Second law of Thermodynamics.

Carnot or Reversible or Ideal heat engine:

• When the working substance is an ideal gas and it is subjected to cyclic process consisting of isothermal expansion, adiabatic expansion, isothermal compression and adiabatic compression, then such heat engine is called Carnot engine. The cyclic process is called Carnot cycle.
Carnot Cycle: Carnot cycle consists of the following four stages (i) Isothermal expansion (process AB), (ii) Adiabatic expansion (process BC), (iii) Isothermal compression (process CD), and (iv) Adiabatic compression (process DA).

The P-V diagram of the cycle is shown in the figure. In process AB heat Q1 is taken by the working substance at constant temperature T1 and in process CD heat Q2 is liberated by the working substance at constant temperature T2. The net work done is the area enclosed by the cycle ABCDA. After doing the calculations for different processes we can show that:

$\large \frac{Q_2}{Q_1} = \frac{T_2}{T_1} $

Therefore, efficiency of the Carnot engine is ,

$\large \eta = 1 – \frac{Q_2}{Q_1} = 1- \frac{T_2}{T_1} $

As T2 is always less than T1 , i.e., the value of η can never be equal or greater than 1. When the temperature of sink T2 = 0 K, then η can be 1 or 100%. But it is impossible.

• For Carnot engine η is independent of the nature of working substance. It depends on only the temperatures of source and sink.

• The efficiency of an irreversible engine is always less than or equal to that of reversible engine when operated between the same temperature limits.

Refrigerator:

The refrigerator is just the reverse of heat engine. In refrigerator the working substance extracts an amount of heat Q2 from the cold reservoir (Sink) at a lower temperature T2. An amount of external work W is done on the working substance and finally an amount of heat Q1 is rejected to the hot reservoir at a higher temperature T1.

• Coefficient of performance of a refrigerator

$\large \beta = \frac{Q_2}{W} = \frac{Q_2}{Q_1 – Q_2} $ ; [∵ W = Q1-Q2]

• For Carnot refrigerator

$\large \frac{Q_2}{Q_1} = \frac{T_2}{T_1} $

Thus , $\large \beta = \frac{T_2}{T_1 – T_2} $

• The relation between efficiency of a heat engine (η) and coefficient of performance of a refrigerator (β) working between the same temperature limits is

$\large \beta = \frac{1 – \eta}{\eta} $

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