Entropy and Third Law of Thermodynamics

1. Entropy and Third Law of Thermodynamics

2. 2nd law of thermodynamics • Kelvin-Planck Statement • It is impossible to construct an engine which operating in a cycle has the sole effect of extracting heat from a reservoir and performing an equivalent amount of work • Clausius’s Statement • It is impossible for a self acting machine working in a cyclic process unaided by external agency to transfer heat from a body at a lower temperature to a body at a higher temperature.

3. Carnot’s Theorem • From the 2nd law of thermodynamics two important results are derived: these conclusions are taken together to constitute Carnot’s theorem which may be stated as follows: • No engine can be more efficient than a perfectly reversible engine working between the same two temperatures. • The efficiency of all reversible engines, working between the same two temperatures is the same whatever the working substance.

4. (Reversible) and two engines • is more efficient than

5. Gain of heat by the source at • Loss of heat by the sink at • External work done on the system is zero • Thus the coupled engines forming a self-acting machine unaided by any external agency transfer heat continuously from a body at low temperature to a body at a higher temperature. • Heat cannot be transferred from one body to another at a higher temperature by a self-acting machine. Hence our assumption is incorrect and we conclude that no engine can be more efficient than a perfectly reversible engine working between the same temperature.

6. The second part of the theorem may be proved by the same arguments as before. For the purpose, we consider two reversible engine R1 and R2 and assume that R2 is more efficient than R1. proceeding in the same way we can show that R2 can not be more efficient than R1. Therefore, all reversible engines working between the same two temperatures have the same efficiency. • The efficiency of all reversible engines, working between the same two temperatures is the same whatever the working substance.

7. Problems • Find the efficiency of the carnot’s engine working between the steam point and the ice point • [Ans: 26.81%] • A Carnot’s engine whose temp of the source is 400 K takes 200 calories of heat at the temperature and rejects 150 calories of heat to the sink. What is the temperature of the sink? Calculate efficiency. • [Ans: 300K, 25%]

8. A carnot’s refrigerator takes heat from water at 0 deg-C and discards it to a room at 27 deg C. 1 kg of water at 0 deg-C is changed into ice. How many calories of heat are discarded to the room? What is the work done by the refrigerator in this process? What is the coefficient of performance of the machine. • [Ans: 87900 Cal; 31.83kJ; 10.13]

9. A carnot engine whose low temp reservoir at 7 deg-C has an efficiency of 0.5. It is desired to increase the efficiency to 0.7. By how many degree should the temp of reservoir • [Ans: 280K]

10. Change in Entropy • Reversible Carnot’s cycle bounded by • Two same adiabatic L and M and • Isothermal T1, T2 and T3 • Let ABCD and DCEF represent reversible cycle • During ABCD • During DCEF • Thus is constant • and is constant, this constant ratio is called change in entropy • Thus

11. Change of Entropy in Reversible Cycle • Isothermal expansion AB: • Isothermal compression CD: • Adiabatic expansion / compression : • ; • For a reversible Carnot’s Cycle

12. Principle of Increase of Entropy • Consider an engine performing irreversible cycle of changes in which the working substance absorbs heat Q1 at temperature T1 from the source and rejects heat Q2 to sink at temperature T2 • According to Carnot’s theorem efficiency is less than that of a reversible engine working between the same two temperatures • There is an increase in entropy of the system during an irreversible process.

13. T-S Diagram

14. In going from A to B • In going from B to C, D to A • In going from C to D • Thus Area of the rectangular in TS diagram External work done in the cycle

15. Physical Significance • Although it is very difficult to conceive the idea of entropy as there is no physical method to demonstrate it. • Important Properties of Entropy • Just like T remains constant in isothermal process, S remains constant in the adiabatic process. • In every natural process (irreversible), there is always an increase in entropy. • The 2nd law of thermodynamics can be stated in terms of entropy. • Due to increase in entropy, unavailable energy increase. • According to Free man Dyson, S is a measure of disorderness. This disorderness can be evaluated by

16. 3rd Law of Thermodynamics • 1906, Nernst • Heat capacities of all solids tend to zero as the absolute zero of temperature is approached and that the internal energies and entropies of all substances become equal there, approaching their common value asymptotically tending to zero • This theorem is useful in explaining the nature of bodies in the neighbor hood of absolute zero temperature. Its importance lies in the fact that it permits the calculations of absolute value of entropy, Helmholtz and Gibbs free energies, etc. In terms of entropy, • At absolute zero temp, the entropy tends to zero and the molecules of a substance or a system are in perfect order. • In all heat engines there is always loss due to conduction, radiation and friction, thus is not equal to . Thus

17. When cycle after cycle is repeated, the entropy of the system increases and attains a maximum value. When the system has attained the maximum value, a stage of stagnancy is reached and no work can be done by engine.

18. Zero point energy • According to Kinetic theory, the energy of a system at absolute zero should be zero. It means the molecules of the system do not possess any motion. But according to the modern concept, even at absolute zero, the molecules are not completely deprived of their motion, hence possess energy. The energy of molecules at absolute zero is called zero point energy.