Thermal & Kinetic Lecture 13 Calculation of entropy, Introduction to 0 th law

1. Thermal & Kinetic Lecture 13 Calculation of entropy, Introduction to 0th law LECTURE 13 OVERVIEW Recap…. Calculation of entropy: reversible process Some abuses of the 2nd law Section 4: Thermodynamics: the 0th law, equilibrium and isotherms

2. Last time…. Reversible and quasistatic changes. The relationship of temperature and the rate of change of entropy wrt energy. Reversible and irreversible processes: entropy changes.

3. 3 4 2 1 373 K 293K 373 K 293K Reservoir at 373K Reservoir at 373 K Reversible and irreversible processes: calculation of entropy OK, now I’m confused. One of the statements of the 2nd law you’ve given us says that there’s no entropy change for a reversible process. Yet, the expression you’ve written down on the previous slide works ONLY for a reversible process……..?! Irreversible process (large temperature differences). BUT the water at the start and at the end of the process is in equilibrium, with well defined entropies. Imagine a reversible process that takes the water between the same two end points……………………

4. 293.1 K 293.2 K Reservoir at 293.1 K Reservoir at 293.2 K T 373 K dQ=CPDT Reservoir at 373 K Reservoir at T+DT Reversible and irreversible processes: calculation of entropy We break the irreversible process down into a series of reversible steps (just as we did when we added weights to the piston). If we calculate the entropy change for this process it’s the same as that for the irreversible process.

5. ? What do we now need to do to evaluate the total change in entropy? ANS: Integrate. Reversible and irreversible processes: calculation of entropy When the water is at a temperature T and it’s heated to T + DT, the heat entering (reversibly) is dQ = CPDT. From the entropy change of the water at each reversible step is:

6. ? Calculating changes in entropy: examples • Calculate the change in the total entropy of the Universe for the following processes: • A block of mass 1 kg, temperature 100°C and heat capacity 100 JK-1 is placed in a lake whose temperature is 10°C. (ii) The same block at 10°C is dropped into the lake from a height of 10 metres. (iii) The same block at 10°C absorbs a photon of light (l = 600 nm).

7. Calculating changes in entropy: examples The change in entropy of the block, DSBlock, is given by: 27.61 JK-1 The entropy gain of the lake is: = 100 x 90/283 = +31.80 JK-1 (Lake acts as a thermal reservoir which is so large there’s no change in its temperature).

8. Calculating changes in entropy: examples (ii) The block is in the same state (at the same temperature) before and after the process. However, this is obviously an irreversible process. Although the temperature of the lake remains constant because it is a thermal reservoir, the kinetic energy of the block is transferred as heat energy into the lake. So there’s a positive change of entropy for the lake: 1 x 9.81 x 10/283 = +0.35 JK-1

9. Your sock drawer (or bedroom) does not become disordered due to ‘entropy’ – the change in thermodynamic entropy here is zero (we aren’t changing the number of accessible microstates). (Same thing applies to playing cards!). OK, then how does a fridge work? Heat is ‘taken out’, therefore entropy decreases! Some ‘abuses’ of the 2nd law and the concept of entropy “The entropy of a body never decreases – it always increases.” Humans and animals are complex, ordered beings. Entropy is a measure of disorder. 2nd law states disorder always increases. Therefore order can’t ‘evolve’ from disorder – theory of evolution can’t be correct…….

10. As usual, before starting a new section, let’s recap some of what we’ve covered in previous sections: S= k ln (W); p = a exp (-DE/kT); Entropy is a measure of the no. of accessible microstates; At thermal equilibrium entropy is maximised (greatest no. of microstates); Entropy does not change in a reversible process; Temperature is related to the rate of change of entropy with energy; The entropy of a closed system tends to increase. Section IV: Thermodynamics from macroscopic and microscopic perspectives “….classical thermodynamics has made a deep impression upon me. It is the only physical theory of universal content which I am convinced, within the applicability of its basic concepts, will never be overthrown”. Albert Einstein, 1949

11. We’ve defined temperature in a couple of different ways, but how do we measure temperature in the real world? How is an experimental measurement of T related to the theoretical definition of temperature? Why can’t we build perpetual motion machines like that shown to the left? How are heat and work related? How do heat engines work and what fundamental limits are there on their efficiency? …and what is Maxwell’s demon?! Introduction to thermodynamics ….but, we still need to address the following questions:

12. Why should we care about thermodynamics? Surely if we can understand the behaviour of matter from an atomic viewpoint using statistical mechanics and quantum theory, that’s enough?! Thermodynamics is not dependent on a particular microscopic model – as such it acts as an important check on our microscopic description of matter. For example, the concept of entropy was developed with no consideration of the atomic structure of matter and before the the first law – the conservation of energy – was known!! Equilibrium and the zeroth law Thermodynamics was developed before an understanding of the atomic structure of matter was achieved. Thermodynamics is concerned with the large scale properties of a system: volume, temperature, pressure, specific heat etc…..

13. System Piston E.g. gas isolated from surroundings. Gas will reach equilibrium state: (i) properties spatially uniform (ii) properties don’t change with time. Gas Adiabatic wall Equilibrium and the zeroth law First, some revision: Surroundings Wall may be adiabatic or diathermal

14. Gas Equilibrium and the zeroth law P, V uniform and constant. Gas in in the equilibrium state (P,V). Specifying P, V, and the total number of molecules fixes all the macroscopic properties of the gas (e.g. thermal conductivity). When thermal equilibrium is reached there are no flows of energy in the system, i.e. no temperature gradients. To have thermodynamic equilibrium, we must have reached thermal, chemical, and mechanical equilibrium. Mechanical equilibrium – no unbalanced forces; Chemical equilibrium – number of particles in each phase remains constant (eg. no change in proportion of water and ice as a fn. of time)

15. C B A C Equilibrium and the zeroth law If two systems are put in thermal contact with each other, they will reach thermal equilibrium (eg blocks considered in earlier lectures on entropy). 0th law of thermodynamics: if each of two systems is in thermal equilibrium with a third, they are in thermal equilibrium with each other.  A TA = TB. B Remember from Section 3: two systems in thermal equilibrium have the same temperature.

16. Isotherms Returning to a consideration of the gas + piston system: • Gas is originally in an equilibrium state (P,V) and is in thermal equilibrium with another reference system. • Plot point on a graph of P vs V. • Push piston in to take gas to new equilibrium state (P’, V’) which is also in equilibrium with the reference system. • Again, plot this point on a graph of P vs V. • Repeat 1 – 4, always keeping the temperature constant. The curve is known as an isotherm.

17. Isotherms A functional relationship between P, V and T exists: T = f(P,V) or P = f (T,V) or V = f(P, T) i.e. of the three measureable variables only two are independent and one may be expressed in terms of the other two.

18. ? ? Write down the equation of state for an ideal gas in the forms P = f(V,T) and T = f(P,V). Write down an equation which describes the isotherms of an ideal gas. Function of state Equation of state Functions of state When a system is in thermodynamic equilibrium the properties of the system only depend on the thermodynamic variables (P, V, T) – the pathway by which equilibrium was reached is irrelevant. We say that T, V, and P are functions of state. For example, in equilibrium, T = f (P,V). ANS: P = nRT/V, T = PV/nR ANS: PV = constant or PV = nRT or T = PV/nR etc….

19. Second, we need a temperature scale based on the change in the physical property. How do we assign a scale? Let’s assume that the physical property (X) varies linearly with temperature (T) so that: X = c TX The value of c is fixed by choosing a certain well defined and reproducible TX and assigning it a certain value. Temperature scales To experimentally measure T first we have to find a physical property of our reference system that varies with temperature: e.g. length of a column of mercury in a glass capillary, voltage of a thermocouple junction.

20. Ermmm…….?! Temperature scales Now, which point do we choose as our fixed point? We choose the point at which ice, water, and water vapour coexist in equilibrium – the triple point of water. (More on phases later…….) Then we assign TX at the triple point of water the value 273.16 K (i.e. 0.01°C) Yes, I know – we’ll explain that 273.16 K figure soon. For now, take on board the following: • X = cTx implies that as Tx  0, X 0 – not necessarily. • Temperatures on the X scale are only defined in certain regions. • Different thermometers based on different variables will agree only at fixed points (deviations from linearity), except for…….