Exercises

1. (a) The derivation of the ideal gas law in the text assumed a monotomic gas yet the ideal gas law is valid in more generality. Explain why this universality holds. Explain also the role of interactions.
   (b) View the simulations of Brownian motion in Ref.[4] for various parameters. Qualitatively, how do you expect the Brownian motion to change with temperature and the relative size of the suspended and fluid particles.
   (c) In Boltzmann's time, many people did not believe in atoms. Apparently Boltzmann, who gave the molecular interpretation of entropy, became depressed by the many attacks on his work and eventually took his own life in 1906; [6]. Do you believe in atoms ? Check out Ref.[5].
   (d)If you are interested in the story (and wager) behind those pictures see Feynman's book quoted in Ref.[8].
   (e) Brownian motion is often simulated as a random walk. Do you remember what a random walk is?
2. (a) How does one know where the solid, liquid and gas regions are in an unlabled $P-T$ plot of a substance?
   (b) What is the significance of the triple and critical points?
   (c) How does the $P-T$ plot for water differ from that of a generic substance?
   (d) Do you know why the "anomalous" $P-T$ property of water is wonderfully important ? (Hint: see Ref.[9]).
3. Show that for a thermodynamic system with parameters $P$, $V$ and $T$, the heat capacities are
   (a) $C_V \equiv \left({ \partial Q \over \partial T}\right)_V = \left({ \partial U \over \partial T}\right)_V$.
   (b) $C_P \equiv \left({ \partial Q \over \partial T}\right)_P = \left({ \partial H \over \partial T}\right)_P$ , where $H= U + PV$ is called the enthalpy (not to be confused with the entropy!).
   (c) Obtain $C_v$ and $C_P$ for an ideal gas. Why is $C_p$ larger than $C_v$ ?
4. Use the definitions (4.19) and (4.20) together with the equation of state of an ideal gas to determine the entropy of an ideal gas.
5. Show that heat conduction, along a metal bar between two heat reservoirs at temperatures $T_1$ and $T_2$ leads to an increase of entropy of the whole system. The result again conforms to our experience that heat flow is an irreversible process in the sense that it flows from the hot to cold object and never the other way.
6. Diffusion is a familiar physical example of the Second Law in action. Give some examples of diffusive processes. Where else in this course have you learnt about diffusion in a useful model of nature?
7. The Second Law can be stated in many equivalent ways. One macroscopic statement of the Second Law is as follows (Clausius): In any cyclic transformation throughout which temperature is defined, one has
   

   $$\oint { dQ \over T} \le 0,$$ (4.46)

   
   
   where the integral is over one cycle. The equality holds for reversible transformations. Show that as a consequence of this statement
   (a) For any transformation,
   

   $$\int_{A}^{B} {dQ \over T} \le S(B) - S(A)$$ (4.47)

   
   
   (b) The entropy of a thermally isolated system never decreases (i.e. recover the original statement of the second law in the text).
8. Two useful state functions are the Helmhotz free energy F, and the Gibbs free energy G, defined by $F=U-TS$ and $G= F + PV$. Show using (4.47) that
   (a) For a mechanically isolated system kept at constant temperature, the Helmhotz free energy never increases (thus the state of equilibrium of a mechanically isolated system at constant temperature is one of minimum $F$.) Does this statement reduce to something familiar in the limit of zero temperature ?
   (b) For a system kept at constant temperature and pressure, the Gibbs potential never increases (thus the state of equilibrium of a system kept at constant temperature and pressure is the state of minimum Gibbs free energy.). Convince yourself that a limiting case of this reproduces (a).
   (c) The above are examples of extremum principles, applicable to more general situations than the first version of the Second Law mentioned in the text (which applied only to isolated states, $U$ and $V$ constant). These principles determine the thermodynamically stable equilibrium states. Convince yourself that in limiting cases the above extremum principles are equivalent to the first version of the Second Law stated in the text.
9. (a)Show that both sides of equation (4.27) have units of heat energy.
   (b) Show that melting and boiling result in an increase of entropy of the system.
10. (a) Water vapour, where the atoms are free to move more randomly, clearly corresponds to a state of larger entropy than the state of liquid water. So why does liquid water at any fixed temperature remain liquid and not spontaneously vaporize to increase its entropy ?
    (b) The above example highlights why things do not spontaneously 'disintegrate' or 'decay', as suggested by misinterpretations of the Second Law. Can you now explain why paper does not undergo spontaneous combustion into a disordered mess ? (Hint: Check out Ref.[8])
11. Show that in an infinitesimal reversible transformation $dG = -SdT + VdP$, and thus deduce the relations
    

    $$S = -\left({\partial G \over \partial T}\right)_{P}$$ (4.48)

    $$V = \left({\partial G \over \partial P}\right) _{T}$$ (4.49)

    $$C_P = -T \left({ \partial^2 G \over \partial T^2}\right)_P$$ (4.50)

    
    
    These relations will be used in the section on phase transitions in ferromagnetic systems.
12. (a) Notice that near the crtitical point of a second order phase transition one obtains power law behaviour of relevant quantities. Where else in this course have you come across power laws ?
    (b) Compare the concepts of "characteristic length" and "scale invariance" used here with their use earlier in the course.
    (c) Compare the universality in second-order phase transitions with its earlier usage in the course.
    
13. There are many Ising Model simulations freely available on the web.
    (a) Run the program at Ref.[13] for various temperatures, trying both hot and cold initialisations, and looking at various plots. Do the results appear reasonable ?
    (b) Try also the smaller online simulation at Ref.[14].
14. (a)Explore the two dimensional percolation online simulation package at Ref.[16]. Try various lattice sizes and probability values. Check for the cluster sizes.
    (b) Estimate the critical probability and compare with the theoretical value $0.5928$ (Remember that the crtical probability refers to a lattice of infinite extent. In practise this means you should explore larger and larger lattices and try to find upper and lower bounds on your $p_c$).